Implied Binomial Models

Implied Binomial Models

Master’s thesis

Alexander Herbertsson

October 10, 2001

Abstract

In this paper we extend the standard binomial model to obtain a more flexible model thatwill be calibrated with market data on European puts and calls.

We also discuss how this extended model can be used to incorporate an investor’s viewof the future behaviour of the stock market.

Contents

0.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Introduction 3

2 The Black-Scholes Model 52.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 The discrete time model 113.1 The standard binomial model . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Martingales and complete market models . . . . . . . . . . . . . . . . . . . 17

4 Implied volatility trees 234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 A generalized binomial model . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Using a generalized binomial model for option pricing . . . . . . . . . . . . 274.4 Arrow-Debreu prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.5 The implied volatility tree model . . . . . . . . . . . . . . . . . . . . . . . 304.6 Building an implied volatility tree . . . . . . . . . . . . . . . . . . . . . . 314.7 Bad transition probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 364.8 Some numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.9 Valuing barrier options in the presence of the smile . . . . . . . . . . . . . 42

5 Implied binomial trees 485.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Extracting the risk-neutral probability distribution . . . . . . . . . . . . . 495.3 Building an implied binomial tree . . . . . . . . . . . . . . . . . . . . . . . 505.4 Some numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.5 Using investor’s biases in an implied binomial tree . . . . . . . . . . . . . . 57

References 59

A Appendix A 61A.1 Centering condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1

0.1 Acknowledgements

This master’s thesis was conducted at the Financial Engineering & Risk Management groupat Handelsbanken Investment Banking in Stockholm, Sweden. I am very fortunate to havegot the opportunity to work in this group. I would like to thank all the members in thisteam, especially senior quantitative analyst Dr. Carl-Johan Rehn, quantitative analyst TorNordquist, quantitative analyst Mathias Hogberg, head Torbjorn Iwarson, Desk ManagerMagnus Fagerang, Portfolio & Pricing Fredrik Edlund. Due to my many discussions withthese persons I have learned a lot about financial mathematics used in practice, often calledfinancial engineering. The experience I have gained during my seven months’ visit in thisgroup is invaluable.

I would also like to thank Prof. Christer Borell for helping me with the mathematicaldetails and for suggesting several improvements of the report for this master’s thesis.

Above all, I would like to thank my dear mother for giving me support and encourage-ment during the preparation of this thesis, as well as always.

Finally I am also very grateful to my cousin and his fiancee for being kind and lettingme stay in their apartment during my visit in Stockholm.

To my mother.

2

1 Introduction

The Black-Scholes capital market model consists of two assets, a stock and a bond. Theprice of the stock at time t is given by the equation

S(t) = S(0)eαt+σW (t)

where (W (t))t≥0 is a normalized Wienerprocess. Furthermore, the price of the bond attime t is given by the equation

B(t) = B(0)ert.

Here α, σ, S(0) and B(0) are real constants where the three last ones are strictly positive.A call option on the stock is the right but not the obligation to buy the stock at the

price K at some future time point T . The price of the call at the time t < T in theBlack-Scholes model is given by the following expression

c(t, S(t), K, T )

= S(t)Φ(

ln S(t)K +(r+σ2

2 )(T−t)σ√

T−t

)

−Ke−r(T−t)Φ(

ln S(t)K +(r−σ2

2 )(T−t)σ√

T−t

)

.(1)

If we let cM(t, S(t), K, T ) denote the corresponding market price of this option, then thereexists a uniqe σ such that the theoretical price and the market price coincide. This quantityσ will be denoted by

σimp(t,K, T )

and is referred to as the implied volatility at time t for a call with strike price K andexercise day T . If the Black-Scholes model gives a good description of a real stock price,then it is natural to believe, in spite of the friction on the market (taxes, transaction costs,. . .) that the interest of arbitrages forces σimp(t,K, T ) to be a constant or almost constantfunction of K and T . However, this is not the case, which has led to an intensive researchfor alternative models. The main part of this master’s thesis is devoted to this subject.

The so called binomial model is achieved by approximating the normalized Wienerprocess with a symmetric random walk with a 2-point distributed increment. If h = T/Nand tn = nh we then write the stock price in the binomial model as

S(tn) = S(0)eαnh+σ√

hPn

j=1 ξj , n = 0, 1, 2 . . . , N

where (ξj)nj=1 is an i.i.d such that

P [ξj = 1] = P [ξj = −1] =12.

Note thatS(tn+1) = S(tn)eXn+1

whereP [Xn+1 = αh + σ

√h] = P [Xn+1 = αh− σ

√h] =

12.

3

Obviously, the binomial model has the same weakness as the Black-Scholes model (at leastfor big N). In this paper we will mainly study a modification of the binomial model suchthat

S(tn) = S(0)eX1+...+Xn

where each Xj, conditioned on X1, . . . , Xj−1 is a stochastic 2-point distributed variable.Below we will try to choose the sequence (Xj)N

j=1 so that we get a better resemblance be-tween theory and market than is given by the standard binomial model or the Black-Scholesmodel. We will here follow articles written by Goldman & Sachs Quantitative StrategiesGroup (see for example [DK1]). A major problem connected with the implementationof this model is the fact that there are quite few (liquid) standard options traded at themarket. However, by interpolating the sparse set of option prices provided by the market,we will receive a great number of ”fictitious” market prices so that the sequence (Xj)N

j=1can be constructed. In this context the Black-Scholes formula will be used. Since the re-quirement of super-fast computations is extremely important on today’s financial marketswe will have to impose yet another condition on the sequence (Xj)N

j=1, namely that therandom variable X1 + . . . + Xn assumes only n + 1 different values for n = 1, . . . , N . Wewill finally show how this modified binomial model is applied to price and hedge exoticoptions consistently with the market. Unfortunately it is beyond the scope of this thesisto test the modified model on the market.

Another type of modified binomial model would be achieved if an investor had a strongopinion regarding the future behaviour of the market. On condition that an investorprovides the bank with a risk-neutral probability distribution at the expiry date T forsome simple contract, it is interesting to derive a reasonable price of this option today andalso inform the investor about the relevant hedging strategy. In this case we will use amodel developed by M. Rubinstein [RM].

To obtain a clear and simple understanding of the ideas that we are going to presentin this thesis, we will only consider the case when there are no dividends on the stock andwe will assume a constant interest rate r (or stated otherwise, the yield curve will be con-stant). It should be mentioned that the implementation that was done at HandelsbankenInvestment Banking incorporated discrete dividends or a continuously dividend yield to-gether with a deterministic non-constant yield curve (for details see [HA2]). Several figuresin this thesis illustrate different characteristics of the S&P 500 index, where, for simplicity,we assume no dividends.

4

2 The Black-Scholes Model

In this section we will give a brief introduction to the celebrated Black-Scholes model foroption pricing. We will also discuss one of the major defects in this model.

2.1 Background

A financial derivative or a contingent claim is a security paper defined in terms of someother security paper, sometimes referred to as the underlying asset. A simple Europeancontingent claim on a stock S with exercise date T is a financial derivative that pays theamount g(S(T )) at some future time point T , where S(T ) is the stock price at T andg(S(T )) is a real valued function. For a European call with strike price K and exercisedate T ,

g(S(T )) = max(S(T )−K, 0).

A European put is the right but not the obligation to sell a stock at the price K at somefuture time point T . Thus, in this case the payoff g(S(T )) is given by

g(S(T )) = max(K − S(T ), 0).

It is natural to ask: what is the fair price, v(t, S(t)) of a simple European claim at timet < T that pays g(S(T )) at time T? Explicit formulas for option prices in terms of knownmathematical functions are of course appreciated by traders since the requirement of fastcomputations is very important on today’s financial markets. One of the most widely usedmodels for a stock price behaviour in continuous time is the so called geometric Brownianmotion model. This means that

S(t) = S(0)eαt+σW (t) (2.1.1)

where α ≥ 0, σ > 0 are constants and (W (t))t≥0 is a so called normalized Wiener process.Here αt is called the drift of the stock log-price and σ is called the volatility of the stockprice. The parameter σ will play a crucial part in this thesis as will soon be seen. Anormalized Wiener process (W (t))t≥0 is a Gaussian stochastic process with the followingproperties:

1. W (0) = 0

2. If t0 < t1 ≤ . . . ≤ tn, then W (t1)−W (t0), . . . , W (tn)−W (tn−1) are stochasticallyindependent stochastic variables.

3. If s < t then W (t)−W (s) ∈ N(0, t− s), i.e. W (t)−W (s) has a normal distributionwith zero mean and variance t− s.

4. W has continuous trajectories.

5

Now, let us return to the problem of finding an appropriate price v(t, S(t)) of a simpleEuropean claim at time t < T that pays g(S(T )) at time T . If we assume that (S(t))t≥0 isgiven by (2.1.1) and that the price B(t) of the bond at time t equals

B(t) = B(0)ert

where B(0) and r are strictly positive constants, arbitrage reasonings together with stochas-tic calculus (Ito’s lemma) lead us to the following PDE, viz.

