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Department of Business Studies Author:
M.Sc. in Finance Camilla Rosenkilde
Master thesis
Academic advisor:
Jochen Dorn
Capital Protected Funds
An Analysis & Valuation with Stochastic Volatility
Aarhus School of Business, Aarhus University
August 2010
CONTENTS
1 INTRODUCTION 1
1.1 Problem statement 1
1.2 Methodology 2
1.3 Delimitations 3
2 CAPITAL PROTECTED FUNDS 4
2.1 Fund terminology 4
2.1.1 Investing in a capital protected fund 6
2.1.2 Fund regulation 6
2.2 Capital guaranteed or capital protected? 7
2.3 Structuring of the protected securities 8
3 THE OPTION-BASED STRUCTURE 8
3.1 Implementation of the option-based strategy 8
3.1.1 Option component 9
3.1.2 Bond component 10
3.1.3 Participation rate 10
3.2 Issuing process 11
4 CONSTANT PROPORTION PORTFOLIO INSURANCE 12
4.1 Implementation of the CPPI structure 13
4.1.1 The CPPI strategy 13
4.1.2 Gap risk 15
4.1.3 Rebalancing algorithm 15
4.2 Issuing process 16
5 OPTION PRICING THEORY 17
5.1 An arbitrage free and complete market 18
5.2 The Black-Scholes partial differential equation 19
5.3 Risk-neutral valuation 21
6 STOCHASTIC VOLATILITY 22
6.1 Empirical findings 22
6.1.1 The volatility smile 24
6.2 Introducing stochastic volatility 25
6.3 The Heston model 25
6.3.1 The Heston PDE and risk neutral processes 26
6.3.3 Heston‟s closed-form solution 28
6.3.4 Dividends 29
7 MONTE CARLO SIMULATION 29
7.1 Simulation procedure 29
7.2 Efficiency of Monte Carlo simulation: Variance reduction 31
7.2.1 Antithetic variable technique 31
7.3 Generating random numbers 32
7.4 Discretization of the model 32
7.4.1 The Euler scheme 33
8 VALUATION OF THE OPTION-BASED STRUCTURE 34
8.1 The chosen product: Scottish Widows Capital Protected Fund 12 34
8.1.1 Payoff structure 36
8.2 Estimating model parameters 37
8.2.1 Continuous interest rates 37
8.2.2 Dividends 38
8.2.3 Calibrating the volatility estimates 39
8.2.3.1 Calibration scheme 40
8.2.3.2 Calibration results 41
8.3 Assumptions 42
8.4 Results 42
8.4.1 Share price of the fund 45
8.5 Sensitivity analysis 46
8.5.1 Risk-free and discount rate 47
8.5.2 Dividend yield 48
8.5.3 Heston parameters 49
9 VALUATION OF THE CPPI STRUCTURE 51
9.1 Implementing the CPPI strategy 51
9.1.1 Pricing gap risk 52
9.1.2 Assumptions 52
9.2 Rebalancing algorithm 53
9.3 Simulation of the standard CPPI strategy 53
9.3.1 Simulation results 54
9.3.2 Changing the multiplier and rebalancing frequency 55
9.4 Extensions to the traditional CPPI structure 56
9.4.1 Exposure constraints 57
9.4.2 Profit lock-in 59
9.5 Results and comparison of the strategies 61
9.6 Sensitivity analysis 61
10 THE OPTION-BASED STRATEGY VS CPPI 63
10.1 Comparison of results 63
10.1.1 Capital protection 64
10.1.2 Return 64
10.1.3 Risk 65
11 CONCLUSION 66
REFERENCES 68
LIST OF FIGURES 71
CD CONTENTS 72
APPENDICES 73
1
1 INTRODUCTION
The recent financial crisis has scared off investors from complex, structured products
incorporating exotic features and unknown exposure to counterparty risk. Instead, the
wish for transparency and regulation has driven the increasing popularity of capital
protected funds. Buying shares in a capital protected fund gives the investor access to
diversification through actively or passively managed portfolios on asset classes such as
real estate, commodities and credit while index trackers in particular have proved
popular. At the same time these funds typically promise 100% capital protection,
meaning that the investor is entitled to receive back the full initial investment if the fund
underperforms. In the light of the last couple of years’ financial turmoil this feature could
seem attractive to a risk-averse investor who, on the other hand, does not want to miss
out on potential gains over an investment at the risk-free rate.
Through time product developers have mainly created the capital protection on structured
products, such as capital protected funds, through the purchase of a zero-coupon bond,
while a call option on the underlying asset provided the upside potential. As interest rates
are currently historically low, the small portion of money left for buying options has
caused the option-based strategy no longer to appear as attractive as before. The Constant
Proportion Portfolio Insurance (CPPI) strategy provides an alternative to the option-
based strategy, and is particularly applicable for creating protection on underlying assets
such as funds. The two strategies differ in how the capital protection is obtained, thus, the
implications of applying each of the strategies also differ.
All of these circumstances make it interesting to examine the two strategies in further
detail, determining how capital protected funds are created and how the shares of each
fund should be priced under the two strategies.
1.1 Problem statement
The purpose of the thesis is to describe how capital protected funds are created using the
two methods: the option-based strategy and the CPPI. More specifically, the creation of
the capital protection will be of main focus. The two most prominent methods for
designing the protection are described and the implications of applying them will be
analyzed. In order to illustrate how the methods work in practice a product using each of
the strategies to create the capital protection is chosen for further analysis and valuation
using the appropriate theory to do so. Furthermore, stochastic volatility is introduced in
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order to reflect the shortcomings of the Black-Scholes model in capturing some of the
market nature. Hence, the analysis and valuation of the products will be based on the
Heston model.
This leads to the following phrasing in the overall problem statement
How is a typical capital protected fund created and valuated with application of
stochastic volatility?
In order to answer this, the following research questions have been defined.
Which parties are involved in the issuing process of a typical capital protected fund?
How is the embedded protection created?
What are the characteristics of the two methods for constructing the protection: The
option-based strategy and Constant Proportion Portfolio Insurance (CPPI)?
How is the fair share price of a typical capital protected fund based on the option-
based strategy with application of stochastic volatility determined?
How is the fair share price of a typical capital protected fund based on the CPPI
strategy with application of stochastic volatility determined?
1.2 Methodology
Given the above statement of research questions the methodology of the thesis can be
accounted for. There are two sides to answering the problem statement. The first part of
the thesis captures the describing and analyzing aspect, which is identifiable in the five
first chapters of the thesis. Chapter 2 introduces the concept of funds and capital
protected funds, and describes the investment process. Chapter 3 and 4 present the
option-based strategy and CPPI, respectively, and account for the characteristics defining
the two methods. In chapter 5, 6 and 7 the theoretical framework for valuing the fund
shares is examined. Chapter 5 describes option pricing theory in a general way and the
approaches to the valuation, namely the PDE approach and risk-neutral valuation
approach, are described. In chapter 6 stochastic volatility is introduced, the Heston model
is described and the implications of modeling stochastic volatility are discussed. Chapter
7 presents Monte Carlo simulation as the method chosen for the numerical evaluation of
the pricing problem. The second overall aspect of the thesis is the evaluating and
independent part, where the theory and methods introduced in the first part of the thesis
are applied in practice and the results of the application are discussed. In chapter 8 the
estimation of model parameters is performed. All of the historical data used are obtained
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from Datastream, the world‟s largest financial database. By using this data provider, the
data is expected to be reliable and accurate. Furthermore, the Heston input parameters are
estimated by calibrating the model to market data. The model implementation is
conducted in VBA, and Excel has been used for representing the data and results. In
chapter 8 and 9 an option-based and CPPI–based product are valuated. The two chapters
also illustrate how the protection in a typical capital protected fund is constructed using
each of the methods and in the CPPI case it is also shown how the structure can be
tailored to adjust the performance of the strategy. Furthermore, a sensitivity analysis for
both of the two products is performed. Based on the above results as well as the earlier
described general characteristics of the two strategies a comparison of the two strategies
is finally made after which the final conclusion can be given.
1.3 Delimitations
With the problem statement in mind this thesis is not intended to provide a
comprehensive analysis of all the specific types of capital protected funds. The creativity
which financial engineers apply in structuring new products makes this an immense task.
Therefore the thesis only deals with the principles behind the most common structures
and strategies. Sometimes variations will be mentioned, but the consequences of them
will not be further analyzed. Moreover, the description of the creation of the protection,
risks and the costs connected with the structures is not detailed, but provides the reader
with knowledge about the main features. Also, the hedging procedure, which the product
developer must carry out to minimize the potential losses on the products, is relevant but
will not be accounted for.
It is assumed that the reader is familiar with standard stochastic calculus and option
theory as well as the Black-Scholes model and framework. The reader is also assumed to
possess basic knowledge about VBA programming. Thus, the coding of the programs
needed for calibration and implementation of the models is not discussed.
Other numerical methods than Monte Carlo simulation do exist, but they will not be
considered or described. Due to the computer intensive simulation of the Heston model,
the array-restrictions of VBA proved to be critical. Consequently, only a limited number
of simulations could be attained. This affects the results negatively so that they might be
imprecise. Nevertheless, the valuations show how the capital protection is created, and
general reflections can be made.
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Furthermore, no in-depth discussion about tax considerations will be carried out, even
though tax issues in practice do affect the attractiveness of the structures. A brief
description of fund regulation is given, however, not in detail and not in the light of the
recent financial crisis.
Finally, the thesis will only be analyzing closed-end funds due to a wish for simplifying
the analyses, and a hypothetical product will be used for illustrating and pricing the CPPI
strategy as no current products were found suitable for analysis. The chapter on valuation
of the CPPI structure includes the valuation of the option on gap risk but in this
connection the antithetic variate method for reducing the variance of the price estimate
will not be applied.
2 CAPITAL PROTECTED FUNDS
The following chapter introduces the terminology that surrounds capital protected funds.
The different types of funds that capital protection might be applied to are defined and an
overview of the investment process is given. Additionally, the regulation and the
difference between capital protected and capital guaranteed funds are outlined.
The concept of capital protected funds is also known as structured funds. A structured
product is a combination of traditional financial instruments and derivatives and it is
engineered to widen the range of investment opportunities for institutional and retail
investors (Das 2001, p.3). In other words, the structured product is a package solution
that allows the investor to invest in a variety of assets, even ones that are not typically
available to the ordinary investor. The products are designed so that they are easily
tailored to investor demand and risk preferences and they can offer risk/return profiles
that are not available to the investor through conventional investment in financial
instruments.
2.1 Fund terminology
The generic term for collective investment schemes is pooled funds or investment funds.
They can be found worldwide, but with varying structures and names. In the US the
funds can be categorized as mutual funds, closed-end funds or unit investment trusts. In
the EU the category Undertakings for Collective Investment in Transferable Securities
(UCITS) is the common term and in the UK investment trusts, unit trusts, open-ended
investment companies and common investment funds among others define the range of
collective investment schemes (Ray 2006, p.17).
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A mutual fund is a form of collective investment, which allows investors to pool their
individual investments and thereby participate in a larger and more diversified portfolio
of investments than would otherwise be accessible (Ray 2006, p.18). In addition, the
participants in a mutual fund enjoy the advantage of specialized professional
management and, in some cases, reduced administration. Depending on the tax
legislation the investors might also obtain some taxation benefits from investing in a
fund. In many cases, the mutual fund is offered by banks, life companies, asset managers
and specialist investment houses. These institutions often use mutual funds as an efficient
way of managing their clients‟ assets.
Individual mutual funds can be further classified according to their asset orientation, such
as stock/equity funds, bond funds, money funds, hybrid funds and so on, or by their
investment objective, such as growth funds, income funds, index or tracker funds. Hedge
funds are another variation of mutual funds, which apply a variety of strategies, including
short-selling and derivative positions across a wide range of asset classes, in order to
make profits in both rising and declining markets. Additional differentiations can be
made in fund structures. For instance, one fund could be investing exclusively in one
other (master fund or feeder fund), or in a number of other funds (fund of funds). The
fund could also have a number of sub-funds (umbrella fund). Furthermore, distinction
between funds can be made according to their legal structure. Funds may be constituted
as companies, as trusts or as partnerships or they may have a joint ownership structure
but no legal personality. The fund may also be an exchange traded fund (ETF), which
tracks an index but is bought and sold as a listed company. Lastly, the mutual fund may
be open-end, where the fund has variable capital and is allowed to issue or redeem shares
on a continuous basis, or closed-end, which is a fund with a fixed amount of capital
issuing a limited number of shares (Ray 2006, p.23).
As well as collective investments pool assets of individual investors, they also need to
divide claims to those assets among the investors. Investors buy protected securities, or
shares, in the collective investment schemes and ownership is proportional to the number
of shares purchased. The value of each share is called the net asset value (NAV). The
NAV is the current market value of a fund‟s holdings minus the fund‟s liabilities divided
by the number of shares issued and it can be determined several times a day, daily,
weekly, or even yearly using the following relationship
(2.1)
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The NAV can also be expressed as the overall fund value, like in the above equation, but
not dividing by the number of shares issued. In the rest of the thesis liabilities are
assumed to be constant, making it possible to ignore their contribution. This simplifies
the valuation and analysis. Furthermore, the term „fund‟ will be referring to any of the
above definitions of funds, however, the most dominantly used types of funds employed
in connection with capital protection are funds tracking indices or basket of indices,
ETFs or hedge funds.
2.1.1 Investing in a capital protected fund
The protected securities, or shares, that investors must buy in order to invest in a capital
protected fund represent equity interests in fund portfolios, which could be all types of
funds described earlier in section 2.2, with guaranteed or protected return of the
principal, the investment amount, at maturity. The return is created by an investment in
risky assets and the protection is obtained through the purchase of risk-free assets.
Hence, at any given time during the investment period
Shares value = Risk-free Assets Value + Risky Assets Value (2.2)
In the event of a closed-end fund launch, that is, the offer of units or shares in a newly
established fund to investors, the investor can invest in the fund during a fixed period
(Ray 2006, p.51). In this period the marketing material and fund constructions are only
preliminary and may be subject to change after the funds to be invested are received. The
objective of the fund, the fee structure, the investment limitations, if any, and other
relevant information are included in the marketing material. At the end of that period, or
a little bit prior to end, the fund managers then assess the mass of funds invested, and
arrange the final portfolio construction and other administrative and operating
procedures. After the issuing of the fund has taken place the investors are typically
demanded to hold the shares for a predefined investment period, usually between 3-5
years. Should the investor wish to sell his shares during this period the capital protection
most commonly does not apply and the investor may not receive back the full initial
amount. Furthermore, a buyback fee is typically imposed. At the end of the investment
period, the investor is usually given the choice between reinvesting the value of the
shares or selling his shares to the product issuer to capitalize the gains, if any.
2.1.2 Fund regulation
Funds are subject to strict regulations, which partly explain the 2009-2010 investor
demand for capital protected funds (Dickinson 2010). Before the firm wishing to
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establish a fund becomes approved by the regulation authorities almost all of the aspects
of their business such as financial resources, competencies and likelihood of failure are
scrutinized. Once the firm is authorized they must meet all the applicable rules and laws
and must provide financial information, maintain the required levels of financial
resources, keep records of their trading activities, and so on (Ray 2006, p.43).
Furthermore, the fund to be marketed to the public must provide a detailed description of
its objectives, policies, structure, management and other relevant operating information.
