Pricing of Basket Options

1Pricing of Basket Options

Radhika NangiaMATH 608D87053120

"The four most dangerous words in investing are: this time its different."- Sir John Templeton

Copyright (c) 2013, All rights reserved by Radhika Nangia. Duplication is strictly forbidden.

21 Introduction

In the fast growing financial markets, it has become imperative to get accurate prices under uncertainties.Options are the essential tools in hedging and risk management. Hence, numerical methods are employedto price accurately and prevent any arbitrage opportunities.An option on a stock A is a financial device that we can buy at time t0 for a price V that allows us toeither buy or sell the stock A at a fixed price K at later times. By saying the price is fixed, we meanthat the price is already known at time t0.Here we will consider the so-called European Call Option which allows us to buy one unit of the stockat a fixed time t1 t0. A Basket Option is one which is based on a basket of stocks, for example, theindex option. An option is only exercised if it is in-the-money i.e. it is in profit. The payoff for theEuropean call option is written in the following manner

u(S, T ) = max(ST K, 0)where ST is the stock price at time T , and K is the exercise price of the option.For pricing simple options on one underlying, the Black and Scholes model leads to a closed formsolution since the stock price at a fixed time follows a lognormal distribution. However, using thefamous Black and Scholes model for a collection of underlying stocks, does not provide us with a closedform solution for the price of a basket option. The difficulty stems primarily from the lack of availabilityof the distribution of a weighted sum of non-independent lognormals, a feature that has hamperedclosed-form basket option pricing characterization. Indeed, the value of a portfolio is the weightedaverage of the underlying stocks at the exercise date.Hence, we need numerical methods to price the equations. There are two ways to go about it, usingMonte-Carlo methods which is a much easier implementation or using Finite Differences which becomesmuch more complicated for this higher degree simulation. We will price the S& P 100 Index Optionwhich is popularly known by its ticker symbol XEO, which as its name suggests has 100 underlyingstocks.

1.1 Some Terminology

This section introduces us to some of the "jargon" in finance. It is necessary to clarify some of the termsbefore giving a mathematical description.Financial Derivative A financial derivative in simple terms is an instrument which is derived fromvalues of some underlying assets/ variables.Portfolio- A portfolio is the collection of all shares, options and other derivatives owned by a investor.Arbitrage- Arbitrage indicates that it is not possible in a financial market to make risk-free profitslarger than just placing money in the bank.

There are several different types of portfolios. One of them is the arbitrage portfolio where sharesand options are combined in such a way that risk is eliminated and the portfolio makes money withprobability 1 in time. This is an instantaneously risk-free money making price process.Risk-free interest rate- The risk-free interest rate, r(t), is the growth of money, M in time or theprofit one makes from the interest rate when placing money in the bank

M(t) = K exp(t

0

r(t)dt)

Volatility- It is related to the standard deviation of the stock price of a share. It is an indication forthe random behaviour of the market.


1.1 Some Terminology 3

Return- It is the ratio of money gained or lost (whether realized or unrealized) on an investmentrelative to the amount of money invested.Basket Option- A basket option is a financial derivative, more specifically an exotic option, whoseunderlying is a (weighted) sum or average of different assets that have been grouped together in abasket. For example an index options, where a number of stocks have been grouped together in anindex and the option is based on the price of the index.Call Option- Call options are a type of security that give the owner the right to buy N number ofshares of a stock or an index at a certain price by a certain date. That "certain price" is called thestrike price, and that "certain date" is called the expiration date.European Option- This option that can only be exercised at the end of its life, at its maturity.Why buy Options?Options are used for several purposes. The two most important are speculating and hedging.

1. Speculation is betting on the movement of a security. If the holder buys a call, he expects thatthe stock price will increase. The exercise price will be K. If the stock price S is greater than K,the call will be exercised and the net profit of the option will be where S K C, where C is thecost of the option. C is an important parameter in the Black-Scholes analysis. Speculation withoptions involves a greater risk than speculation with assets: the profit and losses are multipliedwhen using options.

2. The second purpose to use options is hedging. It can be used like an insurance policy; just asone would insure their house or car, options can be used to insure ones investments against adownturn. Suppose one wanted to take advantage of technology stocks and their upside, but alsowanted to limit any losses. By using options, you would be able to restrict your downside whileenjoying the full upside in a cost-effective way. So, options to reduce the risk of a portfolio.

An image showing the volatility in stocks which provide profit making opportunities.