∂v∂t

+σ2s2

2∂2v∂s2 + rs

∂v∂s− rv = 0 0 ≤ t < T (2.1.2)

where v = v(t, s). Furthermore,v(T, s) = g(s). (2.1.3)

Equation (2.1.2) is the famous Black-Scholes equation named after Myron Scholes andFischer Black who were the first to publish the equation in the above form in 1973 [BS].Note, that the drift coefficient α does not appear in (2.1.2). The only parameter relatingto the price of the stock, that appears in (2.1.2) (and thus also in the solution to (2.1.2))is the volatility σ. It is possible to show that there exists a solution to (2.1.2) with thefinal condition (2.1.3) if we impose some (rather mild) restrictions on g(s) (a change ofvariables will transform (2.1.2) to the heat equation). Thus we obtain

v(t, s) = e−rτE[g(se(r−σ22 )τ+σW (τ))]

where τ = T − t. Note that the option price at time t then equals

v(t, S(t)) = e−rτE[g(s)e(r−σ22 ))τ+σW (τ))]|s=S(t). (2.1.4)

Once again, note that the only information corresponding to the stock, that contributesto the price of the claim at time t is S(t) and the volatility σ of the stock. By applying(2.1.4) to a European call, we get

Theorem 2.1.1 Let Y be a European call option with strike price K and exercise day T .Then the arbitrage free price of Y at time t < T , c(t, S(t), K, T ), is given by

c(t, S(t), K, T ) = S(t)Φ(d1)−Ke−rτΦ(d2) , (2.1.5)

where

d1 =ln S(t)

K + (r + σ2

2 )τσ√

τand

d2 =ln S(t)

K + (r − σ2

2 )τσ√

τ

Here τ = T − t and Φ(x) is the cumulative distribution function for a standard normal

random variable, i.e. Φ(x) =∫ x−∞

1√2π

e−y2

2 dy.

6

The price of a European put, p(t, S(t), K, T ), at time t with exercise price K and time tomaturity T is obtained in a similar way and is given by

p(t, S(t), K, T ) = Ke−rτΦ(−d2)− S(t)Φ(−d1) (2.1.6)

where d1 and d2 are as in Theorem 2.1.1. Alternatively, (2.1.6) follows from (2.1.5) andthe so called put-call parity for European calls and puts stating that

p(t, S(t), K, T ) = Ke−rτ − S(t)− c(t, S(t), K, T ). (2.1.7)

Equation (2.1.7) is obvious from a simple arbitrage argument. The formula (2.1.5) will beused frequently throughout the rest of the paper, especially in Section 4 where it will beused to calibrate a discrete model to market prices on European puts and calls. Below wesometimes write

c(t, S(t), K, T ) = c(t, S(t), K, T, r, σ) (2.1.8)

to emphasize all the parameters that affect the price of the call. We want to remind thereader that (2.1.8) is the theoretical price of a European call with strike price K and exerciseday T . In reality there is also a market price, cM(t, S(t), K, T ), corresponding to the sameoption at every time point t. For a long time, the difference between theoretical prices andmarket prices, the so called ”dollar”-errors were quite negligible. But since the big stockmarket ”crash” in 1987, there has been an increasing and persistent difference betweentheoretical prices and market prices (see [RM] ). The maybe most striking consequence ofthe ”dollar”-error is the so called implied volatility surface, also sometimes referred to asthe smile which we will discuss in the next section.

2.2 The implied volatility

In order to determine the price of a European call or put in the Black-Scholes model wesee from (2.1.5) and (2.1.6) that we need to know t, T , S(t), r and σ. Of course, the onlyparameter among these that can not be observed directly on the market is the volatilityσ. Thus, σ has to be estimated in some way and one usual way to do so is to look at thehistorical stock price data and use the fact that σ2 is mathematically defined as

σ2 =Var(ln( S(t)

S(s)))

t− s.

where t > s. Another estimation of σ in the Black-Scholes model is obtained as follows.Assume that we want to price a European call with strike price K1 and exercise day T1 andthat we know all parameters except for σ. Furthermore, suppose that we know the marketprice cM(t, S(t), K2, T2) of a European call, with the same underlier and with strike priceK2 and exercise day T2. It is then natural to determine σ so that

cM(t, S(t), K2, T2) = c(t, S(t), K2, T2, r, σ). (2.2.1)

7

In view of the above we know realize that equation (2.2.1) is actually a way of testing if theassumptions in the Black-Scholes model are realistic. Suppose we know t, S(t), T , K, rand the market price cM(t, S(t), K, T ) of a European call with strike price K and exerciseday T and let us define a function c(σ) as,

c(σ) = c(t, S(t), K, T ). (2.2.2)

Since S(t) > 0, we conclude that

dc(σ)dσ

= S(t)1√2π

e−d212√

T − t > 0

where d1 is as in Proposition 2.1.1. Hence, c(σ) is an increasing function, in the strict sensewhich implies that the equation

cM(t, S(t), K, T ) = c(σ) (2.2.3)

has at the most one solution, σ. There is no guarantee that there exists a solution to(2.2.3) for arbitrary values of cM(t, S(t), K, T ) but in practice (i.e. on the market) this willalways be the case and we define

σimp = c−1(cM(t, S(t), K, T )). (2.2.4)

Here the quantity σimp is called implied volatility, to distinguish it from the constantvolatility assumed in the Black-Scholes model. Of course, by analogy with the abovediscussion, there also exists a unique implied volatility for every price of a European putpM(t, S(t), K, T ). Thus every market price of a European put or call has an eqvivalent andunique implied volatility. The uniqueness depends on the put-call parity, which holds ona market without arbitrages. Below, for short, we concentrate on calls when we discussimplied volatility.

Now, for a fixed underlier at a fixed time point t0 and consequently at a fixed spot priceS(t0) the market price cM(t0, S(t0), K, T ) of a European call will be completely specifiedby the strike price K and exercise day T . Therefore, at every time point t = t0, the marketwill provide us with a function, cM(t0, K, T ), corresponding to each traded call optionspecified by K and T , i.e.

cM(t0, K, T ) = cM(t0, S(t0), K, T ).

On the left-hand side, we have dropped the S(t0) term since this is implicitly given by themarket (just as we dropped r and especially σ). The quantity cM(t0, K, T ) is only knownfor finitely many (K, T ) but by an appropriate interpolation we get a value of cM(t0, K, T )for all K and T . Hence, we get a mapping in view of (2.2.4),

σimp|t=t0 : R+ ×R+ 7→ R+ (2.2.5)

8

where σimp(t0, K, T ) is given by

σimp(t0, K, T ) = c−1(cM(t0, K, T )). (2.2.6)

This mapping is sometimes referred to as the implied volatility function, or implied volatil-ity surface at t0 since σimp(t0, K, T ) generates a surface when we plot σimp(t0, K, T ) forK > 0, T > 0. If the Black-Scholes model gives a good description of a real stock price, itis natural to believe that the market would act as if the function σimp(t0, K, T ), K,T > 0is almost constant. Unfortunately, in reality, this is not the case. Not only will σimp benonconstant, it will also have a recognizable pattern when we calculate σimp(t,K, T ) frommarket prices on calls at different time points t = t0, t1, . . .. Thus, at every time pointt = t0 every put-call market for some fixed underlier will have an implied volatility struc-ture generated by the function σimp(t0, K, T ). This structure has mainly two characteristicfeatures: the volatility skew and the term structure of volatility. We will now givea brief description ot these two phenomena.

The volatility skew:

If we fix T and t = t0 and investigate σimp(t0, K, T ) = σimp,T (K)|t=t0 as a function ofthe strike price K, we will often get a characteristic pattern shown in Figure 1.

250 300 350 4000.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

strike price

impl

ied

vola

tility

Figure 1: Implied volatilities of S&P 500 index options (January 2, 1990; 10:00 A.M.)maturing in 164 days. The index level is 349.16.

Note from Figure 1 that out-of-the money puts are traded at a higher implied volatilitythen out-of-the money calls. This phenomena is referred to as the volatility skew. Thevolatility skew was first clearly observed during the stock market crash in 1987 and has

9

ever since then become a characteristic feature on most option markets around the world.There is still no satisfactory theoretical overall explanation for the skew. An attempt toexplain the skew is by using the concept of leverage (see [HC], Chapter 17). Anotherexplanation that fits well to the circumstances that prevailed during the crash 1987, isRubinstein’s ”crashophobia” theory (see [RM]). In short, Rubinstein argues that since thetraders realized that a big crash was forthcoming, out-of-the money puts were valued muchhigher compared to out-of-the money calls. Consequently, the implied volatility for lowstriking options were significantly higher than for high striking options. However, thereis still no good theoretical explanation why this ”crashophobia” has become a persistentfeature on most equity markets and then, according to Rubinstein, will constantly indicatea forthcoming crash.