The above circumstances partly explain why the capital protected funds seem attractive
to the investor as opposed to the usual structured products, which can be opaque and
complex to evaluate. The heavy regulation gives the investor an extra level of security,
especially since counterparty risk has been a main point of focus for the last couple of
years. As a consequence, a lot of the fund prospectuses account for the counterparty risk
tied to the product.
2.2 Capital guaranteed or capital protected?
When examining marketing brochures investors might get confused with the frequent
terms used in relation to structured products. The terms capital or principal „guaranteed‟
and capital or principal „protected‟ are often used about otherwise seemingly similar
products. In reality, the products are alike in anything but in the risk of the investor not
receiving back the principal at maturity. Capital guaranteed funds are guaranteed by the
issuing financial institution, which in most cases is a bank or fund house. The risk lies in
the solvency of the issuing financial institution and the investor is guaranteed the
repayment of the full amount of money that was invested (Chen 2009). In contrast,
capital protected funds are not guaranteed, meaning that the investor might not receive
back the full principal at maturity. If the product developer creates the protection through
bonds the repayment of the invested amount is not a certainty as bond issuer could
default. Thus, this added risk is linked to the bond issuers.
Throughout the rest of this paper the terms „capital guaranteed‟ or „capital protected‟ will
refer to both types of products as the categorization does not have an influence on the
numerical results. The categorization, however, is of significant importance in the
investor‟s risk evaluation of the fund and should be taken into consideration before the
investor decides to invest in a particular fund.
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2.3 Structuring of the protected securities
Two methods for creating the protection on funds, or any other asset classes, are most
often found in the literature and used in practice by large financial institutions: option-
based strategies and threshold structures, also known as CPPI. The methods differ in
terms of participation, exposure, costs, risks as well as volatility. Furthermore, the
option-based strategy is a static structure, which is not altered after the inception. On the
other hand, by applying the CPPI strategy the developer dynamically protects the capital
invested through the adjustment of the exposure to risky and risk-free assets.
The following two sections explain how the product developer of a capital protected fund
can use these two most common methods for constructing capital protection on fund
shares and how the methods work, but the results are easily extended to other asset
classes.
3 THE OPTION-BASED STRUCTURE
The option-based structure is a commonly used method for designing the principal
protection of structured products on liquid assets such as stocks, commodities and
indices. It is simple and relatively straightforward to create. However, it can be complex
to explain to the individual investor and the pricing of the structure is quite opaque due to
the costs embedded in the structure. The following analysis pinpoints the most important
characteristics of the method, and accounts for the creation of the capital protection using
this structure. Finally, an overview of the issuing process helps clarifying how the
structure is structured.
3.1 Implementation of the option-based strategy
The construction of capital protection in the option-based structure is based on two
elements: a bond with no periodic income payments – that is, a zero-coupon bond, which
is used for guaranteeing the principal amount invested in shares on the fund, and an
option, which provides the exposure towards the fund or funds (Godden 2002). At
maturity the investor receives the par amount of the share and the extra amount, if any,
based on the percentage increase of the fund value.
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Figure 1: The option-based structure components
Source: Godden (2002)
Figure 1 above illustrates the components of the structure. The right side of the figure
shows how the return scenario at maturity could turn out. The left side depicts how the
price of a share in the capital protected fund can be decomposed into two categories.
Most of the invested amount is used for the purchase of the bond guaranteeing the
investment, while a smaller part is left for the option premium. As the option cost is
usually priced at 35-50% of the investment value and the zero-coupon element typically
costs between 55-70% of the total amount, depending on the maturity, the option cost is
sometimes slightly higher than the funds left after the purchase of the bond (Godden
2002). Thus, the investor sometimes receives exposure to the underlying of less than
100%. The sum of the two component prices makes up the fair price of the share.
(3.1)
However, the above amount does not include the product costs. As can be seen in Figure
1, the investor usually pays an overcharge to compensate for the product developer‟s and
issuer‟s costs connected to the issuance of the product, illustrated by
(3.2)
Consequently, the total price and fair price of a share are typically not coinciding.
3.1.1 Option component
The option is the component that generates the upside potential for gains. It is usually a
European-type call option, due to the fact that the protection provided by the bond is
valid only at maturity and it is typically written with a strike price of the current fund
NAV. More specifically, the call option gives the issuer the right to a certain appreciation
of the fund. As the option component is used for acquiring the extra return the developer
has the opportunity to choose between several types of funds investing in specific asset
classes to create exactly the exposure that the developer finds a demand for by its
customers.
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Several factors affect the price of the option component, the main drivers being the
current NAV, time to maturity, the volatility of the NAV, and the risk-free interest rate.
A long investment term, a low risk-free rate or a high level of implied volatility increases
the option price, all other things held constant. As will become clear later the interest rate
level has a particularly pronounced effect on the total structure value.
The developer, who has essentially written the option on the fund NAV, has to delta
hedge the option in order to offset the potential loss experienced if the option ends up in-
the-money and the investor thus in effect exercises his right to the appreciation over the
strike. However, this topic will not be further elaborated on in the thesis.
3.1.2 Bond component
The bond component provides the protection of the principal. By purchasing a bond with
nominal value equal to the principal the developer secures the investor the payment of
the bond‟s principal. Zero-coupon bonds are predominantly used because of their lower
price compared to coupon paying bonds. The price PBond of a zero-coupon bond with t
years to maturity using the t-year zero rate r and continuous compounding is given by
(3.3)
where CF is the cash flow equal to the principal. The formula shows that the price of the
bond is determined by the time to maturity and the zero-coupon rate. The interest rate can
be observed in the market for shorter maturities and calculated from traded coupon bonds
using the bootstrap method for longer maturities. In many cases, a money market
instrument is used instead of the bond component.
3.1.3 Participation rate
The participation rate is the percentage share of the return of the fund that the investor
receives. Put differently, it is defined as a measure of how many options on the funds that
implicitly has been bought by the investor. Logically, the higher the participation rate,
the more attractive the product seems to be. With a principal and c as the option price the
participation rate R is given by
(3.4)
However, in many cases the issuer requires the payment of administration costs, issuance
costs, and so on, which will influence the participation rate as follows
(3.5)
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The final participation rate can only be established on the inception date of the product,
but the introductory participation rate is usually determined prior to this date by
estimating the bond and option value.
Equation (3.4) shows the relationship between the variables and the participation rate. On
top of the variables included above the same variables that affect the option price also
affect the participation rate. Lower volatility reduces the price of the option component,
which increases the participation rate. A higher bond interest rate reduces the bond
component price, resulting in a higher participation rate. Furthermore, a longer product
life time reduces the bond price leaving more money for buying bonds, but increases the
option price.
The participation rate is often used to attract investors in the marketing of the product. A
common approach to increasing the participation rate is to construct so that it resembles
an Asian option payoff. This involves the determination of the terminal fund NAV at the
maturity of the product using the average fund value over a prespecified number of
assessment dates, which has the effect of decreasing the volatility. Hence, this lowers
option price. Other exotic options, which are usually cheaper than plain vanilla options
are also used for increasing the participation rate. A basket of funds, where the
correlation between the funds reduces the option price, is one example, but digital,
quanto and barrier options are also used by developers due to their option premium
lowering nature.
Another deciding factor of the participation rate is the bond price, which is closely linked
to the risk associated with the issued debt. The risk is quantified by the debt issuer‟s
credit rating. By choosing a bond with lower credit rating it is therefore possible to obtain
a lower bond price thereby increasing the asset allocation share towards the option and
consequently raising the participation rate. However, the tradeoff is higher credit risk
reducing the security of the investment, which has proved critical in the recent financial
crisis.
3.2 Issuing process
The structuring process of the protected security involves several different parties, which
can be seen in Figure 2. Large financial institutions can sometimes take on both the role
as developer as well as the issuer role in the process, whereas minor actors usually only
take part as issuer of the end product. A lot of the structures offered are developed by a
large financial institution, and issued by the subsidiary branches under that institution.
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Figure 2: The issuing process of the option-based structure
Source: Merchant (2004) and own contribution.
The standard structure issuing process is illustrated above. The product issuer issues the
shares and receives the investment capital from the investors, who in turn receives a
number of shares according to the share price. In some cases, the fund is launched only to
serve as the entity providing the risky asset exposure in the structure, while in other cases
the capital protection is added after the launch of the fund where the issuer enters into an
agreement with the developer. The developer guarantees the repayment of the principal
and the payment of the specified participation rate of the performance of the fund at
maturity. As the bond issuer provides the protection of the principal amount by selling
the zero-coupon bond to the developer the credit rating on the issued shares is a
combination of the bond issuer and the developer‟s rating (Merchant 2004). The
developer also writes a call option on the fund. The option gives the issuer the right to
provide the investors with the appreciation of the fund above its initial NAV. At
maturity, if the NAV of the fund is greater than its initial value, the strike price, the
developer delivers the principal and the payoff of the option to the issuer. Then the
investors can sell back their shares to the issuer and capitalize any gains. Conversely, if
NAV is less than its initial value the developer delivers the principal to the issuer, who
then distributes an amount equal to the initial investment in return for the shares bought
to the investors.
4 CONSTANT PROPORTION PORTFOLIO INSURANCE
The Constant Proportion Portfolio Insurance (CPPI) strategy is a popular alternative to
the option-based structure, especially on funds or other thinly traded assets. It is a
portfolio insurance strategy originally developed in 1986 by Fischer Black and Robert
Jones. The objective of portfolio insurance is to protect the investor against negative
market moves while still retaining upside potential, however, smaller than in an
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investment directly in the risky asset. This is exactly what the CPPI strategy achieves by
adjusting the proportions of risky and risk-free assets in the fund portfolio to react to
market changes.
4.1 Implementation of the CPPI structure
The CPPI strategy is basically a trading strategy designed to adjust the exposure to the
underlying risky asset, which could be almost any traded asset(s), by reducing this
exposure during market downturns and increasing it in upward market trends (Black &
Perold 1992). Hence, unlike the static option-based strategy the CPPI is a dynamic
strategy, which creates the capital protection through a rebalancing algorithm.
Furthermore, the strategy ensures that a predetermined minimum return is achieved at all
times, or more typically, at a given date in the future by always ensuring that the amount
invested in risk-free assets is sufficient to meet the specific target level in the future. In
this aspect the CPPI approximates the payoff from a call option on the underlying asset –
which is in essence the option-based structure. However, unlike the option-based strategy
the CPPI requires the portfolio manager to take a long position directly in the risky asset,
instead of through an option, and to more actively manage the structure.
4.1.1 The CPPI strategy
The below figure illustrates the composition of the CPPI. There are two main
components in the strategy which make up the total net asset value, or the NAV, of the
CPPI: a risky asset, and a secure or risk-free asset in the form of cash deposits, bonds or
money market instruments. Like in the option-based structure the risk-free asset ensures
that the strategy meets the repayment requirement at maturity and the risky asset
provides the exposure to potential gains
Figure 3: The CPPI strategy components
Source: Iversen & Bilslev-Jensen (2009).
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The floor, F, is the amount currently needed to guarantee that the principal sum can be
returned at maturity and can be thought of as the present value of the principal at
maturity. If, for instance, the investor invests a total of £100 in a capital protected fund
that promises 100% protection at maturity the floor should yield £100 at maturity. The
asset that achieves this capital protection is usually a zero-coupon bond, or equivalently a
coupon-bearing bond or a basket of fixed income or money market instruments. Hence,
the floor is computed as the discounted price of this bond. Assuming continuous
compounding of interest rates the initial value of the floor, F, will be given like equation
in (3.3), the price of a zero-coupon bond. Thus, the floor is a function of interest rates
and time remaining to maturity, and it increases with the passage of time.
The investment level is the amount invested in the risky asset, which provides the upside
potential of the structure. It takes the form
(4.1)
Here, is the so-called cushion, C. The cushion is the amount needed to
prevent the NAV from falling below the floor. The multiplier m is a leverage factor that
represents how much leverage is allowed in the structure. Usually the size of the
multiplier is within the range of 2-5 and it is chosen to reflect the expected performance
of the fund as well as the risk preferences of the investor. A higher multiplier means that
more funds are invested in the risky asset, which increases the potential for gains.
However, a high multiplier also means that the portfolio will approach the floor faster,
should the NAV experience a sustainable decrease. After having determined the
investment level to be invested in the risky asset, the remaining amount is invested in the
risk-free asset, the proportion B, which is equal to the lower part of the above figure,
NAV-IL.
Figure 4: Illustration of a possible CPPI strategy scenario
Monte Carlo simulation using the Heston model. n=7,500,
T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442,
v0=0.038, q=3.35%, ∆t=6/312, trading boundary=2%.
15
The illustration of a path of the CPPI strategy shows an example of how the strategy
could evolve through time. When the risky asset price S rises a larger proportion of funds
is allocated towards the risky asset, which is represented by the investment level IL. On
the other hand, when the market is declining as illustrated at the very end of the time
horizon more funds are moved to the risk-free asset B in order to protect the floor. The
illustration also shows the main weakness of the CPPI: it is strongly path-dependent. The
performance of the strategy is not just given by the terminal NAV but depends on how
the asset price and thus the allocation of risky and risk-free assets have developed since
inception.
4.1.2 Gap risk
The gap risk is the risk of the NAV falling below the floor. It is a critical factor as the
developer usually provides the guarantee of the repayment of the investment. Should the
CPPI structure fail to obtain the promised amount at maturity the developer must usually
pay the difference between the actual portfolio NAV and the guaranteed amount. In this
case the developer has essentially written a put option, which must be priced and hedged,
on the gap risk (Cont & Tankov 2009). The developer can build the cost of the option
into fees and premiums charged to the investor, but can also try to sell off the gap risk by
packaging up the gap risk in securities and selling it (Pain & Rand 2008).
4.1.3 Rebalancing algorithm
The CPPI strategy can be boiled down to the following trading algorithm. The procedure
itself has no immediate effect on the NAV of the portfolio but only allocates the funds to
the two components. This practice is repeated until maturity with rebalancing of the
portfolio according to a certain predefined frequency.
Step 1: The total portfolio value NAV of the CPPI is calculated using the formula
(4.4)
where S denotes the underlying asset value.
Step 2: Now G is introduced as a constant representing the principal protected amount.
This is generally the initial amount of the portfolio NAV0. The floor can then be
computed, where r is the t-year zero rate at time t. With continuous compounding is thus
equal to
(4.5)
Based on the floor, the cushion is given by
16
(4.6)
Step 3: After having determined the cushion, the investment level is defined by
multiplying the multiplier by the cushion.
(4.7)
This is the proportion to be invested in risky asset.
Step 4: Finally, the rest of the funds are invested in the risk-free asset
(4.8)
The risk-free component is invested in zero-coupons of the same maturity as the CPPI,
Bt, with dynamics
(4.9)
where r is the continuous bond rate.
Step 5: Defining the rebalancing dates as fixed dates the CPPI
portfolio is rebalanced at these dates. At the next rebalancing date the procedure is
performed again, starting at step 1.
The fair price of a share is made up of the two below components
(4.10)
where POption denotes the option on gap risk. Analogously to the option-based strategy the
total share price is given by
(4.11)
When it comes to measuring the performance of the CPPI strategy this is usually done by
assessing the CPPI NAV in terms of the underlying asset performance (Boulier &
Kanniganti 1995), so that
(4.12)
4.2 Issuing process
Like in the option-based structure the issuing of a capital protected fund that uses CPPI
to obtain the capital protection involves different parties. The shares are distributed to
investors according to a scheme similar to that of the option-based strategy. The
developer then invests the capital in the fund, which invests a portion of the capital in
risky assets and the other portion in risk-free assets. This allocation is determined by the
CPPI investment level formula (4.1). Thereafter, the developer manages the allocation of
17
assets of the fund between the risky and risk-free assets according to the trading
algorithm from section 4.1.3.1
Figure 5: The CPPI issuing process
Source: Merchant (2004).