42 The Black-Scholes Model

2.1 Assumptions

The assumptions to derive the Black-Scholes differential equation are as follows:

Geometric Brownian Motion- The stock price follows a geometric Brownian motion withconstant drift and volatility. Infinite Divisibility-Stock can be sold in arbitrary non-integer amounts. Infinite Credit- Market participants can borrow or lend arbitrary amounts of money at arisk-free interest rate. No Transaction Costs- The money paid by the buyer is exactly equal to the money receivedby the seller. No Dividends- There are no dividends during the lifetime of the derivative. No Arbitrage- There are no risk-less arbitrage opportunities. Short Selling- Market participants can borrow arbitrary amounts of stock, for an arbitrary

amount of time, with no interest. Constant Risk-Free Rate- The risk-free rate of interest, r(t) is a constant and the same forall maturities i.e. r(t) = r. Infinite Liquidity- Market participants can buy or sell a unit of stock at any time. Infinite Depth- The sale of a unit of stock does not affect sale price of other units of stock. Discrete Time- The time variable t increases in discrete steps of size dt. No Storage Costs- Market participants can hold onto arbitrary amounts of stock, for anarbitrary amount of time, at no cost.

2.2 Stochastic Model

There are two sources of return in a price process modelled as a stochastic process, namely, a deterministicand a stochastic contribution. If is the average rate of growth of the asset price, also known as the"drift", the deterministic contribution in time dt is found to be dt. The other contribution relates tothe random change in the asset prices. With the volatility related to the standard deviation of thereturns and dX a sample from a normal distribution, the contribution is assumed to be dX. Theresulting equation reads

dS

S= dt+ dX (1)

where dSS is the return. This is the stochastic differential equation. The normal distribution used is aWeiner process with the following properties : E(dX) = 0 and E(dX)2 = dt. Also

E(dS) = E(Sdt+ SdX) = E(Sdt) + E(SdX) = Sdt; E(dX) = 0

V ar(dS) = E(dS2) (E(dS))2 = E((Sdt+ SdX)2) (Sdt)2 = (2S2dX2)since E(S2dXdt) = 0. The standard deviation is the square-root of the variance, so is proportionaltoV ar(dS)/S.


2.3 Parabolic Equation 5

2.3 Parabolic Equation

The Black Scholes Model is a parabolic differential equation which is given as follows:

u

t+ 12

2S22u

S2+ rS u

S ru = 0 (2)

where, r is the riskfree interest rate and is the volatility of the underlying asset. The key financialinsight behind the equation is that one can perfectly hedge the option by buying and selling theunderlying asset in just the right way and consequently eliminate risk. This hedge, in turn, impliesthat there is only one right price for the option, as returned by the Black Scholes formula.


63 Problem Formulation

3.1 Monte Carlo

The Monte Carlo Approach is a relatively simple technique to implement this otherwise very complicatedhigher dimensional problem in numerical methods. The other advantages are that it does not requireboundary conditions and other parameters unlike the other equations. For pricing a basket option weare interested in the following stochastic differential equation

dSi = rSi +dj=1

ijSidWj , i = 1, 2, , d (3)

where ij is the covariance and dWj is a Weiner process and can be expressed as dWj =dtgj , with gj

being independent and normally distributed random numbers.Now to get a form we will use for Monte Carlo, I will use Itos Calculus covered in the last day of class:TheoremIf Xt solves an s.d.e. dXt = udt + dWt & g : R R, g C2 and Y := g(Xt) solves dYt =g(Xt)dXt + 12g(Xt)(dXt)2 We will assume that Wj are independent B.Ms

dSiSi

= rdt+dj=1

ijdWj

d(logSi) =dSiSi 12S2i

(dS2i )

From the problem we have

dSi = rSidt+dj=1

ijSidWj

(dSi)2 =dj=1

2ijS2i dt

as Wi and Wj are independent. So,

d(logSi) =rSidt+

dj=1 ijSidWj

Si 12S2i

dj=1

2ijS2i dt

logSt logS0 = (r dj=1

2ij)dt+dj=1

ijdWj

St = S0 exp ((r dj=1

2ij)t+dj=1

ijWj)

This is modelled as

St = S0 exp((r dj=1

2ij)(T/N) +dj=1

ij

T

Ngj)

where T is the expiry of the option and N the number of intervals at which I evaluate for each iterationand gj the uncorrelated normal r.v.In a lot of places this is modelled differently by taking correlated B.Ms and separating them down


3.2 Finite Difference 7

to uncorrelated set of equations. This would need us to evaluate the Cholesky decomposition. TheCholesky factorization says that every symmetric positive definite matrix A has a unique factorizationA = LL where L is a lower triangular matrix and L is its conjugate transpose. In my case, since Idid not have a perfect correlation matrix, so the lower triangular matrix turned out to be zero. Hence Idid it like mentioned above.