The term structure of volatility:

If, on the other hand, we fix the strike price K and t = t0, and study σimp(t0, K, T ) =σimp,K(T )|t=t0 as a function of time to expiration T , we will receive a pattern that we referto as the term structure of volatility. The term structure for an option is also non-constantbut in this case it is harder to observe a common pattern for different options.

Since we are interested in the total, combined effect of the term structure and theskew, we will from now on refer to them collectively as the smile at t = t0. Thus, fora given put-call market at time t0, with the same underlier, the smile is received whenwe plot the surface that is generated by the implied volatility function, σimp(t0, K, T ) forK > 0 and T > 0. Note that in literature, traditionally it is often only the skew that isreferred to as the ”smile”. This springs from the fact that previously the skew was muchmore studied than the term structure. Furthermore, the skew often did have (and still have)a pattern similar to a grin.

10

3 The discrete time model

In this section we will give a brief introduction to the basic theory of option pricing indiscrete time. It is by no means a full description of the subject and for a more extensiveintroduction we suggest [BC1],[BC2] (lecture notes in swedish) and [BT]. We will firstdiscuss the binomial model which is the simplest model of a financial market, and at thesame time a model often used by practitioners.

3.1 The standard binomial model

Below we will give a quite detailed description of the binomial model mainly for two reasons.Firstly, this model, which sometimes is called the standard binomial model gives us afast, simple and intuitive introduction to the complex theory of option pricing. Secondly,we will later present a generalized binomial model. The study of a generalized binomialmodel calibrated with some market data is one of the main purposes of this paper.

Let us assume we have a financial market with only two assets, a bond and a stock.The variable t will denote time and we consider the case where t only can assume one ofthe values 0, 1, 2, . . . , T − 1, T . For short, below we set N b

a = a, a + 1, . . . , b − 1, b, a <b, a, b ∈ N. The price processes (S(t))T

t=0 and (B(t))Tt=0 for the bond and the stock are

given by the equationsS(t + 1) = S(t)eXt+1 (3.1.7)

andB(t + 1) = B(t)er (3.1.8)

respectively, where t ∈ NT−10 and B(0) > 0, S(0) > 0. Here r denotes the interest rate

which we assume is a positive constant and Xt is a stochastic variable which only canassume the values u or d, where u, d ∈ R and u > d for every t ∈ NT

1 . Furthermore, thedistribution of Xt is independent of the choice of t and is given by

P [ Xt = u ] = pu

andP [ Xt = d ] = pd

where 0 < pu < 1 and pd = 1 − pu. We also assume that the stochastic variablesX1, X2, . . . , XT are stochastically independent, i.e.

P [ X1 = i1, X2 = i2, ...., XT = iT ] = P [X1 = i1]P [X2 = i2] · . . . · P [XT = iT ]= pi1pi2 · . . . · piT

where ik ∈ u, d for every k ∈ NT1 . These assumptions imply that the sequence (Xi)T

i=1 isa so called i.i.d. (independent and identically distributed). Because of the recursive natureof S(t) defined by (3.1.7) we can rewrite S(t) as

S(t) = S(0)eX1+X2+...+Xt . (3.1.9)

11

From this we see that S(t) is a deterministic function of the family of stochastic variables(Xi)t

i=1. Another important feature that follows from (3.1.9) is the following fact. At eachtime point t ∈ NT

0 the possible values of S(t) are

Skt = S(0)eku+(t−k)d k ∈ N t

0. (3.1.10)

Furthermore, for fix k and t such that k ∈ N t0 there are

(tk

)

ways in which the stock pricemay reach the value Sk

t at time t starting at 0, since there are(t

k

)

solutions of the equationX1 +X2 + . . .+Xt = ku+(t−k)d. The standard pictorial representation of the stochasticprocess (S(t))T

t=0 given by (3.1.9) is by means of a so called recombining binomial tree(often just called a binomial tree). A binomial tree with 4 time steps (i.e. T = 4) isillustrated in Illustration 1.

0 1 2 3 4

Illustration 1

Definition 3.1.1 A combination of hS(t) stocks and hB(t) bonds at time t is called a port-folio (at time t), and is represented by the vector h(t) = (hS(t), hB(t)), where hS(t), hB(t) ∈R.

We interpret h(t) = (hS(t), hB(t)) as the portfolio that we buy at time t− 1 and then keepuntil time t. The economic interpretation of the inequalities hB(t) > 0, hB(t) < 0 andhS(t) > 0 are obvious (in financial economics a positive hS(t) is often called a long positionin the stock). If hS(t) < 0, this means that you sell hS(t) stocks at time t− 1 and have ashort position in the stock.

12

We are now ready to introduce concepts such as porfolio strategy, value process andself-financing strategies.

Definition 3.1.2 A sequence of portfolios h = (h(t))Tt=0 = (hS(t), hB(t))T

t=0 such thath(0) = h(1) and

hS(t) = hS(X1, X2, . . . , Xt−1)

hB(t) = hB(X1, X2, . . . , Xt−1)

are deterministic functions of X1, . . . , Xt−1 for every t ∈ NT0 , is said to be a portfolio

strategy or just a strategy. Furthermore, the stochastic process Vh = (Vh(t))Tt=0 where

Vh(t) = hS(t)S(t) + hB(t)B(t), t ∈ NT0

is said to be the value process corresponding to the strategy h = (h(t))Tt=0.

We will sometimes writeVh(t) = Vh(t; X1, X2, . . . , Xt) (3.1.11)

to emphasize that Vh(t) is a deterministic function of the sequence (Xi)ti=1. From the

previous discussion it is obvious that Vh(t) is the value of the portfolio h(t) (that wasconstructed/created at time t− 1) at time t.

Definition 3.1.3 Let h = (h(t))Tt=0 be a strategy. If

hS(t)S(t) + hB(t)B(t) = hS(t + 1)S(t) + hB(t + 1)B(t)

or, eqvivalently,Vh(t) = hS(t + 1)S(t) + hB(t + 1)B(t)

for every t ∈ NT−11 , then h is said to be a self-financing portfolio strategy, or just

self-financing.

The natural interpretation of a self-financing strategy h, is that once we have set up h att = 0 there is no additionally cost to maintain managing the portfolio at t = 1, 2, . . . , T −1.

Definition 3.1.4 A self-financing portfolio h with the following properties

Vh(0) = 0, Vh(T ) ≥ 0 and E[Vh(T )] > 0

or, stated eqvivalently,

Vh(0) = 0, Vh(T ) ≥ 0 and Vh(T ) 6= 0

is said to be an arbitrage possibility, or just an arbitrage.

It is natural to ask under which conditions on the parameters the binomial model is freefrom arbitrage. The answer to this question is given by the follwing proposition that westate without any proof.

13

Theorem 3.1.5 The binomial model is free from arbitrage if and only if

r ∈]u, d[ . (3.1.12)

From now on, unless explicit stated, we will always assume that our model is free fromarbitrage.

A financial derivate or contingent claim is a security paper defined in terms of otherassets. Next we want to extend our primitive financial market by adding a family offinancial derivates. The underlying asset of these derivates is the stock.

Definition 3.1.6 A financial derivate or contingent claim Y with time of maturity T, isa stochastic variable such that

Y = Ψ(S(1), S(2), . . . , S(T ))

or, eqvivalently,Y = Ψ(X1, X2, . . . , XT ).

where Ψ, Ψ : RT 7→ R are deterministic functions.

If Y only depends on S(T ), i.e. Y = Ψ(S(T )), then Y is called a simple claim. The time ofmaturity T is sometimes called exercise date or exercise day. The financial derivate definedin Definition 3.1.6 is often called a European claim to emphasize that it is only possible toexercise at the time T . Contingent claims are sometimes referred to as options. Note thatfor a European call option, Y = max(S(T )−K, 0).

The main problem in option pricing theory is to define the price process (ΠY (t))Tt=0 for

the claim Y . Clearly, we define ΠY (T ) = Y . The pricing problem is discussed in the restof this section. We start by considering the problem in the one-time step case i.e. T = 1.The case T > 1 is then treated by induction.

Theorem 3.1.7 Let t ∈ 0, 1 and let Y = Ψ(S(1)) be a contingent claim. Then thereexists a unique porfolio strategy h, such that

Vh(1) = Y.

Proof. Consider a portfolio strategy h = (h(t))1t=0 = (hS(t), hB(t))1

t=0 with h(0) = h(1) =(hS, hB). Assume that Vh(1) = hSS(1)+hBB(1) = Ψ(S(1)). If the stock rises (i.e. X1 = u)we have

hSS(0)eu + hBB(1) = Ψ(S(0)eu)

and if the stock falls we get

hSS(0)ed + hBB(1) = Ψ(S(0)ed).

From these two equations we see that we can determine hS and hB and get

hS =Ψ(S(0)eu)−Ψ(S(0)ed)

S(0)eu − S(0)ed (3.1.13)

14

and

hB = e−r euΨ(S(0)ed)− edΨ(S(0)eu)B(0)eu −B(0)ed . (3.1.14)

So by forming the portfolio h(0) = (hS, hB) at t = 0 , where hS and hB are given by(3.1.13) and (3.1.14), then Vh(1) = Ψ(S(1)), which proves Theorem 3.1.7.