The above illustration shows that the primary difference from the option-based structure
is that the developer does not directly obtain the capital protection by a static investment
in the bond, but from an indirect asset allocation carried out by the developer or fund
portfolio manager.
The developer guarantees to the investors the repayment of the principal at maturity, and
the payment of 100 %, or other levels of protection, of the performance of the fund.
Hence, the rating on the protected securities is determined by the developer‟s rating,
which is generally double-A or below (Merchant 2004). The developer then hedges its
risk of repayment through the application of the CPPI strategy. If the developer fails in
this task it is exposed to gap risk.
5 OPTION PRICING THEORY
This chapter will outline the fundamentals of continuous-time arbitrage theory. The
chapter thus provides the background needed for performing the valuation of the capital
protected fund shares in chapter 8 and 9. Two approaches to pricing a financial derivative
can be taken, either a PDE approach or a risk-neutral valuation approach. Both of the
approaches will be described as they will both be applied in the thesis; the PDE approach
is used for deriving the closed-form solution in the Heston model, which is used for the
calibration in section 8.2.3, and risk-neutral valuation theory is used in the Monte Carlo
simulations in chapter 8 and 9.
1To simplify the example the fund/portfolio manager and the developer are assumed to be the same. In
reality, they may be separate entities.
18
In the first sections the assumption of no dividends is made. This assumption will be
lifted at the end.
5.1 An arbitrage free and complete market
Before it is possible to derive any option pricing results the definition of an arbitrage free
market must be made. The arbitrage free market is by far the most important assumption
in the option pricing framework. In the following, the market is assumed to be
frictionless, short-selling is allowed, trading is continuous and transaction costs and taxes
are non-existing. Furthermore, the market is considered only to consist of two assets,
which simplifies the description. Define the asset price process S as a geometric
Brownian motion (GBM)2 given by
(5.1)
where is the mean rate of return, is the volatility of S, and Wt is a Wiener process.
The second asset is the risk-free asset with price process B and with dynamics
(5.2)
This is the numeraire, which is assumed to be strictly positive,3 and it is often defined as
a money market account.
Now the definition of an arbitrage free market can be derived. Arbitrage is defined by the
creation of wealth through risk-free profit, that is, arbitrage is making money out of
nothing without taking on risk. The price of financial instrument is fair if and only if
there exists no arbitrage opportunities, or there would be mispricing in the market.
Formally, an arbitrage opportunity is defined by a self-financing portfolio h with value
process V (Björk 2004, p.92), where a self-financing portfolio is defined as a portfolio
where the purchase of a new portfolio is financed solely by selling assets already in the
portfolio. Thus, no adding of extra funds is needed (Björk 2004, p.81) and an arbitrage
opportunity would be such that
(5.3)
The market is said to be arbitrage free if there are no arbitrage opportunities. Now all
self-financing portfolios must yield the risk-free rate since if a self-financing portfolio h
and a process k exist such that
(5.4)
2 The reader is referred to Appendix A.1 for further description of the GBM process.
3 A numeraire is the measure against which other assets are measured.
19
then it must be that kt = rt for all t or an arbitrage opportunity would arise. Thus, it must
hold that
(5.5)
If it would be possible to borrow money for buying the portfolio and earn riskless
profit and if riskless profit could be made by short-selling the portfolio and
investing at the risk-free rate. The important thing to notice is that the above portfolio is
riskless as its dynamics does not contain a driving Wiener process and so the important
result that a riskless portfolio must yield the risk-free interest rate is obtained.
A contingent claim Г can be replicated, or is attainable, if there exists at least one
attainable portfolio such that
(5.6)
where the trading strategy h is a replicating strategy for Г (Björk 2004, p. 111). The
market model is said to be complete if any contingent claim is attainable. Now, if this is
the case, holding the portfolio and holding the contingent claim are equivalent, thus, in
the absence of arbitrage opportunities the price process of the contingent claim
must satisfy
(5.7)
A rule of thumb is that the model is complete and arbitrage free if the number of random
sources, or Wiener processes, is equal to the number of traded assets excluding the risk-
free asset.
5.2 The Black-Scholes partial differential equation
Having determined the basic market model characteristics the objective is now to find the
price process f of a contingent claim Г with maturity T using these previously given
conditions. This could for instance be a European call option. The price process
must be such that the market consisting of St, Bt and does not
contain any arbitrage opportunities. This can be obtained by the replicating portfolio
argument, which states that the price function of the derivative can be replicated with a
self-financing trading strategy by forming a risk-less portfolio and so the price of the
strategy must be the price of the derivative. This is the essential argument used by Black
& Scholes (1973) in their derivation of the famous Black-Scholes (BS) partial differential
equation (PDE).
20
Define a derivative with dynamics like in (5.1), for which the price at time t is
known, and an underlying asset with dynamics like in (5.2). By Itô‟s lemma4 the change
in the variable during an infinitesimal small time interval is then some function
of St and t, time.
(5.8)
A portfolio with price process of the underlying asset and the derivative can then be
constructed by a self-financing trading strategy so that the Wiener process is eliminated
and the portfolio becomes riskless. The portfolio consists of +1 derivative f and a fraction
of - underlying assets, where the plus denotes a long position minus and the minus a
short position. It follows that the portfolio value is given by
(5.9)
The change dΠ in the value of the portfolio at any time t is given by both the option
value and the asset value changes:
(5.10)
Substituting equations (5.8) into equation (5.10) yields:
(5.11)
Now the portfolio will be riskless if the stochastic term in form of dSt is eliminated. This
is obtained by letting
(5.12)
The above is also known as a delta-hedge strategy where the portfolio is continuously
rebalanced so that it stays riskless by choosing the in (5.12). The assumptions
discussed in section 5.1 explain that the portfolio must instantaneously earn the risk-free
rate. If it earned more or less than this return arbitrageurs could make riskless profit.
Hence, it must hold that
(5.13)
where r is the risk-free interest rate. If equations (5.9), (5.11) and (5.12) are now
substituted in equation (5.13) it follows that
(5.14)
Dividing by dt and rearranging yields
4 The reader is referred to Appendix A.2 for a description of Itô‟s lemma.
21
(5.15)
which is the Black-Scholes PDE, which every derivative depending on St and t must be
priced according to, or arbitrage opportunities would exist (Hull 2009, p.285). The
important thing to notice is that the rate of return of the underlying asset has been
suppressed. This means that the price of a derivative does not depend on the return on the
underlying asset; only the volatility is important in the pricing of the derivative. Now
the PDE in (5.15) can be solved either directly or numerically and the arbitrage-free price
of the derivative is then given by the solution formula. However, this will not be of
relevance in this thesis.
5.3 Risk-neutral valuation
Instead of explicitly solving the PDE to obtain the arbitrage-free price of a derivative the
risk-neutral valuation approach can be taken. The rationale behind risk-neutral valuation
is that the price of a financial instrument is not affected by the expected return on the
underlying asset, which was what Black & Scholes (1973) showed in their derivation of
the PDE. This is the same as the investors in a risk-neutral world being indifferent to risk
and therefore they do not require compensation for taking on risk (Hull 2009, p.289).
Thus, the expected return on all securities, including the self-financing portfolio, is just
the risk-free interest rate and the market price of risk, which will be elaborated on later, is
equal to zero under the risk neutral measure.
Returning to the PDE in (5.15) it must hold that the solution to the PDE is the same both
in the risk-neutral world and the real world, thus, the two approaches must yield the same
price . The connection between the two is given by the Feynman-Kač theorem,5
which states that if f is the solution to a PDE of the form in (5.15) with the boundary
condition that at maturity is equal to the derivative payoff, the following risk-
neutral pricing formula holds
(5.16)
where St follows
(5.17)
and where denotes a Wiener process under the risk-neutral measure . From
inspecting the above it is clear that the risk neutral process of St is the same as the GBM
in (5.1), except that the drift term has been replaced with r. The important thing to note
5 The theorem is described in Appendix A.3.
22
is that under the probability measure the discounted asset price process is a
martingale, which means that the expected value of an observation in the future is just
equal to the previous observation value, conditional on all previous observations. In other
words, today‟s asset price is obtained as the expected value of the future asset price under
the -measure discounted at the risk-free rate. Otherwise, arbitrage opportunities would
exist.
The risk-neutral martingale measure is also known as the equivalent martingale
measure (EMM). The measure is equivalent to the real world probabilities ℙ in that the
probability of impossible events is equal under both measures ℙ
, and vice versa for certain events ℙ . Letting denote a
price function of an asset St and Bt be a riskless asset with dynamics as in (5.2), the
results are formally written as in (5.18), which results in the risk-neutral pricing formula
(5.18)
Thus, the price of the derivative at any time t can be found by using a risk-free bond as
numeraire, which transforms the asset prices into martingales. This is a powerful result,
which will be applied throughout chapter 8 and 9.
The only problem left is to determine the expected payoff at maturity of the derivative.
This can be done by Monte Carlo simulation, which is introduced in chapter 7, and
thereafter discounting the payoff to obtain the fair price of the derivative.
6 STOCHASTIC VOLATILITY
The BS model is known to suffer from several shortcomings, as some of the assumptions
in the framework do not correspond to certain market observations. This is shortly
demonstrated in the next sections. To reflect the real world most precisely and obtain
applicable results the model choice will therefore not be given by the BS model. Before
the numerical analysis of the two strategies can be performed the stochastic volatility
framework is presented, and the implications of it are defined.
6.1 Empirical findings
Even though the famous BS model framework is theoretically robust, the assumptions
underlying it do not fit market observations. This thesis focuses on relaxing the
23
assumption concerning volatility.6 The BS model assumes that volatility is constant over
the life of the derivative, thus, it assumes that the price of the derivative is unaffected by
changes in the price level of the underlying asset. However, empirical studies have
shown that log-returns that is assumed in the BS model are not normal distributed and the
studies have described the long-observed features of the implied volatility surface such as
volatility smile and skew, which indicate that implied volatility does tend to vary with
respect to strike price and time to maturity.
Figure 6: Absolute log-returns on the FTSE100
Absolute log-returns on the FTSE100 from 01-07-2005
to 01-07-2010. Source: Datastream and own contribution.
The above graph depicts how the returns on the FTSE100 index during specific periods
seem to exhibit volatility, which was especially pronounced during the height of the
recent credit crisis in 2007-2008. The concept of volatility clustering is observed due to
significant positive autocorrelation of squared returns, which Scott (1987) among others
points out. Additionally, Fouque et al. (2000) observe negative correlation between
volatility and the return on an asset, known as the leverage effect. In a BS world this
correlation would not exist even though Fischer Black already introduced it back in 1976
(Black 1976) and concluded that market participants do not exhibit a symmetric response
to news in the market. Instead, the participants cause higher volatility after bad news
rather than good news. Consequently, the leverage effect implies that returns seem to be
mean reverting. Gatheral (2006, p. 2) supports this finding stating that economical
considerations imply that future volatility will lie within a certain, limited interval.
Empirical evidence therefore suggest that volatility should be modeled using a mean
reverting process.
6 The reader is referred to Appendix A.4 for the assumptions in the BS framework.
24
6.1.1 The volatility smile
A standard tool for proving the inconsistency between BS and market volatility is the
volatility smile, which is obtained using the concept of implied volatility. Implied
volatility, here denoted by , is the BS volatility embedded in market prices
of options, where K denotes the strike price and T the time to maturity. It is
formally expressed as the value of volatility in the BS formula such that
(6.1)
where is the BS price. In practice the value of is obtained by applying
a numerical root-finding algorithm and keeping all other parameters constant.7 Plotting
the implied volatility as a function of strike indeed shows that volatility is not constant.
The volatility smile takes different forms depending on the asset type at hand and for
stocks the implied volatility will typically resemble a smirk (Hull 2009, p.389). As can
be seen in Figure 7a) the implied volatility of FTSE100 index call options across
maturities is not constant, but varies with strike and time to maturity.
Figure 7a) and 7b): FTSE100 call option implied volatility and volatility surface
Option prices of 19 Oct 2009. S=Spot=5281.5, r=risk-free rate=1.2% and q=dividend yield=3.35% for a
variety of strikes and maturities up to 1 year. Source: Datastream and own contribution.
Under the assumptions of the BS model, the implied volatility would be a constant
function, indicated by straight, horizontal lines in Figure 7a) above. Instead, the figure
shows that this is not the case. For many types of options the volatility smile is different
depending on the time to maturity and consequently the smile can be extended to also
incorporate this feature. This yields the above volatility surface in Figure 7b), which
shows how the implied volatility of in-the-money options varies, so that the volatility
surface is not a plane surface.
7 The bisection method for finding implied volatility has been applied. The reader is referred to Appendix
A.5 for a description of this method.
25
Given the above empirical evidence it seems correct to conclude that volatility should not
be modeled using constant volatility as the market does not price options according to the
BS model. Instead, alternative models that incorporate non-constant volatility could be
considered.
6.2 Introducing stochastic volatility
In order to capture the varying volatility reflected in Figure 7a) and 7b), the volatility of
the underlying asset is often modeled as following a stochastic process rather than as a
constant. In that case, the following stochastic differential equations for the asset price St
and its variance vt are assumed to be satisfied
(6.2)
(6.3)
where µ is the drift of asset returns and ,η, are volatility process parameters. By
selecting different functions for and in (6.3) different models for the volatility can be
obtained. The two Wiener processes, WS and Wv are often modeled to be correlated with
, which is the correlation between random asset price returns and
changes in vt. The relation brings yet another dimension to the pricing of an option as the
PDE for models of stochastic volatility now also includes a term for the correlation
between the volatility and the underlying asset.
Since stochastic volatility is not a traded asset an inspection of (6.2) and (6.3) makes it
clear that a model of stochastic volatility includes more stochastic sources than there are
traded assets. The market is in fact incomplete and so the BS portfolio replicating method
from chapter 5 cannot be applied directly in the deriving of a PDE that the price function
must satisfy. The incomplete market implies that the financial derivative no longer can be
perfectly hedged by taking positions in the underlying asset and a risk-free asset, and that
a unique EMM does no longer exist. However, all of the previously given results still
hold. Then the prices obtained using results derived assuming a complete market will still
be arbitrage free, but several EMMs might yield the prices.
6.3 The Heston model
A model that has become very popular and is relatively easy to executed in VBA is the
Heston (1993) model, which has been chosen for further use. What separates the Heston
model from other models of stochastic volatility is that it allows for a fast and easily
implemented closed form solution for European options. This feature is particularly
26
useful when it comes to calibrating the model to known option prices later in section
8.2.3.
The dynamics in the Heston model corresponds to choosing and
in equations (6.2) and (6.3)8 and these equations then become
ℙ (6.4)
ℙ (6.5)
The above asset price process is identical to the geometric Brownian motion (GBM)
underlying the BS model, except that the volatility now is allowed to be time-varying.