Finally, after running the simulation multiple number of times, the payoff is computed as

max(di=1

iSi K, 0)

3.2 Finite Difference

I will briefly write about this method to give an idea of how complicated it gets when solving the sameequation as a parabolic differential equation for n underlying assets. The equation is given as

u

t+ 12

ni,j=1

2i 2j ijSiSj

2u

SiSj+ r

ni=1

Siu

Si ru = 0 (4)

where ij is the correlation between two assets. Correlation is the extent to which assets perform inrelation to one another. For instance, its widely considered good practice to reduce volatility in yourportfolio by investing in a variety of assets whose values rise and fall independently of one another, i.e.negatively correlated with each other.

3.2.1 Principal Component Analysis

Solving the differential equation in its original form is a near impossible task in a short time, so weneed to adopt some short-cuts. First, we need to find the assets most responsible for determining priceof the option, which is done by PCA. Principal component analysis (PCA) is a mathematical procedurethat identifies the direction of maximum variance or the principal component.

I will briefly illustrate the method for the problem in hand: We first need to find the eigenvectors of thecovariance matrix of stocks and denote it by Q. Then, we use the transformation by using PCA and get

x = QS (5)

This transformation leads to a very cumbersome equation, hence we make another transformation. Thistransformation to the principal components can be written as

x = Q ln(S) + b (6)

where = T t and bi = dj=1 qij(r 2j2 ). By applying change of variables to the Black ScholesEquation we get,

u

= 12

di=1

i2u

x2i ru (7)

where (x, ) Rd (0, T ) and i is the eigenvalue number i of the covariance matrix.The payoff for the basket option is,

u(x, 0) = max(di=1

i exp(dj=1

qijxj), 0) (8)

where x Rd


3.2 Finite Difference 8

3.2.2 Asymptotic Expansion

Now the equation (8) may be truncated to any number of dimensions. But to account for all dimensionswithout solving the original higher dimensional problem, each of the non-principal dimensions areapproximated by a linear asymptotic expansion, similar to taylor series expansion. This asymptoticexpansion is given by

u = u(1) +dj=2

ju

j|=(1) +O(|| (1)||2)

where u(1) is the solution in the principal axis, is a parameter vector of eigenvalues and (1) is aparameter vector when truncating the equation to dimension 1. Using a finite difference approach, theterms in the sum maybe approximated by:

u

j|=(1) =

u(1,j) u(1)j

+O(2j )

where u(1,j) corresponds to the solution of the 2D problem on the plane spanned by the principal axisand variable j.

The following is the figure I got from running the finite difference method with controlled inputs.


94 Implementation

4.1 Monte-Carlo

For this implementation some of the concepts in class were very vital. I will describe stepwise how Iimplemented this and some of the theory behind it.

1. Calculate the covariance matrix from the daily returns of each stock.

cov(i, j) =nd=1(Ri Ri)(Rj Rj)

n 1for stock i, j and where n is the number of time periods.

2. Correction for invalid correlation matrix. Many times the data that one gets does not complywith the requirement of symmetry and positive semi-definiteness. The invalid correlation matrixhas negative eigenvalues. I used the method in Jckel [3] called Spectral Decomposition.The method is as follows:

(a) Calculating the eigenvalues i and the right hand side eigenvectors si of C, a real andsymmetric matrix.

C.S = S. with = diag(i)

(b) Setting all negative i to 0.

: i =

i : i 00 : i < 0(c) Muliplying the column vectors si by the square roots of their associated corrected eigenvalues

i and arrange them as columns of B.

T : ti = [m

s2imm]1

B = S

B =TB =

TS

(d) Finally normalizing the row vectors B to unit length gives us B.(e) The acceptable correlation matrix is given by

C = BB

3. Generate uncorrelated uniformly distributed random numbers. As all pseudo-random generatorsare flawed, I use two of them here:

(a) Mersenne Twister: Period of this sequence is a Mersenne number i.e a prime number thatcan be written as 2n 1

(b) Congruential Generator

4. Convert uniformly distributed random numbers to normally distributed random numbers byZiggurat method (inbuilt in Matlab)

5. Simulate the Brownian motion by a random walk and calculate payoffs.


4.2 Finite Differences 10

4.2 Finite Differences

As discussed above the d dimensional problem is reduced to one 1D problem which is spanned by theprincipal axis and a 2D problem in the plane spanned by the principal axis and one other dimensioncorresponding to the variable j giving the solution u(1,j). So we have to repeat the second step for theevery dimension we have. In my case, we have to solve one 1D problem and 99 2D problems.The scheme used is BDF2 method to solve the problem as it is unconditionally stable and second orderaccurate in both space and time. Most of the error introduced in this calculation is through dimensionreduction approximation.