If Y and h are as in Theorem 3.1.7 we now define ΠY (0) = Vh(0). Using (3.1.13) and(3.1.14), and the fact that Vh(1; X1) = Ψ(S(1)),

Vh(0) = e−r[quVh(1; u) + qdVh(1; d)] (3.1.15)

where

qu =er − ed

eu − ed (3.1.16)

andqd =

eu − er

eu − ed . (3.1.17)

Since r ∈]u, d[, qu, qd ∈]0, 1[. Furthermore, we see that qu + qd = 1. Hence, introducing anew probability measure Q such that Q[X1 = u] = qu and Q[X1 = d] = qd,

Vh(0) = e−rEQ[Vh(1; X1)]. (3.1.18)

Now consider the general claim Y = Ψ(S(1), S(2), . . . , S(T )). In the next step we willdefine the price ΠY (t) for the claim Y at time t ≤ T . If t = T it is natural to defineΠY (T ) = Y . Now, let us assume that we are standing at the time point t and that thecurrent spot price of the stock is S(t). Furthermore, assume that we have already definedΠY (s) for s = T, T − 1, . . . , t+1, as a deterministic function of the sequence (Xi)s

i=0. Now,write ΠY (t+1) = Πj

Y (t+1) if Xt+1 = j, j = u, d. From the previous discussion in Theorem3.1.7 we know that it is possible to set up a unique portfolio h(t+1) = (hS(t+1), hB(t+1))at time t such that

hS(t + 1)S(t + 1) + hB(t + 1)B(t + 1) = ΠY (t + 1) (3.1.19)

where

hS(t + 1) =Πu

Y (t + 1)− ΠdY (t + 1)

S(t)eu − S(t)ed (3.1.20)

and

hB(t + 1) = e−r euΠdY (t + 1)− edΠd

Y (t + 1)B(t)eu −B(t)ed . (3.1.21)

Furthermore, we note that the corresponding value of the portfolio h(t + 1) = (hS(t +1), hB(t + 1)) at time t is given by hS(t + 1)S(t) + hB(t + 1)B(t). We now define

ΠY (t) = hS(t + 1)S(t) + hB(t + 1)B(t). (3.1.22)

15

Note that defining ΠY (t) recursively as above, where hS(t + 1) and hB(t + 1)) are as in(3.1.20) and (3.1.21) and h(0) = (hS(0), hB(0)) = h(1), then (ΠY (t))T

t=0 is nothing morethan the value process corresponding to the self-financing portfolio strategy, (h(t))T

t=0.Hence,

ΠY (t) = Vh(t) t ∈ NT0 .

The cost of setting up this portfolio at t = 0 is ΠY (0) but after this it is possible to maintainmanage this portfolio at zero cost, so that Vh(t) = ΠY (t) for t ∈ NT

1 . We will from nowon often write Vh(t) instead of ΠY (t) and V i

h(t) = ΠiY (t), i = u, d. Then from (3.1.15) we

conclude thatVh(t) = e−r[quV u

h (t + 1) + qdV dh (t + 1)]. (3.1.23)

Now, by using (3.1.23) inductively for t = T − 1, . . . , 1, 0 we obtain

Vh(0) = e−rT∑

it∈u,dt=1,2,...,T

qi1 · . . . · qiT Vh(T ; i1, . . . , iT ). (3.1.24)

We writeVh(0) = e−rT EQ[Vh(T ; X1, . . . , XT )] (3.1.25)

which is consistent with our earlier notation. This defines the so called martingale measureQ in T time steps. The preceding discussion is summarized in the following theorem.

Theorem 3.1.8 Let Y = Ψ(S(1), S(2), . . . , S(T )) be a contingent claim with time of ma-turity T . Then there exists a unique self-financing porfolio strategy h, such that

Vh(T ) = Y.

The existence of a self-financing portfolio strategy that replicates a claim at t = T is thecornerstone in the binomial model. This is the reason why banks (in the hypotheticalbinomial model) sell options without taking any risk. To see this, consider the followingimportant example. A customer wants to buy a derivate from a bank at t = 0. Thederivate will payout Y = Ψ(S(1), . . . , S(T )) at t = T . What should the bank charge thecustomer for the claim at t = 0 ? And more importantly, is it possible for the bank tohedge the risk that is involved in selling the claim ? The answer to both of these questions(in the binomial world) is of course given by the above discussion. So the price that thecustomer has to pay for the claim Y at t = 0 equals ΠY (0). Furthermore, with this amountit is possible for the bank to set up a self-financing portfolio strategy that will replicatethe price process of the claim for every possible value of S(t) at every time point t ∈ NT

1 .In other words, the bank completly hedge themselves against the risk involved in writingthe option.

The binomial model is just a first step towards a more complex theory of option pricing.Later in Section 4.2, we will present an extended binomial model where Xt depends ont − 1 as well as on S(t − 1) for every t ∈ NT

1 . This model will be called a generalizedbinomial model in order to distinguish it from the standard binomial model.

16

3.2 Martingales and complete market models

We now want to discuss some general results of great interest in option theory. We assumethat the reader is familiar with some concepts of advanced probability theory. For a moredetailed approach, we refer to the litterature, for example [WD], [BC1], [BC2], [SH], [EK]or [KF].

We will work in a so called finite market model, that is Ω in the underlying proba-bility space (Ω, P,F) is finite, F = 2Ω, P [ ω ] > 0, ∀ω ∈ Ω, and the time set equals0, 1, . . . , T where T is a positive integer. Here P is called the market measure. Themodel contains a stock and a bond and we will not impose any particular restriction onthe probabilistic nature of the stochastic price process (S(t))T

t=0. The price process of thebond is deterministic and given by the relationship

B(t) = B(0)ert (3.2.1)

at every time point t ∈ NT0 , where r is a strict positive constant. Concepts like portofolio,

self-financing portfolio strategy, arbitrage etc. extend without any change to this moregeneral situation.

Let Y be an arbitrary stochastic variable defined on the measure space (Ω,F). ThenEP [ Y ] denotes the expected value of Y with respect to the measure P . For short, we oftenwrite E[ Y ] = EP [ Y ]. Now, let Fk be the σ-algbra generated by (S(t))k

t=0 for k ∈ NT0 , i.e.

Fk = σ(S(0), S(1), . . . , S(k)). (3.2.2)

Here Fk is called the information generated by the stochastic variables S(0), . . . , S(k)known at time k. Note that Fk ⊆ F for every k ∈ NT

0 . Let X be an arbitrary stochasticvariable defined on the triple (Ω, P,F) and let G be an arbitrary sub-σ-algebra of F , i.e.G ⊆ F . The conditional expectation of X given G is denoted by

E[ X | G ]. (3.2.3)

It is natural to think of E[ X | G ] as the expected value of Y given the ”information” G. IfX and Y are arbitrary stochastic variables on (Ω, P,F) it is simple to prove that

E[ αX | G ] = αE[ X | G ] , ∀α ∈ R, (3.2.4)E[ X + Y | G ] = E[ X | G ] + E[ Y | G ]. (3.2.5)

Furthermore, assume that G0, G1 are arbitrary sub-σ-algebras of the σ-algebra F such thatG0 ⊆ G1. Then the so called Tower Property of Conditional Expectation says that

E[ E[ X | G1 ] | G0 ] = E[ X | G0 ]. (3.2.6)

Let X be an arbitrary stochastic variable defined on the triple (Ω, P,F) and let G = σ(X),where σ(X) is the σ-algebra generated by the stochastic variable X. Then

E[ X | σ(X) ] = X. (3.2.7)

17

We are now ready to introduce so called martingales. Let (Mk)Nk=1 be a sequence of

stochastic variables defined on the probability space (Ω, P,F). Furthermore, let (Gk)Nk=1

be a sequence of σ-algebras where Gk ⊆ F for k ∈ NN1 . The sequence (Mk,Gk)N

k=1 is calleda martingale if the following conditions hold for every k, 1 ≤ k ≤ N , viz.

1. Gk ⊆ Gk+1

2. σ(Mk) ⊆ Gk

3. E[ |Mk| ] < ∞

4. Mk = E[ Mk+m | Gk], m ≥ 1, k + m ≤ N .

The sequence (Gk)Nk=1 is sometimes referred to as a filtration with respect to the σ-algebra

F . In this paper we will only consider filtrations of the type (Fk)Tk=1 where the Fk are as in

(3.2.2). Thus, for notational convenience, we will from now on call the sequence (Mk)Nk=1

a martingal if (Mk,Fk)Nk=1 is a martingale. In general, a probability measure defined on F

that makes the process (Mk)Nk=1 into a martingal is called a martingal measure.