Additionally, the correlated Wiener processes are introduced
ℙ
ℙ (6.6)
Thus, the Heston model incorporates correlation between the volatility and the
underlying asset price, which in most cases will be negative – an effect, which he calls
the leverage effect. Recalling from section 6.1 this should ensure that the model is more
in accordance with observed market dynamics when it comes to capturing the volatility
clustering described as the asset price will tend do go down when volatility goes up. The
leverage effect also causes the distribution of the asset returns to have a fat left tail. A fat
left tail means that the likelihood of obtaining negative returns increases, which seems to
explain some of the deviations from the BS framework discussed in section 6.1.
The above equation in (6.5) is also known as the Cox, Ingersoll, and Ross (1985) square
root process, where the term describes the evolution in the variance on the
underlying asset. The part ensures that the process exhibits mean reversion
towards the long-run mean of the variance at speed . That is, the higher , the faster
the process will tend towards the long-run mean, which the variance revolves around.
Thus, the Heston model also includes the earlier mentioned mean-reversion effect.
Finally, is defined as the volatility of the variance, which determines how much the
random component in the Brownian motion affects the variance .
6.3.1 The Heston PDE and risk neutral processes
The fact that the market is incomplete in a stochastic volatility model, such as the Heston
model, introduces a problem since it is not sufficient to create a riskless portfolio by
eliminating the uncertain, random term WS. Another Wiener process Wv is also present,
which makes the derivation of the PDE different from in the BS framework. The issue is
8 Note from now on the notation .
27
solved by introducing a second benchmark derivative depending on the same
variables as f, the price of which other contingent claims will be priced in terms of. This
makes it is possible to derive the PDE in the same manner as in the derivation of the BS
PDE. The reader is referred to Appendix A.7 for the actual derivation, which yields the
PDE
(6.7)
The boundary conditions that must be used to determine the final price of a financial
derivative is then given by the given payoff of the derivative. However, to be able to use
the results the market price of volatility risk in the PDE in (6.7) must be
defined. The economic interpretation of is that it is that is the risk premium that
measures the expected mean excess return over the risk-free rate compared to the risk
taken on. In relative terms it is also known as the Sharpe ratio in the settings of CAPM
theory. Thus, the term is also referred to as the market price of risk. Since volatility is
not a traded primary asset like S, the market premium on risk is not straightforward to
define. Fortunately, one can use the fact that the price of the financial derivative
in an incomplete market is determined by aggregate risk aversion on the
market, liquidity and other factors. Thus, the price of risk is determined by the market.
By calibrating the model to observed market prices, an assumption about the preferences
on the market is made (Björk 2004, p. 221). Letting the market price of risk
, which corresponds to choosing a specific probability measure throughout the rest of
the thesis, it is implicitly assumed that market agents are risk-neutral. This means that the
results from the calibration can be used for option pricing.
Now the following risk neutral processes of the Heston model can be defined
(6.8)
(6.9)
(6.10)
where
denotes the risk-adjusted -Wiener processes. Hence, the transformation of
the asset price GBM process is in effect carried out by substituting the risk-free rate r
with the mean rate of return in the drift in (6.4), like in the BS case. Since the market
premium on risk was assumed to be zero the variance process reduces to the
exact representation above as the difference between the objective and risk neutral
processes is indeed the market price of risk.
28
6.3.3 Heston’s closed-form solution
Imposing boundary conditions according to the payoff function of interest, the above
PDE can be solved to obtain a formula for the price of the given financial derivative. In
his paper Heston (1993) proceeds with deriving a semi-closed form solution to the PDE
for European call options. However, since the focus of this thesis lies on pricing the
capital protected fund shares the details in the derivation are not given here and the
reader should look to Appendix A.8 for the full derivation. Instead the main results are
summarized. The price of a European call option is given by
(6.11)
The represent the -adjusted probabilities that the option expires in-the-money
conditional on , the logarithm of the asset price process, and today
(6.12)
These probabilities are not available in a closed-form solution. Instead, Heston (1993)
derives the solution for the characteristic functions fj using the fact that they are
derivatives depending on the same state variables as Pj,
(6.13)
where is the imaginary unit, and with terminal conditions
and
Thus, in order to solve the characteristic functions and must be found.
The derivation of the solutions is considered beyond the scope of this thesis and the
reader is referred to Heston (1993) for the procedure. Therefore, the solutions are
provided here without further detail
(6.14)
(6.15)
where
and
The risk-neutral pseudo-probabilities are found by an inverse Fourier transform
(6.16)
29
where Re designates a real number. The two complex integrals in (6.16) must be
evaluated in order to find the option price and this can only be done by numerical
methods, which explains why the solution is only semi-closed. The solution is found by
plugging the solution of (6.16) in (6.11). The actual integration will not be described in
detail, but will instead be based on the numerical integration techniques in Rouah &
Vainberg (2007). These results are applied later in the Calibration section.
6.3.4 Dividends
Reducing the risk-neutral drift with q, where q is the continuous annual dividend yield on
the underlying asset, the dynamics of the asset price process can be modified to
incorporate continuous dividend payments. This holds for all the previous defined results
(Björk 2004, p.237). Inserting q in the asset price process in (6.8) yields the below asset
price process, which will be applied throughout the rest of the thesis, along with the
variance process in (6.9)
(6.17)
Furthermore, to incorporate continuous dividend payments in Heston‟s closed-form
solution, equation (6.11) can be adjusted to account for this by replacing the spot price S
with .
(6.18)
7 MONTE CARLO SIMULATION
Based on the previous sections, the model for pricing capital protected fund shares has
been developed, and will now be implemented. The products to be analyzed are all path-
dependent, as the final payoff depends not only on the terminal value of the underlying,
but also on the observations up until that time. Monte Carlo simulation is a simple tool
for simulating and pricing such path-dependent instruments or other instruments based
on particular complex asset price dynamics. Hence, this thesis proceeds with Monte
Carlo simulation as the chosen tool for pricing the option component in the SWCPF12
and for simulating and pricing the CPPI strategy.
7.1 Simulation procedure
Given the process dynamics of the financial derivative Monte Carlo (MC) simulation
provides a technique of obtaining numerical solutions to the pricing problem. As has
been shown earlier this is exactly what is needed for applying risk-neutral valuation to
pricing of options as it was described how the price of a derivative can be expressed as
30
the discounted value of its expected payoff. Hence, Monte Carlo simulation is a natural
tool for this.
Consider an option with a payoff at time T given by a function f of the underlying asset
prices. In order to price the option the dynamics of the underlying asset is modeled under
the risk-neutral measure, which ensures that discounted asset prices are martingales. The
estimator of the price of the option Y is given by
(7.1)
Recall that equation (7.1) is of the same form as the risk neutral valuation formula in
(5.16). To evaluate this expectation a number of paths of the underlying asset over the
time interval [0,T] are simulated according to its risk-neutral dynamics. At the end of
each path the discounted payoff of the option is calculated, and the average across paths
is the estimate of the option price.
(7.2)
From the Law of Large Numbers, which is described in Appendix B.1, the estimate
converges to the true price as n tends to infinity, or
with probability 1 as .
The variance of the simulation can then be used as an estimate for and is given by
(7.3)
The standard error is equal to
(7.4)
The confidence interval of the estimate gives the interval that contains the true value with
a given amount of certainty 1 – . The interval can then be calculated as given below,
where denotes the fractile of the standard normal distribution.
(7.5)
Letting =5% a 95% confidence interval is then given by
(7.6)
since 1.96 is the 97.5% fractile in the standard normal distribution. This significance
level will be applied throughout the rest of the thesis.
31
7.2 Efficiency of Monte Carlo simulation: Variance reduction
When evaluating the results later derived it is of interest to find out how precise the
approximation in equation (7.2) is. The error estimate of a Monte Carlo simulation was
given above in (7.4). The smaller the standard error of the estimate, the more accurate the
results and the narrower the confidence interval will be. To improve the precision of the
estimate the interval can be reduced by increasing the number of simulated paths n, or
reducing the variance. Unfortunately, in order to approximate a reasonably small
standard error a huge amount of simulations is usually necessary and this is the main
disadvantage of the MC simulation procedure, referred to as crude Monte Carlo. To
minimize the problem a number of techniques that help reduce the variance are available.
It should, however, be noted that the techniques are many and the most effective ones are
often also complex to implement. Therefore, this thesis will only consider antithetic
variance reduction, but methods such as control variates and moment matching could
also be applied.
7.2.1 Antithetic variate technique
The intuition behind the use of antithetic variate variance reduction is that the variance of
the MC estimate can be reduced by balancing the values of the paths. Hence, for every
simulated path one also calculates the price of the option using the opposite path – that is,
the path that is created using the same randomly drawn numbers, but with negative sign.
The variables form an antithetic pair in that a large value is accompanied by a small
value, which suggests that an unusually large or small output computed from the first
path will be balanced by the value computed from the antithetic path resulting in a
reduction in variance. Given the pairs of observations, where denotes the antithetic
path, the antithetic variates unbiased estimator is the average
of all 2n observations
(7.7)
In the calculation of the standard deviation of the estimator it should be noted that and
are not independent, only the pairs of observations are. Thus, the standard deviation
must be calculated using the above average of the path and its antithetic path – i.e. each
term in the sum above in (7.7) – and not the 2n payoffs that is given by calculating the
payoffs of each path separately. Now the confidence interval can be found using (7.6)
like in the crude Monte Carlo case.
32
7.3 Generating random numbers
One of the components of the Monte Carlo simulation is the random number generator,
which produces the stochastic component, that is, the Wiener processes in (6.8) and (6.9).
The increments over a time period of ∆ are normal distributed, so that ,
which can be represented by where . Thus, a sequence of
uniformly distributed random variables Ui and methods for transforming those variables
to the normal distribution are needed to simulate the asset paths.
The sequence of random variables U1, U2,… should have the property that
Each Ui is uniformly distributed between 0 and 1
The Ui are mutually independent.
The first property is an arbitrary normalization. Numbers in another interval are also
applicable but the normalization U(0,1) ensures that the distribution can be transformed
to many other distributions, which makes the generation of N(0,1) random numbers
straightforward. The second property is of greater importance as it implies that all pairs
of values should be uncorrelated and that the value of Ui should not be predictable from
U1,…,Ui-1. In other words, this property ensures that the numbers are random. An
effective random number generator produces values that seem to be consistent with the
two properties above (Glasserman 2004, p.39). If the numbers are not truly random, the
Monte Carlo estimate preciseness and convergence speed will be somewhat poorer.
In reality, the computerized random number generator produces pseudorandom numbers
as it only mimics randomness, and this should therefore be kept in mind when the
financial derivative price estimates are analyzed. However, no further discussion of the
different pseudo-random number generators will be carried out, and the VBA function
Rnd will be applied. To transform the random U(0,1) numbers to the N(0,1) distribution
the Marsaglia-Bray algorithm is used.9
7.4 Discretization of the model
The stochastic differential equations previously given are continuous-time and must be
discretized in order to simulate the paths of the asset and variance processes. Given an
arbitrary set of discrete time points t, which a derivative‟s life time is divided into, the
discretization generates random paths for the logarithm of the asset price, S, and the
variance v for every change in discrete time t. Both of the processes are path-dependent,
as the asset price process depends on the stochastic value of the variance process. Hence,
9 The reader is referred to Appendix B.2 for a short description of the method.
33
the whole path must be simulated. The risk-neutral processes given in (6.8)-(6.10) in the
Heston model must be used for the simulation in order for the risk-neutral valuation
results to hold. In their discrete form they are written as
(7.8)
(7.9)
(7.10)
By Itô‟s lemma the log process of the asset can be written in discrete form as
(7.11)
where
. As mentioned in section 7.3 and . When n
correlated samples from the normal distribution are required, the usual procedure is to
obtain the samples using Cholesky factorization. However, in this case there are only two
correlated samples, which means that the correlation structure can be obtained using a
simpler approach (Hull 2009, p.430). Two independent samples from the standard
normal distribution can be transformed into two correlated samples and , using the
below formula
(7.12)
where are independently and identically distributed standard normal variables that
each have zero correlation with .
The approximation of a continuous-time process by a discrete-time process introduces
discretization bias, which should be kept in mind when evaluating the results. The bias
makes it difficult to obtain valid confidence intervals, as well as it causes a higher
number of time steps and simulations in order to reduce the standard error to be
necessary (Broadie & Kaya 2006). Several other improvements to overcome
discretization bias have been proposed, and the reader is referred to Appendix B.3 for
further discussion.
7.4.1 The Euler scheme
The Euler scheme provides a simple approximation of the true continuous time dynamics
in (6.8)-(6.10), where the processes are approximated exactly as written in (7.8) and
(7.9). In this setting, time T is divided into m subintervals of the length
and the
approximation of the form
(7.13)
34
can be applied to the asset price process, where Z is a standard normal random variable
and . The variance stochastic process can be approximated in the same way,
which yields
(7.14)
Shocks to the volatility, , are correlated with the shocks to the asset price, ,
as can be seen in (7.12).
A serious shortcoming of this method is that the processes will often become negative as
a consequence of sampling the random number Z. To avoid the negative variance
practitioners usually adopt one of two approaches (Gatheral 2006, p.21): Either the
absorbing assumption, where if vt < 0 then vt = 0, or the reflecting assumption, where if
vt < 0 then vt = – vt. The absorbing assumption will be applied in the simulations.
Equipped with (7.12)-(7.14) the Monte Carlo simulation of the asset price and variance
processes can be performed.10
8 VALUATION OF THE OPTION-BASED STRUCTURE
Recently offered capital protected funds are typically tied to the indices across the world.
The issuers emphasize diversification, actively managed portfolios and capital protection
when marketing the products to the retail or individual investor. Hence, the capital
protected fund chosen for analysis in this thesis is index-linked and incorporates the
Asian tail feature, as these characteristics dominate the market. The fund has been chosen
in order to illustrate one of the methods that the developers make use of when these types
of products are created. Products with other types of exotic features, such as the often
used quanto-feature where the option is denominated in one currency but pays out in
another, or products that are based on a basket of assets will not be analyzed, even
though those characteristics are also common.
8.1 The chosen product: Scottish Widows Capital Protected Fund 12
Scottish Widows plc is a Scottish investment company, now a subsidiary of the major
UK financial institution, Lloyds Banking Group plc. After the acquisition of HBOS in
2009 Lloyds is now the largest retail bank in the UK, and holds the position as market
leader in a number of areas (Lloyds Banking Group, 2010a). Scottish Widows is the
group‟s specialist provider of life insurance, pensions and investment products,
10
In practice, it is advisable to test the model and determining the optimal number of time steps and
simulations by comparing simulation results to the analytical price of the option. Due to page limitations
this test is not performed.
35
distributed through Lloyds branch network. The Scottish Widows Capital Protected Fund
12 (SWCPF12) is one out of a number of funds offered by Scottish Widows, and the
fund return is based solely on the UK FTSE100 index.
Table 1: SWCPF12 product characteristics
Effective date Termination date Offer price Protection Participation Cap Payoff
19 Oct 2009 5 Oct 2015 £1/share 100% 150% 65% Asian
Source: Prospectus enclosed on CD-ROM.
As can be seen above, the investment horizon of the product is 6 years, and the product is
100% capital protected. The product consists of a zero-coupon bond, which constitutes
the capital protection, and an option, which provides the upside potential.11
However, the
product is not constructed with an option component directly written on the fund NAV,
but indirectly through an option of the FTSE100. The main risks connected to the
product are the counterparty risk associated with the single option seller, and the credit
risk of the bond component. Should the counterparties fail to meet their obligations, the
capital protection fails, and the investor might not receive back his full investment.
However, Scottish Widows seeks to reduce this risk by demanding collateral from the
counterparties.