4.2.1 Scheme

The BDF-2 is a two step method, and hence we require another scheme to do the first step. I usedBackward Euler for the first time step.The Scheme is given by the following equations

uni =23tP (u

ni ) +

43u

n1i

13u

n2i (9)

where the spatial difference operator P in the 1D case is given by

P (uni ) =12uni1 2uni + uni+1

x2 runi (10)

and in the 2D case it is given by

P (uni ) =121

uni1 2uni + uni+1x2 +

12d

uniNp 2uni + uni+Npx2d

runi (11)

The logarithmic domain must be specified. It was arbitrarily chosen as x1 [0, 2x01] for the principalaxis and xd [x0d 3, x0d + 3] for the other axes. Here x0 corresponds to the transformation of S0, thecurrent price of underlying assets.

The boundary conditions that were used were:

u = 0

u =di=1

ien

j=1 qij(xjbj) Ker

2u

x2d= 0

at x1 = 0at x1 = 2x01at xd = x0d 3, x0d + 3

(12)

with the linear boundary conditions discretized as

uni = 2uni+Np uni+2Np ,uni = 2uniNp uni2Np ,

i = 1, 2, , Npi = 1 + (Nd 1)Np, , NdNp

(13)


11

5 Results

The following two figures illustrate how previously uncorrelated sequences of random numbers undergometamorphosis after Cholesky decomposition.


12

The following figure illustrates the correlation matrix

The following is the 3 simulated paths of the price process for the XEO option.


13

I drew the following figures taking a basket of only two underlying assets, Apple Inc and Google. Thisfollowing figure shows the price process followed by the two assets in 1 iteration.

This figure shows the price process followed by the basket of two options:


5.1 Greeks 14

5.1 Greeks

The Greeks are quantities representing the sensitivity of the price of derivatives such as options to achange in underlying parameters on which the value of an instrument or portfolio of financial instrumentsis dependent. The name is used because the most common of these sensitivities are often denotedby Greek letters. Collectively these have also been called the risk sensitivities,risk measures or hedgeparameters. The Greeks are vital tools in risk management. Each Greek measures the sensitivity of thevalue of a portfolio to a small change in a given underlying parameter, so that component risks maybe treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see forexample delta hedging.

1. Delta -, measures the rate of change of option value with respect to changes in the underlyingassets price. Delta is the first derivative of the value V of the option with respect to the underlyinginstruments price S. Using finite differences it can be approximated as

= vS0

v(S0 + S0) v(S0)S0The following figure illustrates the Delta calculated for two stocks:

The implication of Delta is that if for example, the delta of a portfolio of options in XYZ (expressedas shares of the underlying) is +2.75, the trader would be able to delta-hedge the portfolio byselling short 2.75 shares of the underlying. This portfolio will then retain its total value regardlessof which direction the price of XYZ moves.

2. Gamma- , measures the rate of change in the delta with respect to changes in the underlyingprice. Gamma is the second derivative of the value function with respect to the underlying price.Using finite differences it can be approximated as

= 2v

S20 v(S0 + S0) 2v(S0) + v(S0 + S0)S20


5.1 Greeks 15

The following figure illustrates the Gamma calculated for two stocks:

The values I worked with are :

Risk free rate r Expiry Time T Strike Price K0.16% 1/12 yr 805

The results were as follows:

Method Basket size & contents Value of option No of iterations Time TakenMonte Carlo XEO option : 100 stocks 28.681807 5 104 3549.1475 secsMonte Carlo 2 basket option: AAPL & GOOG 254.0287 106 1564.1293 secs

Finite Difference 2 basket option: AAPL & GOOG 251.21133 301 1146.297641 secs


16

The graph between value of option and value of the underlying portfolio : XEO option

6 Conclusion

Monte Carlo is heavily used in the Financial Markets to deal with higher dimensional data, infact thestatistics say that Monte Carlo is used 60% of the time, finite difference methods 30% of the time, andother methods the rest of the time. Finite difference methods are preferred for data that has smallerdimensions 7 , usually Forex options (FX) because they have a faster convergence rate , but MonteCarlo needs averagely about 107 iterations. It is however preferred for its ease of use when dealingwith higher dimensional data. There are many modifications that could have been done here in thisproject if I had time. For e.g. instead of pseudo random numbers one can use low discrepancy numbers,Sobol numbers to perform this simulation. Also, I could have used variance reduction techniques to getbetter results.

References

1. Bjrk, T. Arbitrage Theory in Continuous Time. Oxford University Press, 2009.2. Brandimarte, P. Numerical Methods in Finance and Economics. John Wiley & Sons,Ltd, 2002.3. Jckel, P. Monte Carlo Methods in Finance. John Wiley & Sons,Ltd, 2002.4. Reisinger, C., and Wittum, G. Efficient hierarchical approximation of high-dimensional option

pricing problems. SIAM Journal on Scientific Computing 29, 1 (2007), 440458.


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