Theorem 3.2.1 Let (Vh(t))Tt=0 be the value process for an arbitrary self-financing portfolio

strategy, h, in our market model. Assume that there exists a probability measure Q, definedon the measurable space (Ω,F) such that the discounted stochastic process (e−rtS(t))T

t=0 isa Q-martingale, i.e.

e−rtS(t) = EQ[ e−r(t+m)S(t + m) | Ft], m ∈ NT−t0 , t ∈ NT

0 . (3.2.8)

Then (e−rtVh(t))Tt=0 is a Q-martingale.

Proof. First note that

EQ[ e−rVh(t + 1) | Ft ] = EQ[ e−r (hS(t + 1)S(t + 1) + hB(t + 1)B(t + 1)) | Ft ]= EQ[ e−rhS(t + 1)S(t + 1) + hB(t + 1)B(t) | Ft ]= hS(t + 1)EQ[ e−rS(t + 1) | Ft ] + hB(t + 1)B(t)= hS(t + 1)S(t) + hB(t + 1)B(t)= Vh(t). (3.2.9)

Now, let k = t + m and note that Ft ⊆ Fk−n when n = 1, 2, . . . , m. We then get thefollowing relations

18

EQ[ e−r(t+m)Vh(t + m) | Ft ] = EQ[ e−rkVh(k) | Ft ]= e−rkEQ[ Vh(k) | Ft ]= e−rkEQ[ EQ[ Vh(k) | Fk−1 ] | Ft ]= e−r(k−1)EQ[ e−rEQ[ Vh(k) | Fk−1 ] | Ft ]= e−r(k−1)EQ[ Vh(k − 1) | Ft ]= e−r(k−2)EQ[ e−rEQ[ Vh(k − 1) | Fk−2 ] | Ft ]= e−r(k−2)EQ[ Vh(k − 2) | Ft ]= . . . . . . . . .

. . . . . . . . .= e−r(k−m)EQ[ Vh(k −m) | Ft ]= e−rtEQ[ Vh(t) | Ft ]= e−rtVh(t) (3.2.10)

and we are done.

Note that if t + m = T , then (3.2.10) implies the following relationship

Vh(t) = EQ[ e−r(T−t)Vh(T ) | Ft ], for every t ∈ NT0 . (3.2.11)

In other words this expression states that the value of the self-financing portfolio strategyh at time t is equal to the discounted expected value of h at time T with respect to themeasure Q, given that we know the information generated by the sequence (S(k))t

k=0.

Definition 3.2.2 Equivalent measures. Let P and Q be probability measures definedon the measurable space (Ω,F). We say that P and Q are equivalent measures if they havethe same null-sets, i.e.

P [A] = 0 ⇔ Q[A] = 0, A ∈ F .

A martingale measure Q equivalent to the market measure P that makes the process(e−rtS(t))T

t=0 to a Q-martingale is often called a risk-neutral measure.

Theorem 3.2.3 No arbitrage condition. Assume that there exists a risk-neutral mea-sure Q. Then our market model is free of arbitrage.

Proof. Let (Vh(t))Tt=0 be the value process for an arbitrary self-financing portfolio strategy

such that Vh(0) = 0 and Vh(T ) ≥ 0. Theorem 3.2.1 then implies that (e−rtVh(t))Tt=0 will be

a Q-martingal. Hence,e−rT EQ[ Vh(T ) | F0 ] = Vh(0) = 0.

19

This implies that

Vh(T ) = 0 a.s. [Q] ⇒ Vh(T ) = 0 a.s. [P ] ⇒ Vh(T ) = 0

since P [ ω ] > 0, ∀ω ∈ Ω. Hence, there are no arbitrages in our market model, which isexactly what we want to show.

In Theorem 3.2.3 we assume that there exists a risk-neutral measure Q. It turns outthat if there are no arbitrage possibilities in our model then it actually exists such a risk-neutral measure (see e.g. [EK]).

Theorem 3.2.4 Existence of risk-neutral measure. Assume that there are no arbi-trage possibilities in our market model described as above. Then there exists a risk-neutralmeasure.

Corollary 3.2.5 A market model is free of arbitrage if and only if there exists a risk-neutral measure.

Note that Theorem 3.2.4 only guarantees the existence of a risk-neutral measure. We willsoon state a condition which will quarantee that the risk-neutral measure is unique in anmodel free of arbitrage.

Definition 3.2.6 Reachable claim. Let Y = Ψ(S(1), S(2), . . . , S(T )) be a contingentclaim. If there exists a self-financing porfolio strategy h, such that

Vh(T ) = Y,

then Y is said to be reachable or attainable. Furthermore, the portfolio is then said tobe a hedging portfolio or a replicating portfolio for Y .

Theorem 3.2.7 Suppose that there are no arbitrage possibilities in our market model andlet Q be a risk-neutral measure. Let Y be a reachable contingent claim and let h be ahedging portfolio for Y . Then

Vh(t) = e−r(T−t)EQ[ Y | Ft ]

for every t ∈ NT0 .

Proof. Since h is a hedging portfolio, h necessarily must be self-financing. Hence, Theorem3.2.1 implies that (e−rt(Vh(t))T

t=0 is a Q-martingale. We then get

Vh(t) = e−r(T−t)EQ[ Vh(T ) | Ft ]= e−r(T−t)EQ[ Y | Ft ]

for every t ∈ NT0 . This is exactly what we want to show.

20

Corollary 3.2.8 Suppose there are no arbitrages in the model and let Q and Q′ be risk-neutral measures on the measurable space (Ω,F). Let Y be a reachable claim. Then

EQ[ Y | Ft ] = EQ′ [ Y | Ft ]

for every t ∈ NT0 .

Definition 3.2.9 Arbitrage free price of reachable claim. Assume that there areno arbitrage opportunities in our market-model. Let Y be a reachable contingent claim withexercise day T . Then the arbitrage free price ΠY (t) of Y at time t is defined as

ΠY (t) , e−r(T−t)EQ[ Y | Ft ]. (3.2.12)

for every t ∈ NT0 where Q is a risk-neutral measure.

Definition 3.2.10 Complete market model. If all contingent claims in our marketmodel are reachable we say that the market model is complete.

We now state the condition that makes a market model complete

Theorem 3.2.11 Condition for completness in an arbitrage free market model.Suppose the market model is free of arbitrage. Then it is complete if and only if the risk-neutral measure is unique.

Proof. Suppose the market is complete and let Q, Q′ be equivalent martingale measures.Now if A ∈ FT , then Corollary 3.2.8 implies that

Q[ A ] = EQ[ 1A | F0 ] = EQ′ [ 1A | F0 ] = Q′[ A ].

Hence, Q = Q′. For the proof of the other direction we refer to [EK].

We know that if there exists a risk-neutral measure Q in our market, then e−rt(S(t))Tt=0

will be a Q-martingale, that is

e−rtS(t) = EQ[ e−rt′S(t′) | Ft ], t ≤ t′, t, t′ ∈ NT0 .

A rewriting of this expression gives

er(t′−t)S(t) = EQ[ S(t′) | Ft ], t ≤ t′, t, t′ ∈ NT0 (3.2.13)

A certain forward contract Yf on the stock S obligates the holder to buy the stock S atthe exercise day t′ at the price K that is determined at the inception day t, t ≤ t′. Hence,Yf = S(t′) −K. The forward price F (t, t′; S) of the stock is the value of K that makesthe claim Yf worthless at the inception day t i.e. ΠYf (t) = 0. Thus, if we assume that ourmarket is free of arbitrage and then use Theorem 3.2.7,

EQ[ S(t′)−K | Ft ] = 0

21

orK = EQ[ S(t′) | Ft ].

Hence, by definition,F (t, t′; S) = EQ[ S(t′) | Ft]. (3.2.14)

and (3.2.13) yieldsF (t, t′; S) = er(t′−t)S(t). (3.2.15)

We thus have the following important

Theorem 3.2.12 Risk-neutral formula for forward price. Assume that there existsa risk-neutral mesure Q in our market model. Let F (t, t′; S) be the forward price of thestock S contracted at t with delivery date t′. Then

F (t, t′; S) = EQ[ S(t′) | Ft ]. (3.2.16)

22

4 Implied volatility trees

4.1 Introduction

There are many continuous time models in finance that will account for the volatilitysmile. Most of these models will be incomplete or contain to many parameters that arehard to estimate. In this section we will introduce a so called implied volatility tree modelfor option pricing. This model was developed by Emanual Derman and Iraj Kani at theQuantitative Strategies Group of Goldman Sachs & Co (see [DK1] ). In short, the impliedvolatility tree uses market data on standard European puts and calls to price arbitraryoptions consistently with the smile. Moreover, the model is complete. But before wepresent this model we have to introduce the so called generalized binomial model.