The cash investment period is the period in which the fund is made available for
subscription and it ran from 24 July 2009 to 5 October 2009. If an investor invested in
the fund during this period, the cash would earn interest. The amount invested determines
the number of shares that the investor is entitled to. For instance, the offer price of £1 per
share results in 10,000 shares received given an investment of £10,000. Any interest
earned in the cash investment period will be added to the investment amount. At the
termination date the investor receives back the NAV of his total investment by selling the
shares to Scottish Widows.
The fees incorporated in the structure amount to a sole initial charge of 7.70% of the
investment amount. This fee is deducted from the amount used for buying shares. If the
investor buys 1 share at a price of £1, he effectively only receives a fraction of 1-0.077 =
0.923 of the share. This is an important implication when the total share price is
estimated later on. No other fees are incurred, except for a switching out fee, which will
be ignored in the analysis, assuming that the investor does not withdraw his investment
during the investment period.
11
The prospectus does not explicitly state that zero-coupon bonds are used for providing protection, but
this is assumed to be true. The prospectus and the marketing brochure can be found on the enclosed CD.
36
8.1.1 Payoff structure
The option component associated with the SWCPF12 can be classified as an Asian
option as the option payoff is based on the average of the underlying index at several
assessment dates. The fund invests only in the UK FTSE100 index comprised of the top
100 capitalized UK companies listed on the London Stock Exchange and is not actively
managed, but tracks the index. This allows the valuation to be carried out by assessing
the FTSE100, thereby indirectly the NAV of the fund.12
Had the SWCPF12 consisted of
an actively managed fund based on a variety of underlyings the NAV of the fund should
be modeled, which could turn out to be quite tedious. The terminal value is calculated
as the arithmetic average of the index price on every trading day. This can be represented
as
(8.1)
where Fti is the value of the index at assessment date i = 1,2,3,…,n. The assessment dates
are specified as the UK stock market trading days over the last 12 months of the
investment period. Assuming 252 trading days per year the pricing of the option
component is thus based on the FTSE100 index price on these 252 days and the formula
can thus be represented as
(8.2)
The fact that the assessment dates are placed in the end of the investment term is a so-
called Asian tail feature. Compared to a sampling period across the whole investment
period the Asian tail does not have as much influence on the option price as if the
sampling period was to begin at the inception date, but still, the tail lowers the option
price.
Given a cap δ, a floor η and a participation rate γ, the promised percentage return on the
fund at maturity (Bennett et al. 1996) can be expressed as
Payoff
(8.3)
where is the terminal average value of the FTSE100 and is the value of the
FTSE100 at the inception date, t0. In the case of the SWCPF12 the value of the floor is
0%, because of the fact that the investor is assured no loss of principal. A cap of 65% is
enforced to lower the option cost, as the cap cuts off some of the upside potential. With a
12
Recall that NAV=Assets-liabilities divided by outstanding shares. By ignoring liabilities, is becomes
possible to price a SWCPF12 unit directly through the FTSE100.
37
share price of £1 the analysis proceeds with a principal of £1. Consequently, the payoff is
presented as
Payoff
(8.4)
From the above formula it can be seen that the payoff is linked to return on the FTSE100,
not the actual index price. Not taking other characteristics of the payoff structure into
account the investor is promised £1.5 per 1% increase in the FTSE100 if the percentage
rise in the index is less than 65%. If the percentage rise is bigger than 65% the investor
receives 65%. Should the return on the FTSE100 turn out to be negative the option pays
off zero and the investor only receives the protected initial investment.
The above formula for the payoff on a share in the SWCPF is now determined and will
be used for determining the fair and total price of a share.
8.2 Estimating model parameters
In order to price the capital protected fund shares the model needs some input estimates.
All of these input parameters are non-observable directly in the market and the final
parameters are consequently estimated with a degree of uncertainty.
8.2.1 Continuous interest rates
The risk-free continuous interest rate is needed for determining the option component
value. As the SWCPF12 is written on the FTSE100 the straightforward choice would be
to use rates on UK government bonds as the benchmark for the true risk-free rates and
extract the risk-free rate from the UK Treasury Gilts. However, some market participants
argue that Treasury rates are too low, because they carry different tax considerations and
regulations and that LIBOR is a better proxy for the risk-free rate (Hull 2009, p. 74). The
higher LIBOR rates incorporates the fact that a financial institution most often has to be
at least AA-rated in order to borrow at LIBOR and AA-rated entities have a small risk of
bankruptcy. The spread over Treasuries thus reflects this credit risk, and consequently
LIBOR is used as a proxy for the risk-free rate in this thesis.13
LIBOR is the rate that
banks demand for borrowing funds to another bank on the interbank market, however,
there are no LIBOR rates available for a maturity of more than a year. Therefore UK
interest rate swaps are used as proxy for LIBOR for maturities of up to 6 years. LIBOR
will be used as a proxy for the risk-free rate in the drift of the asset price and for
discounting the option component. As discussed above, the proxy will be a bit higher
13
The data used for this can be found on the enclosed CD in the spreadsheet ”LIBOR and swap rates”.
38
than the true risk-free rate demanded by investors in the risk-neutral world. Thus, the
option component estimate might be slightly lower than its true value. Furthermore, it is
assumed that the risk-free rate is deterministic. Alternatively, stochastic interest rates
could be used for modeling the short interest rate.
The proxy for Scottish Widows‟ costs related to buying the bond component must also be
estimated. It is assumed that the bond is bought from Lloyds Banking Group. Hence,
because of Lloyd‟s AA-rating (Lloyds Banking Group, 2010b), the bond component
interest rate of the protected security can also be assumed to approximate LIBOR.
Accordingly, the assumption that the credit spread between the swap rates and Scottish
Widows borrowing rate is zero and that this spread is independent of the maturity of the
option is made.
LIBOR is calculated by the money market convention (actual/360), thus, the rate must be
transformed to bond market conventions (actual/actual) by adjusting the accrual factor
(Jensen 2005, p. 70). Linear interpolation, where one calculates the unknown rate
between two maturities by assuming that it lies on a straight line between the two closest
known rates, can then be applied in order to find the LIBOR rate from the
closest in time LIBOR rates and
. The formula for the interpolation
and transformation to bond market conventions with continuous compounding is given as
(8.5)
The above formula will be applied throughout the rest of the thesis when proxies for risk-
free and bond rates are determined.14
8.2.2 Dividends
Incorporating dividend payout in the Heston model only concerns the asset price process
and so the volatility process is left unaffected. When pricing options on underlyings that
pay dividends, the risk-neutral stochastic process of the underlying must be adjusted to
reflect the dividend payments by subtracting the dividend yield from the drift, which was
shown in (6.17). The dividend yield is assumed to be continuous and constant during the
investment period. The first assumption is reasonable as the underlying FTSE100 index
is comprised of a large number of individual stocks that can be realistically assumed to
payout dividends throughout the year, whereas the second assumption is more critical.
14
An example of the calculations is given in on the enclosed CD in the spreadsheet “LIBOR and swap
rates”.
39
The data is obtained from Datastream15
using the DY variable, which expresses the
anticipated annual dividend yield on the FTSE100. A historical estimate could also have
been applied, but this is not consistent with the forward-looking nature of the option.
Therefore, the expected annual dividend yield on 19 October 2009 is assumed to equal
3.35%.
8.2.3 Calibrating the volatility estimates
The Heston model needs to return the prevailing price of European options on the
FTSE100 so that the volatility structure of the model reflects market conditions. The
parameter estimates are usually obtained by minimizing the error between
model prices and market price, so that the parameters are fitted to the market volatility. If
the parameter values produce a volatility smile or surface similar to that of the market, it
can be assumed that the model will reflect the market volatility well. Furthermore, by
calibrating with market prices the obtained estimates are risk-neutral, which makes it
possible to simulate the option payoff directly using the calibrated parameters. Recall
that this was also the condition for assuming the market price of risk to equal zero as
discussed in section 6.3.1.
Several loss functions can be applied in order to minimize the squared error. Suppose
there are N market options prices and that the model prices
depend on a set of parameters . The root mean squared error loss function (Rouah &
Vainberg 2007, p. 283) is defined as
(8.6)
where are the estimation errors . The RMSE
parameters are those that minimize the RMSE loss function above, which entails that this
loss function minimizes the raw difference between market and model prices. This is
where the semi-closed form solution from section 6.3.3 is useful. Hence, the model
prices are computed using Heston‟s semi-closed solution.16
Alternatively, the %RMSE, which minimizes the relative difference between market and
model prices could be applied. However, the this method puts more weight to cheaper
options (OTM options) and less weight to in-the-money options and long-term options
due to the fact that the model and market price difference becomes amplified when the
15
The data can be found in the spreadsheet “FTSE100 dividend yield and option prices”. 16
The reader is referred to Appendix A.8 for the derivation of the solution to the PDE as well as a brief
explanation of the VBA code used for implementing it.
40
market price is small. Since the time horizon of interest is 6 years and the SWCPF12
option component can be said to be ITM, the RMSE is chosen. This is also the choice of
Baksi et al. (1997), and several others have made use of this technique for similar
purposes.
8.2.3.1 Calibration scheme
The minimization problem is not easy to solve as the loss function is not convex and
usually many local extrema exist. Two approaches to address the problem can be taken:
using local algorithms or stochastic algorithms. Stochastic algorithms do not depend on
the initial guess of parameters, but in general these algorithms are computationally more
burdensome than local optimizers (Mikhailov & Nögel 2003). Hence, the local optimizer
approach is chosen. More specifically, the Nelder-Mead algorithm, which searches for
the minimum value of two or more variables, is applied. The algorithm is fast and easy to
implement and is standard in many mathematical packages such as Matlab.
Consequently, the technicalities behind the algorithm will not be discussed in further
detail in this thesis and the VBA code thus follows Rouah & Vainberg (2007).
Remembering from section 6.3 on risk-neutral valuation, the market price of risk has
effectively been eliminated, which leaves five parameters to be estimated. Consequently,
the following must hold
ℙ (8.7)
The initial guesses are restricted to constraints they must fulfill. Setting the constraint
provides a lower bound for the volatility process so that it never becomes
negative, also known as the Feller constraint (Cox et al. 1985). All of the parameter
values that have to be calibrated are restricted to be positive numbers, except for rho, the
correlation. Instead the correlation is constrained in the interval -1 to 1. As the Nelder-
Mead algorithm only searches for local minima the calibration the initial guesses of
parameter values are essential. Here, the same initial guesses as in Mikhailov & Nögel
(2003) are applied as initial guesses of parameter estimates.
Table 2: Initial parameter estimate guesses
v0
-0.3 2 0.1 0.2 0.1
Source: Mikhailov & Nögel (2003)
Furthermore, the market data used in the calibration has to be chosen. As the SWCPF12
uses the FTSE100 index as underlying the option prices on this index on 19 Oct 2009,
41
the inception date of the SWCPF12, are obtained.17
Put options are excluded from further
analysis as Bakshi et al. (1997) notice how calls and puts imply the same u-shaped
volatility patterns across strike prices for a fixed term to expiration. This relationship is
mainly due to the put-call parity, and consequently the exclusion of put options should
not affect the conclusions. The range of options is selected on the basis on their
moneyness (S/K), to include out-of-the-money, at-the-money, and in-the-money options.
Ideally, options of different maturities should be included in the calibration. However,
because of computational restrictions, it has been chosen to only calibrate options with
0.164 years to maturity as options with short time to maturity are more liquid.
8.2.3.2 Calibration results
Running the calibration using VBA yields the following results..
Table 3: Calibration results
ρ κ θ σ V0
-0.303 1.180 0.083 0.442 0.038
Source: Own contribution, cf. the spreadsheet “Calibration”.
The risk-neutral parameters are now directly given by the above results in Table 3, as all
the parameter estimates are calibrated from market prices. The results become
and . The spot volatility of in the risk-neutral process
then varies around the long-term level of with mean reversion speed of
1.180. In figure 8a) and 8b) the volatility smile and surface of the FTSE100 using the
estimated Heston parameters are presented.
Figure 8a) and 8b): Volatility smile an volatility surface using calibrated parameters
FTSE100 call option implied volatility and volatility surface at 19 Oct 2009 with S=Spot=5281.5, r=risk-
free rate=1.2% , q=dividend yield=3.35%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038 for a variety of
strikes and maturities indicated by individual lines. Source: Own contribution.
17
The data can be found in the spreadsheet “FTSE100 dividend yield and option prices”.
42
Comparing Figures 8a) and 8b) with the market implied volatilities of Figure 7a) and 7b)
it is easily seen how the whole structure has shifted and flattened out. Additionally, the
volatility surface shows that the Heston model fails to capture the implied volatility of
options with short time to maturity and ITM strikes, but fits quite well for longer
maturities. This is a general flaw of the Heston model and implies that the Heston model
should not be used for pricing options with short maturities. The loss function of 6.265
implies a mean error of £6.265 between market and model prices, which are denominated
in 1,000s. This suggests that the calibrated model parameter estimates yield prices that
are quite close to the observed market prices and that the calibration thus provides
parameter estimates that are rather good at capturing market prices.
It should be noted that the fit of the model probably would be even better if the
parameters were calibrated across time to maturity.
8.3 Assumptions
Before the actual valuation of the SWCPF12 is conducted, some assumptions are made.
During the Cash Investment Period of the SWCPF12 the cash that an investor invests
will earn interest and this interest gain will be added to the initial investment to increase
the capital protected value. However, this interest depends on when the investor invests
in the fund, and will thus be ignored in the valuation.
Time to maturity is 6 years as mentioned in the prospectus. The averaging of the
FTSE100 value takes place the last year of the option life time. Due to simplification
actual dates are ignored and the product life time is divided into 6 years of 252 trading
days with no regards to the actual number of trading days. The averaging is then based
on the number of trading days in that period.
8.4 Results
Using the risk-neutral valuation theory, where the price of an option is the expected
payoff discounted at the risk-free rate, the valuation of the option-based product can now
be performed in VBA using Monte Carlo simulation for approximating the -dynamics
of the Heston model. The SWCPF12 can, as mentioned earlier, be decomposed to a zero-
coupon bond and an option with an Asian tail and by equation (3.1) these values together
make up the fair value of the product. The offer price per share was £1 and this unit price
is used as basis for the following calculations. In order to price the option component, the
following input parameters are needed.
43
Table 4: Input parameters for the pricing of SWCPF12
Input parameters
Spot price (S0) 5281.5 Kappa (κ) 0.180
Risk-free rate (r) 3.036% Theta (θ) 0.083
Dividend yield (q) 3.35% Rho (ρ) -0.303
Discount rate (rd) 3.036% Volatility of Variance (σ) 0.442
Time to Maturity (T – t) 6 Current variance (v) 0.038
Participation rate (γ) 150% n simulations 16,000
Cap (δ) 65% 3.275%
Floor (η) 0% 3.473%
Source: Own contribution.
The simulations are comparable, meaning that it is ensured that the simulations paths are
comparable between the strategies by starting the same place in the list of random
numbers used for simulating the Wiener processes. In every iteration in the simulation
the risky asset price path and volatility path are simulated using the above parameters.