4.2 A generalized binomial model

The standard binomial model that we studied in Chapter 3.1 is achieved by approximatingthe normalized Wienerprocess with a symmetric random walk with a 2-point distributedincrements. Thus if we let h = T/n and ti = ih, i = 1, . . . , n then

S(ti) = S(ti−1)eXi

for i ∈ Nn1 , where

P [Xi = αh + σ√

h] = P [Xi = αh− σ√

h] =12

and where (Xi)ni=1 is an i.i.d. Obviously, the binomial model has the same weakness as

the Black-Scholes model (at least for big n). There is however a very natural and simpleextension of the standard binomial model (i.e. the discrete Black-Scholes model), that willstill be complete. Here S(ti) is given by

S(ti) = S(0)eX1+...+Xi

for i ∈ Nn1 , where each Xi, conditioned on X1, . . . , Xi−1, is a 2-point distributed stochastic

variable. Let us now give a formal definition of this generalized binomial model.

Definition 4.2.1 Generalized binomial model. Let t = t0 < t1 < . . . < tn = T be anarbitrary discretization of the interval [t, T ] into n different time steps. Let S(ti) denotethe price of the stock S at ti, i ∈ Nn

0 . Then

S(ti) = S(ti−1)eXi (4.2.1)

or, stated eqvivalently,S(ti) = S(t0)eX1+...+Xi (4.2.2)

for every i ∈ Nn1 . Here (Xk)i

k=1 is a sequence of stochastic variables such that

23

1. S(t0) is a known real-valued strictly positive scalar and S(t0) = S0,0.

2. Conditioned on X1, . . . , Xi−1, Xi is a real-valued 2-point distributed stochastic variablefor every i ∈ Nn

0 .

3. The stochastic variable X1 + . . . + Xi only takes on i + 1 different values for everyi ∈ Nn

1 .

4. In view of 3, let (Si,j)ij=0 denote the possible outcomes for S(ti) at ti for every i ∈ Nn

1 .Then

0 < Si,0 < Si,1 < . . . < Si,i−1 < Si,i (4.2.3)

andSi+1,j < Si,j < Si+1,j+1 (4.2.4)

for every j ∈ N i0 and every i ∈ Nn

1 .

5. If S(ti−1) = Si−1,j then the possible values that S(ti) assumes are Si,j or Si,j+1 forevery j ∈ N i

0 and every i ∈ Nn1 .

Note that the generalized binomial model can be represented as a recombining binomialtree which follows from Properties 4 and 5 in Definition 4.2.1. Thus, Illustration 1 at page12 may still represent a generalized binomial model as in Definition 4.2.1. In this casehowever, we will refer to it as a flexible recombining binomial tree or, for short, aflexible tree.

Theorem 4.2.2 The generalized binomial model is free of arbitrage if and only if

Fi−1,j − Si,j

Si,j+1 − Si,j∈ ]0, 1[ (4.2.5)

for every i ∈ Nn1 and every j ∈ N i

0. Here Fi−1,j denotes the forward price of the stockS contracted at ti−1 with delivery date ti, given that S(ti−1) = Si−1,j, that is Fi−1,j =Si−1,je(ti−ti−1)r.

Theorem 4.2.2 can be proved exactly as for the standard binomial model. In this casehowever,

Xi(ω) ∈ u(S(ti−1), ti−1), d(S(ti−1), ti−1). (4.2.6)

where u(S(ti−1), ti−1) > d(S(ti−1), ti−1). By simple means, one proves that the generalizedbinomial model is comlpete (cf. Theorem 3.1.7).

Note that Theorem 4.2.2 provides us with a constraint on the sequence (Si,j)ij=0, i ∈ Nn

1that will make the generalized binomial model free of arbitrage. Before we continue wewant to emphasize some properties of (S(ti))n

i=0 that strongly relates to Theorem 4.2.2 andat the same time we introduce some notation that will be frequently used in the rest ofthe paper.

24

Assume that the generalized binomial model is free of arbitrage. Theorem 3.2.4 willthen imply that there exists a risk-neutral measure Q. It is simple to show that thesequence (S(ti))n

i=0 is a Markov-process w.r.t. Q, i.e.

Q[ S(ti) ∈ A | S(t0), . . . , S(ti−1) ] = Q[ S(ti) ∈ A | S(ti−1) ]

for every Borel set A and i = 1, . . . , n. Furthermore, according to Theorem 3.2.12

F (tk, t′; S) = EQ[ S(t′) | Ftk ] (4.2.7)

where F (tk, t′; S) denotes the forward price of the stock S contracted at tk with deliverydate t′ and Ftk = σ(S(t0), S(t1), . . . , S(tk)). Now, let Fi−1,j denote the forward price of Scontracted at ti−1 with delivery date ti, given that S(ti−1) = Si−1,j, i.e

Fi−1,j = F (ti−1, ti; S)|S(ti−1)=Si−1,j (4.2.8)

Since (S(ti))ni=0 is a Markov process

EQ[ S(ti) | Fti−1 ] = EQ[ S(ti) | S(ti−1), S(ti−2), . . . , S(t0) ]= EQ[ S(ti) | S(ti−1) ]. (4.2.9)

Thus, in view of (4.2.7), (4.2.8) and (4.2.9)

Fi−1,j = EQ[ S(ti) | S(ti−1) = Si−1,j ]. (4.2.10)

In view of property 5 in Definition 4.2.1

EQ[ S(ti) | S(ti−1) = Si−1,j ] = Q[ S(ti) = Si,j+1 | S(ti−1) = Si−1,j ]Si,j+1

+ Q[ S(ti) = Si,j | S(ti−1) = Si−1,j ]Si,j

= qi−1,jSi,j+1 + (1− qi−1,j)Si,j. (4.2.11)

In the last equality in (4.2.11) we have used the notation

qi−1,j = Q[ S(ti) = Si,j+1 | S(ti−1) = Si−1,j ] (4.2.12)

where qi−1,j denotes the the risk-neutral transition probability between the two states Si−1,j

at ti−1 and Si,j+1 at ti. Hence,

Fi−1,j = qi−1,jSi,j+1 + (1− qi−1,j)Si,j (4.2.13)

and we obtainqi−1,j =

Fi−1,j − Si,j

Si,j+1 − Si,j. (4.2.14)

Finally, Theorem 4.2.2 impliesqi−1,j ∈ ]0, 1[ (4.2.15)

25

for i ∈ Nn1 , j ∈ N i

0. Note that (4.2.7)-(4.2.14) is an alternative proof of the ”only if” partin Theorem 4.2.2.

In view of 5 in Definition 4.2.1, or equivalently equation (4.2.6), the risk-neutral transi-tion probability qi−1,j defined as in (4.2.14) for every j ∈ N i−1

0 is a function of S(ti−1) andti−1. In the standard binomial model however, we know that Si,j will in view of (3.1.10)be given by

Si,j = S0eju+(i−j)d (4.2.16)

where Xi ∈ u, d, u > d and ∆ti = h for every i ∈ Nn1 . Hence, in this case

qi−1,j =erh − ed

eu − ed = q (4.2.17)

which obviously is a constant expression independent of ti−1 and S(ti−1). Note that (4.2.17)is consistent with the expression for the constant risk-neutral transition probability q thatwe derived for the standard case in Section 3.1.

If X is a 2-point distributed stochastic variable that can take on the two real values aand b with probability pa and pb = 1− pa, respectively, then

Var(X) = pa(1− pa)(a− b)2. (4.2.18)

Now assume that S(ti−1) = Si−1,j. Then

VarQ(Xi|S(ti−1) = Si−1,j) = qi−1,j(1− qi−1,j)(

lnSi,j+1

Si−1,j− ln

Si,j

Si−1,j

)2

(4.2.19)

or, stated equivalently,

VarQ(Xi|S(ti−1) = Si−1,j) = qi−1,j(1− qi−1,j)(

lnSi,j+1

Si,j

)2

. (4.2.20)

This leads us to the following

Definition 4.2.3 Local volatility. Let S(ti)ni=0 be a stochastic price process for a stock

S in a generalized binomial model. Then the local volatility σ(S(ti), ti) of the stock S atthe point (S(ti), ti) equals

σ(S(ti), ti) ,

√

VarQ(Xi+1|S(ti−1))∆ti+1

. (4.2.21)

If S(ti) = Si,j, then

σ(Si,j, ti) =1√

∆ti+1

√

qi,j(1− qi,j) lnSi+1,j+1

Si+1,j(4.2.22)


1 .

26

4.3 Using a generalized binomial model for option pricing

Theorem 4.2.2 states the conditions that the sequence (Si,j)ij=0 for i ∈ Nn

1 must fulfillin order to quarantee that the generalized binomial model will be free of arbitrage. Ifwe assume that these conditions are fullfilled then there exists a risk-neutral probabilitymeasure Q. In view of Definition 3.2.9 it is then possible to price any European contingentclaim Y in our model. Hence, the price ΠY (ti) at time ti of a European contingent claimY with exercise date tn , ∈ Nn

1 equals

ΠY (ti) = e−r(tn−ti)EQ[ Y | Fti ] (4.3.1)

where as usual Fti is the sigma-algebra generated by S(t0), . . . , S(ti), i.e.

Fti = σ(S(t0), . . . , S(ti)) = σ(X1, . . . , Xi).