However, during the first 5 years of the option life no assessment takes place. Due to
computational restrictions it has not been possible to simulate the asset dynamics using
daily time steps in this interval. Consequently, monthly time steps are applied for the first
5 years, which yields 5 x12=60 time steps. This, however, increases the before-
mentioned discretizaiton bias. The last year of the product life time the VBA code
registers every trading day price of the FTSE100, and finally uses these 252 prices for
calculating the arithmetic average. Then the payoff function in equation (8.4) is applied
and the payoff is discounted back 6 years at the risk-free rate. This procedure is repeated
n times. The proxy for the risk-free rate is computed using the interpolated LIBOR rate
for the given time horizon from 19 October 2009 to 5 October 2015, which yields
(8.8)
Thus, the risk-free rate proxy is estimated as 3.036%.
To obtain a precise estimate of the option price, the number of simulations n must be
decided on. The results of running a different number of simulations and calculating the
option price estimate and the related confidence intervals using equations (7.2) and (7.6),
is presented in the below graphical illustration.
44
Figure 9: Crude Monte Carlo estimate of option price
The input parameters used can be found in Table 4.
Source: Own contribution.
Due to Excel/VBA-programming restrictions it has not been possible to run more than
20,000 simulations. Consequently, the option price estimate could have been improved
by increasing the number of simulations in another program than VBA. However, the
above figure illustrates well how the confidence interval shrinks with the number of
simulations. To increase computational speed and the preciseness of the option price
estimate the antithetic variance reduction method is introduced and the procedure given
in section 7.2.1 is thus applied.
Again, running various numbers of simulations yields the results shown in the below
figure.
Figure 10: Antithetic variate estimate of option price
The input parameters used can be found in Table 4.
Source: Own contribution.
The figure shows how the antithetic variable method for reducing variance of the
estimate obtains a much narrower confidence interval faster than the crude Monte Carlo
method. Already by 8,000 iterations the level seems to even out. This is also reflected in
a comparison of the standard errors of the two methods.
45
Figure 11: Comparison of standard errors
The input parameters used can be found in Table 4.
Source: Own contribution.
Inspection of the above illustration reveals how the standard error is reduced when the
number of simulations is increased. Running 500 simulations yields a crude Monte Carlo
standard error that is 46.19% higher than when applying antithetic variance reduction.
Increasing the number of simulations lowers the numerical difference, but the relative
difference between the two methods is rather constant. The computational speed,
however, is higher with crude Monte Carlo, but this issue is quite irrelevant due to the
computational restrictions that only allows for up to 16-20,000 simulations. At this low
number of simulations, the difference in speed is negligible. Hence, the antithetic
variable method will be applied in the conclusive simulation.
8.4.1 Share price of the fund
Running 16,000 simulations the fair option price estimate is obtained. This brings down
the standard error to 0.001 and yields a price estimate of the option component of 0.152.
Table 5: Option component price
Option component Confidence interval:
Option value 0.152 Lower bound 0.150
Standard error 0.001 Upper bound 0.154
Source: Own contribution.
The interest rate used for calculating the price of the bond component is calculated like in
(8.8). Since the time horizon is assumed the same as the option life time the rate is thus
equal to 3.036%. The bond component value is then calculated as given in (3.3):
(8.9)
At this point it is already clear that the last years‟ historically low interest rate levels do
not leave much of the amount invested for the option component. The need for lowering
46
the option price by exotic payoffs is thus very clear in the SWCPF12 case. The total fair
price of a share unit without deduction of fees can be calculated from (3.1):
(8.10)
Comparing the fair price of £0.986 to the offer price of £1 the estimated value is 1.5 %
below the offer price, which can be seen in Table 6. This percentage seems rather low as
emission costs, management costs and likewise must be paid from this amount. The fair
participation R which relates the percentage return according to equation (3.4) on the
share to the percentage appreciation in the FTSE100, is lower than the marketed value of
the participation percentage of 150%. The investor thus in effect receives a return of
109.87% times any gain in the value of the FTSE100, which is substantially lower than
the promised 150%.
Table 6: Scottish Widows Capital Protected Fund 12 (SWCPF) valuation results
SWCPF12 value Comparison data
Fair share price 0.986 Actual/estimated value 1.015
+ 7% fee 0.078 Participation rate, R 1.095
Total share price 0.910
Source: Own contribution.
As mentioned in section 8.1 an initial fee of 7.7% is deducted from the amount to be
invested leaving only a fraction of 0.923 of the share for a unit investment.
Consequently, the total price of a unit including fees is equal to (1 – 7.7%) x 0.986 =
£0.910. Now the share price estimate is 9.9% below the offer price. Concluding the
valuation, a share in the SWCPF12 is overcharged, although not considerably much.
However, characteristics of the structure could have been altered after the subscription
period, and the developer‟s price estimate of the option might not have been determined
on the 19 Oct 2009, which could possibly explain some of the price deviation.
8.5 Sensitivity analysis
In order to evaluate how sensitive the results with respect to the estimated input
parameters are the below sensitivity analysis has been carried out. The analysis is based
on the fair price of the share, which means that fees are not included. The changes are
measured at inception, meaning that the consequences of changes during the life time of
the product, reflected in mark-to-market values, will not be discussed. In the analyses all
of the variables, except for the variable being analyzed, are kept constant. Table 7
provides an overview of the scenarios and the changes in the variables in each scenario
that will be made.
47
Table 7: Sensitivity analysis parameters
Scenario -1 pp Base +1 pp
Risk-free rate 2.036% 3.036% 4.036%
Bond interest rate 2.036% 3.036% 4.036%
Dividend yield 2.350% 3.350% 4.350%
Source: Own contribution.
8.5.1 Risk-free and discount rate
The level of the proxy for the risk-free rate influences the drift of the asset price
dynamics and the discount rate used for discounting the payoff of the structure. The rate
has been estimated using LIBOR as proxy as 3.036% as of 19 October 2009. However,
some uncertainty about whether LIBOR can be used as a proxy for the true risk-free rate,
or not, exists. Therefore the risk-free rate variable is changed by +/- 1 percentage point
compared to the interest rate level 19 October 2009 in Table 8.
Table 8: Risk-free rate sensitivity analysis
Scenario -1 pp Base +1 pp
Risk-free rate Option value -5.30% 0.151 5.30%
Fair share price -0.81% 0.985 0.71%
Participation 5.63% 1.102 -4.72%
Source: Own contribution.
The results in Table 8 show how the option value and the fair unit price are affected in
the same manner by reducing or increasing the risk-free rate. The option value is reduced
when lowering the risk-free rate because the drift of the asset price dynamics is reduced,
which lowers the expected payoff and thus the option price. This effect outweighs the
lower rate for discounting back the payoff, which affects the fair unit price, albeit not by
the same strength. The risk-free rate only influences the option value, and not the bond
component – the capital protection. Hence, the unit price is not affected to the same
extent as the option value. A lower option value increases the participation, while an
increase in the option and unit price as a result of a higher risk-free rate does not reduce
the participation as much as the opposite.
The rate used for estimating the price of the capital protection – the bond component – is
the interest rate that Scottish Widows earns by investing in a zero-coupon bond or a
money market instrument. The protection has been assumed to be bought from Lloyds
Banking Group, which means that the rate is determined by Lloyds‟ credit risk, access to
markets, and likewise. The assumption that LIBOR can be used for discounting seems
reasonable, but might not in fact be true. Thus, the discount rate used for calculating the
bond component price is varied in the sensitivity analysis.
48
Table 9: Bond interest rate sensitivity analysis
Scenario -1 pp Base +1 pp
Bond interest rate Option value 0.00% 0.151 0.00%
Fair share price 5.18% 0.985 -4.97%
Participation -30.94% 1.102 29.15%
Source: Own contribution.
The bond interest rate is only used for estimating the bond component price, and
consequently the option component is not affected by changes in the rate. The share price
is negatively affected when increasing the discount rate, and the participation rate
positively affected by increasing the discount rate. The opposite is true when reducing
the discount rate. The analysis shows how the participation rate is much more sensitive to
the rate used for pricing the bond component than the risk-free rate used for pricing the
option component. This is explained by the fact that the bond component constitutes
almost all of the structure – 83.3% in the base case scenario – and this value of the
component thus affects the fair share price and participation rate seriously. Hence, if the
developer wishes to adjust the participation rate adjusting the bond component would
result in the biggest change. A way to reduce this bond component price could be to
purchase the bond from a counterparty with a lower credit rating, meaning that the risk
premium on the company‟s debt would be lower. However, this would also increase the
risk associated with the structure, reflected in a larger probability of protection failure.
8.5.2 Dividend yield
The forward estimate of the continuous annual dividend yield involves a rather large
degree of uncertainty, as the yield is not directly observable in the market. For instance,
the anticipated yearly dividend yield on the FTSE100 obtained from Datastream has
changed from 5.19% on 1 April 2009 to 3.35% on 19 October 2009. It is therefore of
interest how much a change in the dividend yield affects the results. Varying the
dividend yield with +/- 1 percentage point yields the results below.
Table 10: Dividend yield sensitivity analysis
Scenario -1 pp Base +1 pp
Dividend yield Option value 11.26% 0.151 -10.60%
Fair share price 1.73% 0.985 -1.73%
Participation -10.25% 1.102 12.16%
Source: Own contribution.
A higher dividend yield than originally estimated reduces the drift and this lowers the
expected payoff. Thus, the option value is negatively affected when the dividend yield is
49
adjusted upward and vice versa when adjusted downward. Consequently, the share price
is affected in the same way and the participation rate change is affected with opposite
sign. Like in the changing risk-free rate analysis, only the option value is affected by
adjustment of the dividend yield, but the influence is almost twice as strong. The fair
share price, however, is not seriously affected due to the option component making up a
rather small part of the whole structure. Given the product investment horizon of 6 years,
the actual dividend yield is likely to change regularly over this period. Thus, even though
the parameter has a huge influence on the option estimate a precise estimate of the
continuous dividend yield is difficult to produce.
8.5.3 Heston parameters
The risk of the Heston model parameters being imprecisely calibrated to market values,
could also affect the estimated value of the SWCPF12 through changing the option
component price. The parameters are adjusted by +/- 50% of the base case value, and
results of the sensitivity analysis are listed below.
Table 11: Heston parameter sensitivity analysis
Scenario Low Base High
Kappa (κ) Option value 0.00% 0.151 -0.66%
Fair share price -0.10% 0.985 -0.10%
Participation 0.18% 1.102 0.64%
Theta (θ) Option value -3.97% 0.151 2.74%
Fair share price -0.71% 0.985 0.41%
Participation 4.36% 1.102 -2.67%
Rho (ρ) Option value -3.31% 0.151 -3.80%
Fair share price -0.61% 0.985 -0.57%
Participation 3.72% 1.102 3.95%
Volatility of Variance (σ) Option value -1.32% 0.151 1.90%
Fair share price -0.30% 0.985 0.29%
Participation 1.54% 1.102 -1.87%
Current variance (ν) Option value 0.00% 0.151 0.00%
Fair share price 0.00% 0.985 0.00%
Participation 0.00% 1.102 0.00%
Source: Own contribution.
Changing the volatility of variance σ does not influence the results considerably. A
positive σ makes the volatility stochastic around the long-term volatility parameter, by
which σ is multiplied. Consequently, the probability of large changes in the asset price is
increased when increasing σ, leading to fatter tails and higher kurtosis in the distribution.
50
The higher kurtosis thus leads to a small increase in the option price. The contrary is true
when reducing σ.
Furthermore, κ, the mean-reversion speed affects the components to a very small degree.
The mean reversion speed pulls the initial variance towards the long-term mean θ and
decides the relative weight of the current volatility compared to the long-run volatility.
As the long-run variance parameter value is higher than the current variance, a higher
mean-reversion speed makes the drift of variance process become higher than initially
faster.
The parameters ρ, σ and θ affect the option value, fair share price or participation rate
within a range of 0%-4.36% with ρ, the correlation between the asset and the variance,
being one of the most influential variables of the three. When ρ < 0 a higher absolute
correlation parameter results in lower volatility when the asset price rises, and vice versa.
ρ influences the skewness of the distribution of returns – the probability density
distribution gets a fatter left tail, and a thinner right tail, that is, it becomes negatively
skewed when the negative correlation increases. Hence, more negative extreme returns
occur. Increasing and decreasing ρ both result in a reduction in the option value. When
increasing the correlation, the probability of the asset ending with a low value is
increased and this reduces the option price. On the other hand, reducing correlation
results in a shift in the probability density function so that the mean is reduced. The
results suggest that this shift outweighs the higher probability of high terminal values of
the underlying when the correlation parameter is reduced.
Along with ρ, the long-term level of the variance θ influences the option price most
strongly. θ determines the level of the volatility along with the current volatility vt. The
longer the time to maturity, the more influence θ exerts on the variance process and
because of the relatively long time to maturity of the SWCPF of 6 years this parameter is
the most influential one, affecting the option price in the same way as σ. Consequently,
the current variance v has no effect on the option price.
Concluding the sensitivity analysis, the results show how the components are relatively
mildly affected by the Heston parameter estimate changes compared to varying interest
rates and the dividend yield. The Asian feature of the option component reduces the
impact of volatility on the option value, which might dampen the effects of the change in
parameter values.
51
9 VALUATION OF THE CPPI STRUCTURE
To illustrate how the CPPI is used for creating capital protected funds the next sections
implement the strategy and explain how a hypothetical CPPI product works and can be
valuated. The performance of the CPPI depends largely on the three variables: the
rebalancing frequency, the floor and the multiplier. Consequently, the following
valuation scenarios will show how the alteration of each of the three variables one at a
time affects the expected return on a fund share and the risk of the CPPI structure.
9.1 Implementing the CPPI strategy
The path-dependence and any adjustments of the standard structure of the CPPI strategy
makes the pricing of CPPI products complex, thus, no closed-form solutions are
available. Therefore the structures are usually priced using Monte Carlo simulations,
which will also be applied in this thesis.
Gap risk is especially pronounced when the risky asset in the CPPI structure is fund(s),
which in many cases have trading restrictions tied to them and rebalancing of the
structure is not even close to be continuous. The consequence is that the impact of sharp
market moves is raised, resulting in higher gap risk compared to more liquid assets. An
approach to incorporate this extra risk in the model is to introduce extra random jumps
and jump models are usually preferable when modeling the CPPI structure. However,
due to space restrictions, this type of models is not considered in this thesis. Instead, the
dynamics of the underlying risky asset is in the following assumed to follow the earlier
described Heston model. Allowing the volatility to be stochastic is expected to enhance
the usefulness of the simulation results, compared to constant volatility models, because
of the correlation between asset prices and volatility, which amplifies the volatility in
market downturns.
A critical input to the CPPI strategy is the number of rebalancing dates. In the 6 year
period the standard CPPI product is assumed to be rebalanced once a week, yielding
6x52=312 rebalancing dates, assuming 52 weeks per year. This seems a reasonable
assumption as the portfolio manager obviously will want to keep trading costs relatively
small, but at the same time not take on too much gap risk, which arises from large price
movements in between rebalancing. Furthermore, a trading restriction to lower trading
costs and time spent monitoring the market is imposed and this means that rebalancing
will only take place if the risky asset value has changed with more than 2% since last
rebalancing date. This trading boundary affects the risk of the structure, while not
52
inducing much influence on the NAV. Gap risk is reduced substantially with a lower
trading boundary and vice versa when upward adjusting it. This is because of the smaller
probability of large movements in between rebalancing.
The purpose of the next section is to simulate different CPPI structures and price them to
illustrate how the method is implemented in practice.