Note that since we always know S(t0), σ(S(t0), . . . , S(ti)) then reduces to σ(S(t1), . . . , S(ti))i.e. σ(S(t0), . . . , S(ti)) = σ(S(t1), . . . , S(ti)). Since the generalized binomial model is com-plete there exists a self-financing portfolio h such that Vh(tn) = Y with probability one.Furthermore, the discounted value process for this strategy will be a Q-martingale. Hence,

Vh(ti−1) = e−r∆tiEQ[ Vh(ti) | Fti−1 ] (4.3.2)

orV j

h (ti−1) = e−r∆ti [qi−1,jVj+1h (ti) + (1− qi−1,j)V

jh (ti)] (4.3.3)

where ∆ti = ti− ti−1 and V jh (ti−1) = Vh(ti−1)|S(ti−1)=Si−1,j for every j ∈ N i−1

1 , i ∈ Nn1 . Note

that (4.3.3) is an obvious generalization of the expression (3.1.23) stated in the standardbinomial model. Thus, if we want to find the price ΠY (t0) of Y at t0 it is possible to usethe relation (4.3.3) recursively to compute ΠY (t0). In the next chapter we will show howto compute the price of a simple European claim by using so called Arrow-Debreu prices.

If we let (Si,ki)ni=0 be a realization of (S(ti))n

i=0 and let xki = ln Si,kiSi−1,ki−1

for every i ∈ Nni=1

and finally set ti = t0 then by equation (4.3.1),

ΠY (t0) = e−r(T−t0)∑

(Si,ki )ni=0

q0,k1 · . . . · qn−1,knVh(T ; xk1 , . . . , xkn) (4.3.4)

where the summation is taken over all possible realizations (Si,ki)ni=0, of the stochastic

process (S(ti)ni=0. This sum will contain 2n terms since there are 2n different realizations

of (S(ti)ni=0. For big n, 2n is an enormous number and it would be impossible to compute

ΠY (t0) in a reasonable time period. However, using the Monte-Carlo method one onlycomputes the value

∑

A q0,k1 · . . . ·qn−1,knVh(T ; xk1 , . . . , xkn) for a limited number of differentrealisations A of (S(ti)n

i=0 to obtain a sufficiently accurate value of ΠY (t0) (see [HC], or[WP] for a more detailed discussion about this subject).

So far we have only considered the arbitrage free price of a European contingent claim.However, to determine the price of a simple American option in the generalized binomialmodel is as simple as in the standard binomial model. Hence, we will not pursue this pointat all here.

27

4.4 Arrow-Debreu prices

For any j ∈ N i0, i ∈ Nn

1 let

πi,j = Q[ S(ti) = Si,j | S(t0) = S0,0 ] (4.4.1)

where Q denotes the risk-neutral measure in our generalized binomial model. Furthermore,let us consider an arbitrary simple European claim Y , with time of maturity tn = T . Then,if i ∈ Nn

1 ,ΠY (ti) = v(ti, S(ti))

is a deterministic function of ti and S(ti). Set

vi,j = v(ti, Si,j) (4.4.2)

so thatv(ti, S(ti)) ∈ vi,j : j ∈ N i

0. (4.4.3)

Note thatv(tk, S(tk)) = e−r(ti−tk)EQ[v(ti, S(ti)) | Ftk ] (4.4.4)

where tk ≤ ti, since(

ΠY (tk)e−r(tk−t0))n

k=0 is a martingale with respect to Q. In particular,

v(t0, S0,0) = e−r(ti−t0)EQ[v(ti, S(ti)) | Ft0 ]|S(t0)=S0,0 . (4.4.5)

Now, assume that we know (vi,j)ij=0 and (πi,j)i

j=0 at an arbitrary fixed time point ti, i ∈ Nn1 .

Then

EQ[v(ti, S(ti)) | Ft0 ]|S(t0)=S0,0 = EQ[v(ti, S(ti)) |S(t0) ]|S(t0)=S0,0

= EQ[v(ti, S(ti)) |S(t0) = S0,0 ]

=i

∑

j=0

Q[ S(ti) = Si,j |S(t0) = S0,0 ]v(ti, Si,j)

=i

∑

j=0

πi,jvi,j. (4.4.6)

Thus

EQ[v(ti, S(ti)) | Ft0 ]|S(t0)=S0,0 =i

∑

j=0

πi,jvi,j (4.4.7)

and we get

v0,0 = e−r(ti−t0)i

∑

j=0

πi,jvi,j

or

v0,0 =i

∑

j=0

λi,jvi,j (4.4.8)

28

where λi,j is given by,λi,j = e−r(ti−t0)πi,j. (4.4.9)

Note that λ0,0 = 1. Here λi,j is called the Arrow-Debreu price, or state-contingent pricecorresponding to the node (i, j), or state S(ti) = Si,j for the stock S at time ti. Note thatthe quantity λi,j equals the value of a derivative security at time t0, that pays the amount1 at ti if S(ti) = Si,j and 0 otherwise. The Arrow-Debreu price λi,j given by (4.4.9) willplay an important part when we will present the implied volatility tree model in Section4.5. Therefore, we now want to show how to compute (λi,j)i

j=0 by using so called forwardinduction. To this end, suppose (λi−1,j)i−1

j=0 and (qi−1,j)i−1j=0 are known, where,

qi−1,j = Q[ S(ti) = Si,j+1 |S(ti−1) = Si−1,j]

and1− qi−1,j = Q[ S(ti) = Si,j |S(ti−1) = Si−1,j].

Now, to compute λi,j for some j ∈ N i−11 we use the definition of πi,j stated in equation

(4.4.1) and have

πi,j = Q[ S(ti) = Si,j | S(ti−1) = Si−1,j−1 ]πi−1,j−1

+Q[ S(ti) = Si,j | S(ti−1) = Si−1,j ]πi−1,j

= qi−1,j−1πi−1,j−1 + (1− qi−1,j)πi−1,j. (4.4.10)

Hence,πi,j = πi−1,j−1qi−1,j−1 + πi−1,j(1− qi−1,j). (4.4.11)

Now, using (4.4.9) we get

e−r(ti−1−t0)πi,j = λi−1,j−1qi−1,j−1 + λi−1,j(1− qi−1,j). (4.4.12)

andλi,j = e−r(ti−ti−1)(λi−1,j−1qi−1,j−1 + λi−1,j(1− qi−1,j)). (4.4.13)

Furthermore, in view of this derivation, it is also obvious that λi,i and λi,0 are given by,

λi,i = e−r(ti−ti−1)λi−1,i−1qi−1,i−1

andλi,0 = e−r(ti−ti−1)λi−1,0(1− qi−1,0).

Let ∆ti denote ∆ti = ti − ti−1 for i ∈ Nn1 . Summing up, given (λi−1,j)i−1

j=0 and (qi−1,j)i−1j=0,

λi,j =

e−r∆tiλi−1,i−1qi−1,i−1, j = i,e−r∆ti(λi−1,j−1qi−1,j−1 + λi−1,j(1− qi−1,j)), 1 ≤ j ≤ i− 1,e−r∆tiλi−1,0(1− qi−1,0), j = 0.

(4.4.14)

Of course, it must hold that,λ0,0 = 1. (4.4.15)

29

So, by using (4.4.14) recursively with (4.4.15) as a start value, we see that we can compute(λi,j)i

j=0 for i = 1, 2, . . . , n once we know (qi,j)ij=0 for i ∈ Nn

1 . Note that in a standardbinomial model where ∆ti = h and qi,j = q are all real valued constants for j ∈ N i

0 andi ∈ Nn

0 we then get

λi,j = e−r(ti−t0)(

ij

)

qj(1− q)i−j

for j ∈ N i0 and i ∈ Nn

0 .

4.5 The implied volatility tree model

We are now ready to give a formal formal definition of the implied volatility tree. LetMS be the set of all traded European puts and calls on the same underlying stock S. Letχ(K,T ) denote a traded option (European put or call) with strike price K and exercisedate T . Then

MS = χ(K,T ); (K, T ) listed on the market. (4.5.1)

Furthermore, we also assume that we know the market price vM(t0, K, T ) at time t = t0or the equivalent implied volatility σimp(t0, K, T ), corresponding to each option χ(K, T ) ∈MS. At last, let vTree(t0, K, T ) denote the theoretic price of the option χ(K,T ) ∈ MS att = t0, when χ(K, T ) has been valued on our (yet undefined) implied volatility tree thatspans from t = t0 to tn = T .