9.1.1 Pricing gap risk
From the developer‟s point of view, the quantifying of gap risk is essential, and in this
particular case of the hypothetical CPPI product an estimate of the price of the put option
written on the gap risk will be priced using the same techniques as in the valuation of a
SWCPF12 share. The option price is thus assumed to be priced as a cost or fee in the
total structure like in (4.11). The payoff of the put option is given as
(9.1)
That is, the payoff is the difference between the initial protected investment and the
terminal NAV if the end NAV turns out to be lower than the principal G, which is
essentially the definition of gap risk. The option price estimate is then obtained by
discounting the payoff at the risk-free rate, recalling the risk-neutral valuation procedure
described in section 5.1
9.1.2 Assumptions
The hypothetical standard CPPI product used for further analysis will have maturity 6
years, beginning at 19 October 2009 like the SWCPF 12. This means that the previously
estimated SWCPF12 Heston parameters based on the FTSE100 index will be in
accordance with the used time horizon here. Hence, all of the below results are simulated
using these parameters.
The portfolio value will only be simulated at rebalancing dates, making the simulation
less time-consuming. Additionally, no borrowing constraints are assumed, and the
strategy therefore takes full advantage of market rises. In practice, borrowing constraints
are often applied because of margin limits and collateral (Pain & Rand 2008).
The NAV of the fund is simulated over a time period of 6 years with a starting value at
inception of £100, hence, it is assumed that a share in the fund costs £100. This is in
accordance with most market products. With 100% capital protection this is also the
amount protected. Furthermore, LIBOR is again used as proxy for the risk-free and bond
interest rates. Assuming constant interest rates, the risk-free rate calculated in the earlier
53
section is applicable in this chapter too. This means that the interest rate for a period of 6
years from the inception date is given as 3.036%.
It should be noted that the following simulations are designed to show the average cost to
returns before fees. Thus, no management fees, administration fees, or other sorts of fees
that are popular to impose on the investor when buying into a capital protected fund, are
considered. Furthermore, the trading costs associated with the strategy are left out of the
analysis. Obviously, these trading costs in the form of bid-ask spreads or brokerage fees
could seriously alter the results of the simulations. However, fee structure and trading
costs are different from fund to fund, and depends on rebalancing frequency,
administration costs, and so on. Thus, for simplicity‟s sake, fees and trading costs are not
considered.
9.2 Rebalancing algorithm
To implement the CPPI strategy the rebalancing algorithm from section 4.1.3 is applied
and coded in VBA. Using the same model as in the option-based analysis part of the
thesis the risk-neutral dynamics of the Heston model given by (7.13)-(7.15) are used for
simulating the structure.
With 100% capital protection, and an initial investment of £100, G=100, the floor at
inception is equal to
(9.2)
This amount increases to £100 at maturity. The cushion
multiplied by the multiplier 3 can then be used for calculating the investment level
. This leaves to be invested
in the risk-free asset, which is a proportion of 49.96% of the investment as opposed to the
17.7% of funds allocated to the option component in the option-based strategy. Some
CPPI products impose the investment level weight to equal 100% at inception but to keep
things simple this will not be considered here.
In the following sections all strategies are simulated using 7,500 simulations due to
computational restrictions. The simulations are ensured to be comparable like in the
option-based valuation.
9.3 Simulation of the standard CPPI strategy
Imposing no exposure constraints two simulated paths of the standard CPPI strategy with
a multiplier of 3 are illustrated below. It should be noted that all of the figures in the
sections to come show paths that are simulated using the risk-neutral Heston parameters,
54
and must not be regarded as “real world” paths. The graphical illustrations only serve as
examples helping the reader to visualize the strategies.
Figure 12a) and 12b): Two simulated paths of the standard CPPI strategy
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038, q=3.35%, ∆t=6/312,
trading boundary=2%. Source: Own contribution.
Figure 12a) shows how the CPPI works when the risky asset price drops under the floor.
The floor ensures that the investor receives the principal of £100 at maturity even though
the asset price is far below the floor level. At points a) the structure experiences
deleveraging as the fund price drops. At point b) the CPPI structure is now in a cash-
lock, locking in the risk-free asset proportion such that the strategy is fully invested in the
risk-free asset. As the asset value experiences a rise after the knock-out the investor
completely misses out on any of these gains. Figure 12b) illustrates how the structure
captures the rise in the asset price and the investment level actually grows to exceed the
NAV as it becomes fully invested in the risky asset due to its positive drift. Assuming no
borrowing restrictions, i.e. that it is possible to borrow at the risk-free rate, the risk-free
asset proportion can become negative and NAV can exceed the risky asset value, which
is not depicted in the figure. The illustration also displays how the CPPI is particularly
well-suited if markets exhibit an upward trend.
9.3.1 Simulation results
Running 7,500 simulations of the CPPI strategy and applying equation (4.10)-(4.12)
yields the following results.
Table 12: Standard CPPI valuation results
Fair share price 91.976 Expected return 10.35%
Put option value 0.95 Standard deviation 89.58
Total share price 92.930 Gap risk 0.26%
Participation rate 112.88%
Cash-lock 44.07%
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038, q=3.35%, ∆t=6/312,
trading boundary=2%. Source: Own contribution.
a)
a)
b)
55
The fair price of a share in the fund is estimated as £91.976. Including the option on gap
risk, the total price of a share would amount to £92.93. The expected return of 10.35%
and a participation rate of 112.88%, which measures the CPPI strategy‟s performance
over an investment directly in the risky asset, present a rather good result. However,
these results do not include any other fees, the introduction of which would yield a
poorer result to an investor. Gap risk incurs if the NAV drops by more than
, and it can be seen that the strategy is exposed to gap risk in 0.26% of the
simulations, which is due to the weekly rebalancing and trading boundaries. Likewise,
the cash-lock situations are rather pronounced as they occur in 44.07% of the
simulations. Based on these results, the application of a standard CPPI strategy to a
capital protected fund reveals a quite positive result.
9.3.2 Changing the multiplier and rebalancing frequency
The multiplier can be thought of as the leverage of the structure. With a multiplier of 1
the structure is essentially like a buy-hold strategy, where the initial proportions are not
altered during the investment term.
Table 13: Alternative multiplier results
m=4 m=5 m=6
Fair share price 91.967 94.852 100.014
Put option value 2.46 6.98 31.39
Total share price 94.431 101.837 131.400
Participation 112.86% 116.41% 122.74%
Expected return 10.34% 13.80% 20.00%
CPPI standard deviation 298.20 664.52 2134.26
Gap risk 0.60% 1.02% 1.50%
Cash-lock 60.43% 68.36% 70.21%
n=7,500, T=6, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038,
q=3.35%, ∆t=6/312, trading boundary=2%. Source: Own contribution.
Raising the multiplier to more than 1 increases the return of the CPPI since a larger
proportion is allocated towards the risky asset increasing the possibility of gains. On the
other hand, the risk of the structure also becomes enlarged, and this results in a bigger
risk of the strategy breaking the floor or ending up in cash-lock. The consequences of
applying different higher multipliers are shown in Table 13. When the multiplier is
increased to m=6 the expected return and participation rate increases by almost 10
percentage points compared to the results using a multiplier of 3. However, this comes at
the price of higher risk, which can be seen from the exploding standard deviation,
increased gap risk of 1.50% and the percentage of 70.21% of cash-lock situations.
56
Furthermore, the total price of a share is a staggering 131.40, reflecting the high gap risk
through the put option price. It is obvious how the developer of a CPPI-based capital
guaranteed fund must carefully decide on the multiplier before the issuing of a product.
Another important factor that the performance of the CPPI strategy depends on is the
rebalancing frequency. The decision on this is a trade-off between trading costs and risk,
which is illustrated below. 252 trading days a year, along with 52 weeks a year and 12
months a year are assumed.
Table 14: Alternative rebalancing frequency results
Monthly Weekly Daily
Fair share price 91.994 91.976 93.164
Put option value 5.84 0.95 0.12
Total share price 97.835 92.930 93.283
Participation 111.70% 112.88% 114.53%
Expected return 10.37% 10.35% 11.78%
CPPI standard deviation 86.20 89.58 84.75
Gap risk 1.62% 0.26% 0.02%
Cash-lock 30.67% 44.07% 41.91%
Average rebalancing times 24 74 149
n=7,500 , T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038,
q=3.35%, trading boundary=2%. Source: Own contribution.
As can be seen above the optimal choice would be daily rebalancing if gap risk to be
avoided is of higher priority. The monthly rebalancing scheme results in fewer cash-lock
situations than in the other rebalancing schemes, which is probably due to assets not
being moved to the risk-free investment very often given the few rebalancing dates, but
the number of gap risk situations has risen drastically and the expected return does not
seem to outweigh the higher risk. The daily scheme yields a little bit higher expected
return than the other along with a very small degree of gap risk, which normally would
suggest that this would be a good choice of frequency. However, this must be compared
to the average rebalancing times of 149, which suggests that daily rebalancing would be
trading cost intensive if these costs were included in the simulations.
9.4 Extensions to the traditional CPPI structure
Over time, several extensions to the basic CPPI structuring of a capital guaranteed fund
have evolved in order to enhance the performance of the structure and to adjust it to meet
investor preferences. In the following an overview of these extensions is given below, as
well as illustrations of how the features are incorporated and how they affect the average
NAV and risk of the structure.
57
9.4.1 Exposure constraints
The standard CPPI strategy takes full advantage of any rise in the risky asset value, as
there is no constraint on maximum exposure. In practice, however, unbounded
investment in the risky asset may be undesirable given limitations in margins and
collateral. By imposing the constraint
(9.3)
and letting the exposure equal 100% the strategy‟s investment level IL is restricted. This
also reduces the risk associated with the strategy, since the structure is not hit as hard if
the asset prices dives after the structure has been fully invested in the asset. Thus, the
exposure constraint lowers the upside potential. The constraint can be written as a
minimum function, recalling that m denotes the multiplier
(9.4)
Another alternative to the standard CPPI is imposing a minimum exposure constraint.
Should the underlying asset price fall, the allocation to the risky asset could fall to zero,
resulting in the unwanted situations of cash-lock. To avoid this a minimum level of
investment in the risky asset can be imposed, such as
(9.5)
The above restriction will reduce the number of cash-lock situations since part of the
principal protection is removed in this strategy by not letting the strategy become fully
invested in the risk-free asset. The strategy will therefore be able to regain a larger
exposure towards the risky asset after a decrease in the asset price. In this particular case,
the investment level is not bounded on the upside potential. This could be imposed,
though. Now the above can be written using a maximum function
(9.6)
To illustrate both the minimum and maximum constraint, the two above features are
imposed on each rebalancing date. Figures 13a) and 13b) show a path of the CPPI
strategy, imposing maximum constraint of 100% of NAV and minimum constraint10%
of NAV, respectively, on the investment level.
58
Figure 13a) and 13b): Examples of a maximum and minimum exposure constraint path
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038, q=3.35%, ∆t=6/312,
trading boundary=2%. 100% maximum exposure constraint, and 10% minimum exposure constraint.
Source: Own contribution.
In Figure 13a) it can be seen how the investment level IL never exceeds the risky asset
value due to the constraint on the investment level as opposed to the standard CPPI
structure in Figure 12b). Figure 13b) shows how the strategy does not end up in cash-
lock even though the risky asset value is far below the floor. At the end, the strategy
participates, albeit not much, in the rise of the asset value.
Imposing a minimum constraint of 10% of NAV and a maximum constraint of 100% of
NAV on the investment level yields the following results.
Table 15: Maximum and minimum exposure constraint valuation results
Maximum constraint:
100% of NAV
Minimum constraint:
10% of NAV
Fair share price 92.664 90.745
Put option value 0.692 2.652
Total share price 93.356 93.397
Participation 113.72% 111.36%
Expected return 11.18% 8.88%
CPPI standard deviation 31.40 89.63
Gap risk 0.25% 0.73%
Cash-lock 43.25% 0.00%
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038,
q=3.35%, ∆t=6/312, trading boundary=2%. Source: Own contribution.
Compared to the standard CPPI strategy the expected return is almost the same in both of
the constraint strategies, however, a little bit lower when imposing the minimum
exposure constraint. When the maximum constraint is imposed the number of cash-lock
situations decreases a little bit compared to the standard CPPI strategy. Hence, the
objective of the strategy is fulfilled. The minimum constraint results in an increase in the
number of gap risk situations, but now the cash-lock situations have completely been
59
eliminated, which may outweigh that the expected return in this strategy is smaller than
in the standard CPPI.
9.4.2 Profit lock-in
If the price of the risky asset exhibits an upward trend the investment level will rise
above the NAV and the strategy becomes fully invested in the risky asset, resembling an
ordinary long position in the asset. However, this also entails that the floor becomes
insignificant in relation to the NAV, which means that if the asset price sudden falls, the
structure will experience a faster deleveraging (Boulier & Kanniganti 1995). The profit
lock-in feature, also known as the “ratchet effect”, ensures that a fraction of the
performance is periodically locked-in and captures the performance of the risky asset
(Paulot & Lacroze 2009). When implementing the feature the floor is still set to grow at
the rate specified in the earlier section, securing the capital at maturity. Separating this
strategy from the basic CPPI strategy is that G, the principal protected amount, is no
longer a constant, but a piecewise constant. That is, the floor is variable and will be
raised if the NAV or cushion exceeds some specified trigger also raising the capital
protected amount. Thus, the floor is raised if the asset price increases but left intact
otherwise, which means that the strategy secures profit in favorable times and carries
them through if the market declines.
The profit lock-in strategy is implemented by imposing a trigger function at some
rebalancing dates – usually annually. Typically the trigger function is based on the NAV
or cushion, which must be higher than the previous rebalancing date‟s value, or some
other value, reflecting a positive drift on the risky asset and thus the need for locking in
the profit. If this is the case, the floor is adjusted upwards by increasing G. The profit
lock-in feature, however, can limit the participation in the risky asset so that some of the
potential profit is lost. If the risky asset shows strong positive movement in the beginning
of the period the floor is raised. Thus, the exposure that would have otherwise been
beneficial to the investor is traded in for a larger proportion of investment in the lower
yielding risk-free asset. This is also illustrated below in Figure 14b).
Using the same parameters as in the above simulations and imposing a profit lock-in rate
at 75% of NAV with annual rebalancing results in a simulation path of the profit lock-in
strategy as shown below. If the NAV at the rebalancing date is higher than the NAV was
a year ago, the principal protected amount G is raised by 75% of the difference between
the NAV values, thus, the floor is raised. Figure 14a) illustrates how a decrease in the
60
asset price early in the period prevents the structure from participating in the soaring
asset price later in the period.
Figure 14a) and 14b): Two simulated paths of the profit lock-in strategy
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038, q=3.35%, ∆t=6/312,
trading boundary=2%. 75% of the NAV is locked-in annually if current NAV > NAV previous year.
Source: Own contribution.
In figure 14b) it can be seen how the feature works well in locking in the profit at the
annual verification dates, if the NAV exhibits a positive drift right from the beginning. It
is obvious how the investor benefits from the raising of the floor when the asset price
dives late in the period.
The Monte Carlo simulation results show how the profit lock-in has a positive effect on
the ending NAV value of the strategy.
Table 16: CPPI profit lock-in valuation results
Fair share price 100.857 Expected return 21.01%
Put option price 6.29 Standard deviation 264,36
Total share price 107.147 Gap risk 0.27%
Participation rate 123.77% Cash-lock 48.17%
Average final guarantee, G 119.30
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038,
q=3.35%, ∆t=6/312, trading boundary=2%. 75% of the NAV is locked-in annually if current
NAV > NAV previous year.
The expected return is now almost doubled compared to the expected return on the
standard CPPI strategy. This suggests that the downside of the feature, namely that it can
limit the participation in the risky asset, is more than outweighed by the lock-in of profits
in positive market trends in the given investment scenario. Furthermore, the protected
amount is no longer equal to the invested amount but has increased by 19.3%. The
number of cash-lock situations has increased as expected, but the gap risk is not really
much higher. The participation rate has also improved quite a bit, which might justify the
61
higher total share price. These results suggest that the profit lock-in strategy could be a
good alternative to the standard CPPI structure.
9.5 Results and comparison of the strategies
The previous sections have shown how the CPPI strategy can be adjusted while still
retaining capital protection. However, the different strategies have certain consequences
in relation to the risk of the structures, which must be taken into account when
constructing the product.
Table 17: Comparison of results for standard CPPI, minimum and maximum constraint and profit
lock-in strategies
Strategy Total share price Expected return Gap risk Cash-lock Participation
Standard CPPI 92.93 10.35% 0.26% 44.07% 112.88%
Minimum constraint 93.397 8.88% 0.73% 0.00% 111.36%
Maximum constraint 93.356 11.18% 0.25% 43.25% 113.72%
Profit lock-in 107.148 21.01% 0.27% 48.17% 123.77%
n=7,500, T=6, m=3, r=3.036%, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038, q=3.35%, ∆t=6/312,
trading boundary=2%. Source: Own contribution.
The results clearly demonstrate how the strategies have different consequences for the
average NAV and risk of the CPPI structure. The developer runs a substantial gap risk by
forcing the structure to uphold a minimum exposure towards the risky asset.
Furthermore, the lower expected return and participation rate do not suggest that this is
an attractive feature to impose on the CPPI in the settings of this thesis. Imposing a
maximum exposure constraint increases the expected return and participation rate a bit
compared to the standard CPPI structure, while also lowering the risk of cash-lock
situations. This can be explained by the fact that the structure does not allow the
investment level to exceed NAV and so the risk of becoming completely deleveraged is
reduced. Finally, the profit lock-in strategy yields the highest expected return and
participation rate, but also comes with the highest cash-lock risk. Raising the floor
annually to lock in previously earned profit seems to work well, however, these results
may change if more frequent adjustment of the floor is imposed, which might result in
higher trading costs and a higher number of cash-lock situations.
9.6 Sensitivity analysis
To be able to compare the CPPI strategy results with the options-based strategy results a
sensitivity analysis based on varying the same input parameters as the SWCPF12, while
keeping all other variables constant, is carried out below. However, the varying of the
62
Heston volatility parameters is not performed, and only the standard CPPI strategy is
included in the analysis.
Table 18: CPPI strategy sensitivity analysis
Scenario - percentage change -1 pp Base +1 pp
Risk-free rate Total share price 4,21% 92,93 -3,71%
Expected return -18,36% 10,35% 22,22%
Gap risk 0,00% 0,26% -3,85%
Cash-lock 5,67% 44,07% -5,42%
Participation 4,35% 112,88% -3,86%
Bond interest rate Total share price -2,13% 92,93 1,27%
Expected return -20,39% 10,35% 11,98%
Gap risk 0,00% 0,26% 0,00%
Cash-lock 5,26% 44,07% -1,70%
Participation -1,92% 112,88% 1,12%
Dividend yield Total share price 1,03% 92,93 -1,86%
Expected return 22,22% 10,35% -18,36%
Gap risk -3,85% 0,26% 0,00%
Cash-lock -5,42% 44,07% 5,67%
Participation -3,86% 112,88% 4,35%
n=7,500, T=6, m=3, =1.180, θ=0.083, ρ=-0.303, σ=0.442, v0=0.038, ∆t=6/312.
Source: Own contribution.
Table 18 shows the results from changing the above variables in the standard strategy
with weekly rebalancing, 7,500 simulations and a multiplier of 3. Lowering the risk-free
rate at which the NAV is discounted produces a higher total price both from the
discounting of the NAV itself, which outweighs the reduction in the asset price process
drift, and the discounting of the put option on gap risk. At the same time the number of
cash-lock situations increases, which could be due to a lower drift of the asset price
process. The lower drift also lowers the expected return substantially. The consequences
of upward adjusting the risk-free rate are explained by the opposite reasoning. Changing
the bond interest rate, or the rate at which the floor is determined, by -1 percentage point
leads to a high decrease in the expected return, and thus a decrease in the unit price. A
lower interest rate leads to a higher bond floor needed to obtain the protected amount, G,
which could explain the increased number of cash-lock situations and thus the lower
NAV. Interestingly, the number of gap risk situations do not change when adjusting the
bond interest rate. Varying the dividend affects the drift on the asset price process,
leading to a lower expected return and unit price when the dividend yield is upwards
adjusted, and vice versa when downward adjusted. The same reasoning as in the risk-free
rate case explaining the cash-lock situations can be used.
63
10 THE OPTION-BASED STRATEGY VS CPPI
Though the two methods for constructing the protection of the principal obtain the same
objective, the methods differ in certain aspects. Using the previous results and
descriptions, some of the most important differences are listed below. Since the two
products used for illustration in this thesis are not directly comparable due to the different
leverages and features of the structures, only general reflections are made. However,
these reflections will partly be based on what the analysis of the products has revealed.
10.1 Comparison of results
One of the things that separate the option-based and CPPI strategies is the level of
flexibility tied to the structures. The CPPI is rather flexible in that the investment level is
increased with upward trending markets, and vice versa. The option-based strategy, on
the other hand, is static throughout the life of the product. Therefore, the option-based
approach should perform better when realized volatility is high as the investor has locked
in the implied volatility estimates incorporated in the option price. The CPPI is also
flexible with regards to possible underlying assets and the strategy can provide access to
markets that are otherwise not easily attainable with the option-based approach, as most
options are tied to indices or individual stocks. Writing an option on an illiquid fund
might not be optimal with the option-based strategy as a market for this option is usually
non-existing, should the need for unwinding the fund occur. In general, upward trending
assets with low volatility such as funds of funds, diversified debt portfolios, multi-asset
portfolios, and so on, are among the optimal choices as underlyings for a CPPI strategy.
On the other hand, the CPPI is strongly path dependent. A loss in the beginning of the
investment period means that the strategy moves most of the funds into risk-free assets,
which prevents the investor from participation in gains later on in the period. Another
consequence of the path-dependency is that a great part of the costs of the structure – the
trading costs – are only known retrospectively. Conversely, the option-based strategy
with an Asian option feature is also path-dependent, but the probability of the option
ending up in the money after a plunge in the price of the underlying is considerably
higher than in the CPPI case.
Finally, the CPPI structure is less sensitive to interest rates because they usually have
lower bond exposure than option-based structure, which becomes evident in the
sensitivity analysis, where varying the bond interest rate in the option-based strategy
changes the participation rate dramatically at up to 30% as opposed to a minor change 4-
64
5% in the CPPI strategy. Additionally, the low interest rates in the current economy
causes the bond protection in the option-based strategy to be expensive and this leaves
little room for designing the return component of the strategy, which became clear in the
valuation of the SWCPF12. With this in mind, the CPPI might be the better alternative in
the current market conditions.
The below table provides an overview of the most prominent differences between the
two structures.
Table 19: The option-based strategy vs. CPPI
Feature Option-based CPPI
Principal protection Through bonds Through rebalancing algorithm
Administration costs Low Medium-high, due to constant
supervision and frequent rebalancing
Risk of failure Low, depends on bond issuer Low-high, depends on multiplier and
trading restrictions
Suitable underlying assets Liquid assets, such as indices and
stocks
Upward trending assets with rather low
volatility
Path-dependency Depends on the option type Highly
Exposure Static Dynamic
Transparency Poor, due to complex options and
cost structure
Good, simple to explain and NAV can
usually be observed
Source: Own contribution.
10.1.1 Capital protection
In the option-based strategy the capital protection is created by the developer and the
direct purchase of bond. Conversely, applying the CPPI strategy means that the fund
manager, or developer, invests in risk-free assets continuously and so the protection is
more static. Unlike the option-based strategy the CPPI creates the capital protection by
dynamically changing the allocation between the two asset types depending on their
performance. The protection is thus obtained by ensuring that the NAV of the strategy
stays above the floor. By introducing the profit lock-in feature in the CPPI strategy the
capital protection becomes variable, as opposed to in the option-based strategy, where it
stays constant throughout the investment period. This feature could be used for attracting
investors also reinforced by the quite impressive performance reflected in the expected
return and the participation rate.
10.1.2 Return
One of the advantages of the option-based strategy is that the return to the investors at
maturity does not depend on the delta hedging employed by the developer to protect its
risk connected to the option. On the other hand, this is clearly a risk when the CPPI
strategy is employed, as the returns on the strategy are strongly dependent on the
65
frequency and trading restrictions connected to the rebalancing of the portfolio, and the
investor is thus more directly exposed to the hedging process. However, the dynamic
allocation of the CPPI can result in a proportion of more than 100 % to the risky asset in
a positive market environment, assuming no borrowing restrictions, which could
potentially increase gains.
The participation rate is the only parameter that allows for a direct comparison between
the two strategies. The option-based strategy yields a participation rate of 109.5%,
whereas the standard CPPI obtain a rate of 112.88%. Looking separately at the
participation rate as a performance measure the CPPI turns out to be best in the
comparison.
10.1.3 Risk
The risks of the structures also differ depending on the type of risk. The exposure to
volatility is clearly different between the two structures. In the option-based strategy the
direct exposure to volatility is taken on by the developer through the call option written,
whereas the exposure to volatility in the CPPI strategy is shared by the investor and
developer. The main risk connected with the option-based structure is the credit risk tied
to the zero-coupon bond, or the capital protection component. This risk is usually
transferred to the investor. Conversely, the risk in the CPPI structure is not as directly
tied to the bond issuer and the main issues stems from the gap risk, which is taken on by
the developer and thus not affects the investor. The cash-lock situations means that the
upside potential is lost and so the investor only receives his initial investment back.
Though this is an unwanted situation, it hardly defines as risk in the conventional
definition where the investor forfeits the guaranteed investment amount.
66
11 CONCLUSION
The objective of this thesis was to analyze how capital protected funds are created and
valuated by the use of the option-based strategy and the CPPI. To create the appropriate
foundation for answering the problem statement the first part of the thesis dealt with the
description of capital protected funds as well as the option pricing theory needed to
answer the numerical aspects of the problem statement. Then, in the final part of the
thesis, the valuation was carried out in practice to illustrate how a typical capital
protected fund using the option-based and CPPI strategy are valuated.
A typical capital protected fund is a mutual fund, ETF or likewise, structured by a
developer – usually a large financing institution – who enters into an agreement of the
repayment of the principal and the appreciation in the underlying asset, or fund, to the
issuer. The issuer usually markets the fund and administrates the investor relations. In
order to invest in the capital protected fund, the investors buy shares in the fund
according to the amount invested and the share price. The method that the developer uses
for hedging its repayment risk is where the option-based strategy and CPPI comes into
the picture.
The option-based strategy is a static structure composed of a zero-coupon bond, which
provides the capital protection, and a call option on the fund, which delivers the exposure
to the fund. On the other hand, the CPPI is a dynamic rebalancing portfolio strategy in
which a floor is constructed and the trading strategy, where the NAV is ensured to stay
above the floor, provides the capital protection and the exposure to the fund.
The valuation of the SWCPF12 with application of stochastic volatility using Monte
Carlo simulation determined how the bond and option component are priced individually
and how these values together make up the fair price of the shares of the fund.
Furthermore, the sensitivity analysis made it clear that the structure is heavily dependent
on the cost of the bond component. The valuation of the standard hypothetical CPPI
product illustrated that the fair share price of a CPPI product is estimated by simulating
the path-dependent NAV of the fund and discounting the terminal NAV at the risk-free
rate. Additionally, the pricing of a put option on the gap risk, defined as the risk that the
NAV ends up lower than the required level needed for repayment of the principal, is
needed as this gap risk is usually taken on by the developer and is therefore typically
priced in the final share price of the fund. Thus, the discounted terminal NAV and the put
option price together make up the fair share price of a CPPI-structured fund.
Furthermore, modifications of the structure to illustrate the characteristics of these
67
methods for enhancing the standard strategy can be made and a sensitivity analysis
showed how the CPPI structure responds to changes in the input variables.
Finally, the comparison of the two strategies clarified the features in which they differ or
are alike. The CPPI strategy performs better for upward trending underlying assets with
rather low volatility, which makes it more suitable as capital protection method for
capital protected funds. However, the path-dependency of the structure poses a potential
threat to the performance of the strategy. Thus, various features can be imposed in order
to enhance the strategy. The option-based strategy on the other hand might be attractive
when the underlying assets of the fund are liquid, so that options exist and are traded in
liquid markets, or if interest rate levels are higher than in the current market environment.
In that case, the participation rate is increased from the larger amount left for buying
options because of the lower bond price implied by higher interest rates.
After all, the developer‟s choice of method depends, among other, on the perceived risk-
aversion and demands of investors. If customers want an actively managed strategy that
can respond to market changes quickly the CPPI is superior. If on the other hand, the
“buy high – sell low” and path-dependent characteristics of the CPPI seem unattractive to
customers the option-based strategy might be the chosen method. Moreover, the time
devoted to spend on monitoring the fund and hedging the repayment of the protected
amount is also a factor in the choice.
In conclusion, the thesis presents a tangible analysis and valuation of capital protected
funds, but also the framework for analyzing and pricing other capital protected products.
It provides the reader with a deeper understanding of the characteristics of the two
strategies and the factors that must be considered when capital protected funds are
created and priced. Future research might include analysis of other exotic features of
capital protected funds, including jumps or stochastic interest rates in the modeling of
asset prices as well as the hedging of gap risk in relation to the CPPI strategy.
68
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Webpages
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Prospectuses
Scottish Widows Capital Protected Fund 12. Enclosed on CD.
71
LIST OF FIGURES
Figure 1: The option-based structure components 9
Figure 2: The issuing process of the option-based structure 12
Figure 3: The CPPI strategy components 13
Figure 4: Illustration of a possible CPPI strategy scenario 14
Figure 5: The CPPI issuing process 17
Figure 6: Absolute log-returns on the FTSE100 23
Figure 7a) and 7b): FTSE100 call option implied volatility and volatility surface 24
Figure 8a) and 8b): Volatility smile an volatility surface using calibrated parameters 41
Figure 9: Crude Monte Carlo estimate of option price 44
Figure 10: Antithetic variate estimate of option price 44
Figure 11: Comparison of standard errors 45
Figure 12a) and 12b): Two simulated paths of the standard CPPI strategy. 54
Figure 13a) and 13b): Examples of a maximum and minimum exposure constraint path 58
Figure 14a) and 14b): Two simulated paths of the profit lock-in strategy 60
72
CD CONTENTS
The enclosed CD contains all of the data and calculations used for producing the results
presented in the thesis.
Source data
FTSE100 dividend yield and option prices.xlsx
FTSE100 logreturn.xlsx
LIBOR and swap rates.xlsx
Scottish Widows Capital Protected Fund 12 prospectus.pdf
Scottish Widows Capital Protected Fund 12 brochure.pdf
Calibration
FTSE100 implied volatility.xlsm
Calibration.xlsm
Simulation
CPPI.xlsm
CPPI- alternative strategies.xlsm
SWCPF12.xlsm
By pressing Shift+F9 in the spreadsheets, the functions in the current sheet are
recalculated.
73
APPENDICES
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