Definition 4.5.1 Implied volatility tree. An implied volatility tree at t = t0 is ageneralized binomial model such that

vTree(t0, K, T ) = vM(t0, K, T ) ∀ χ(K,T ) ∈MS (4.5.2)

Thus, our model will generate theoretic prices on European puts and calls which coincidewith their corresponding market prices at t = t0. In other words, the model will becalibrated with the market prices on European puts and calls. In most otheroption pricing models (for example the Black-Scholes model) one makes some axiomaticassumptions about the market and the stochastic process corresponding to the stock andthen make a derivation leading to some pricing formula for puts and calls etc. It is thenrare that the theoretical prices will coincide with the corresponding market prices (thesmile, or implied volatility function is a perfect example of this kind of scenario). In viewof our remark at page 24, the implied volatility tree is a flexible recombining binomial tree.Note that since our model is an generalized binomial model, vTree(t0, K, T ) is computedusing (4.4.8), i.e

vTree(t0, K, T ) =k

∑

m=0

λk,m max (ε (Sk,m −K) , 0) (4.5.3)

where tk = T for some k ∈ Nn1 . Here ε = 1 if χ(K,T ) is a call and ε = −1 if χ(K,T )

is a put. Furthermore, as above, λk,m, m = 0, . . . , k, denote the Arrow-Debreu prices atT = tk.

30

4.6 Building an implied volatility tree

In this section we will describe how to construct an implied volatility tree. Before wepresent the main algorithm we want to say some words about forward prices. First of allwe want to remaind the reader about the relationship

F (t, t′; S) = EQ[ S(t′) | Ft ] = S(t)er(t′−t) (4.6.1)

where F (t, t′; S) denotes the forward price of S contracted at t with delivery date t′, t′ > t.From this relation,


where Fi−1,j denotes the forward price of S contracted at ti−1 with delivery date ti, giventhat S(ti−1) = Si−1,j, i.e

Fi−1,j = F (ti−1, ti; S)|Si−1=Si−1,j

andqi−1,j = Q[ S(ti) = Si,j+1 |S(ti−1) = Si−1,j ].

Note that in view of (4.6.2)

qi−1,j =Fi−1,j − Si,j

Si,j+1 − Si,j. (4.6.3)

This relationship will play a crucial part when we construct our implied volatility tree.We are now ready to show the main algorithm for constructing an implied volatility

tree (see Illustration 2 below).We are going to build our tree by using forward induction. Assume that we have

computed the sets (Si−1,j)i−1j=0 and (λi−1,j)i−1

j=0. Suppose furthermore that we know onevalue, say Si,m of S(ti) at ti and that we also know σimp(t0, K, T ) for all K, T , K, T > 0(cf. p. 8). Then we will derive a recursive method that will provide us with the values(Si,j)i

j=0, i.e. the possible outcomes for S(ti) at time ti. So, let us take a look at Illustration2. Assume that we know the value of Si,j for a fixed j. We now want to compute Si,j+1

using the information at ti−1. Let cM(t0, Si−1,j, ti) be a fictitious price of a European callχc(Si−1,j, ti) at t = t0 with strike price Si−1,j and exercise date ti defined by the equation

cM(t0, Si−1,j, ti) = c(t0, S(t0), Si−1,j, ti, r, σimp(t0, Si−1,j, ti)).

This call is probably not traded on the market i.e. χc(Si−1,j, ti) /∈ MS (if χc(Si−1,j, ti) ∈MS, then cM(t0, Si−1,j, ti) will no longer be a fictitious price but actually a market price).

31

Si−1,j+1

Si−1,j

Si−1,j−1

Si,j+2

Si,j+1

Si,j

Si,j−1

ti−1 ti

Illustration 2

Now, let us return to Illustration 2. Let cji−1,k denote the theoretic price of the European

call χc(Si−1,j, ti) at ti−1 given that S(ti−1) = Si−1,k. From equation (4.3.3) in Section 4.2,

cji−1,j = e−r∆ti [qi−1,j max(Si,j+1 − Si−1,j, 0) + (1− qi−1,j) max(Si,j − Si−1,j, 0)].

But since Si,j < Si−1,j < Si,j+1 (see Illustration 2) we then get

cji−1,j = e−r∆tiqi−1,j(Si,j+1 − Si−1,j). (4.6.4)

Now, using (4.6.3)

Si,j+1 =cji−1,jSi,j + e−r∆ti(Si,j − Fi−1,j)Si−1,j

cji−1,j + e−r∆ti(Si,j − Fi−1,j)

. (4.6.5)

According to this, Si,j+1 is determined by the information known at ti−1 if Si,j and cji−1,j

are known. In fact, below we will tell how cji−1,j can be determined from the information

at time ti−1.Let us return to Illustration 2. Suppose now that we know Si,j+1 and want to compute

Si,j. Let pM(t0, Si−1,j, ti) be the fictitious price of a European put χp(Si−1,j, ti) at t0 withstrike price Si−1,j and exercise date ti. It is possible to compute pM(t0, Si−1,j, ti) in thesame way as we did for cM(t0, Si−1,j, ti) using (2.1.6) instead of (2.1.5) . Furthermore, let

32

pji−1,k denote the price of χp(Si−1,j, ti) at ti−1 given that S(ti−1) = Si−1,k. To determine

pji−1,j we again use (4.3.3) in Section 4.2 and get

pji−1,j = e−r∆t[qi−1,j max(Si−1,j − Si,j+1, 0) + (1− qi−1,j) max(Si−1,j − Si,j, 0)].

Now, since Si,j < Si−1,j < Si,j+1, this expression reduces to (see Illustration 2)

pji−1,j = e−r∆ti(1− qi−1,j)(Si−1,j − Si,j) (4.6.6)

and equations (4.6.3) and (4.6.6) then give

Si,j =pj

i−1,jSi,j+1 + e−ri∆t(Fi−1,j − Si,j+1)Si−1,j

pji−1,j + e−r∆ti(Fi−1,j − Si,j+1)

. (4.6.7)

Thus it is possible to compute Si,j given that we know Si,j+1 and pji−1,j.

It now remains to show how to compute cji−1,j and pj

i−1,j based on the informationknown at ti−1. We start with cj

i−1,j. Remember that cji−1,k is the price of the European call

χc(Si−1,j, ti) at ti−1 given that S(ti−1) = Si−1,k. Hence, looking at Illustration 2 we realizethat

cji−1,k = 0, if k < j

since when k < j, then Si,k < Si−1,j and we obtain

cji−1,k = e−r∆ti [qi−1,k max(Si,k+1 − Si−1,j, 0) + (1− qi−1,k) max(Si,k − Si−1,j, 0)] = 0.

If k > j, then Si,k > Si−1,j for k > j and we get

cji−1,k = e−r∆ti [qi−1,k(Si,k+1 − Si−1,j) + (1− qi−1,k)(Si,k − Si−1,j)] (4.6.8)

or, in view of (4.6.2),cji−1,k = e−r∆ti(Fi−1,k − Si−1,j).

Finally, the case k = j was treated in equation (4.6.4). Thus, in view of the abovediscussion,

cji−1,k =

e−r∆ti(Fi−1,k − Si−1,j), j < k ≤ i− 1,e−r∆tiqi−1,j(Si,j+1 − Si−1,j), k = j,0, 0 ≤ k < j.

(4.6.9)

It is now time to use our sequence of Arrow-Debreu prices (λi−1,j)i−1j=0 that we have computed

at ti−1. Since χc(Si−1,j, ti) is a simple European claim we then in view of Definition 4.5.1and equation (4.4.8) get the following relationship

cM(t0, Si−1,j, ti) =i−1∑

k=0

λi−1,kcji−1,k.

33

But (4.6.9) then gives us

cM(t0, Si−1,j, ti) =i−1∑

k=j

λi−1,kcji−1,k

= λi−1,jcji−1,j +

i−1∑

k=j+1

λi−1,kcji−1,k

= λi−1,jcji−1,j + e−r∆ti

i−1∑

k=j+1

λi−1,k(Fi−1,k − Si−1,j). (4.6.10)

Hence, letting Σci−1,j denote

Σci−1,j = e−r∆ti

i−1∑

k=j+1

λi−1,k(Fi−1,k − Si−1,j) (4.6.11)

we obtain

cji−1,j =

cM(t0, Si−1,j, ti)− Σci−1,j

λi−1,j. (4.6.12)

The put price pjj−1,k is computed in a similar way as the call price cj

j−1,k. We have that

pji−1,k = 0, if k > j

since Si,k > Si−1,j when k > j (see Illustration 2). The case k = j have already been givenin equation (4.6.6) and when k < j we have

pjj−1,k = e−r∆ti [qi−1,k(Si−1,j − Si,k+1) + (1− qi−1,k)(Si−1,j − Si,k)]

orpj

j−1,k = e−r∆ti(Si−1,j − Fi−1,k).

where we again used equation (4.6.2). Hence, we get

pji−1,k =

0, j < k ≤ i− 1,e−r∆ti(1− qi−1,j)(Si−1,j − Si,j), k = j,e−r∆ti(Si−1,j − Fi−1,k). 0 ≤ k < j.

(4.6.13)

Now, recalling that

pM(t0, Si−1,j, ti) =i−1∑

k=0

λi−1,kpji−1,k

and using (4.6.13) we get

34

pM(t0, Si−1,j, ti) =j

∑

k=0

λi−1,kpji−1,k

= λi−1,jpji−1,j +

j−1∑

k=0

λi−1,kpji−1,k

= λi−1,jpji−1,j + e−r∆ti

j−1∑

k=0

λi−1,k(Si−1,j − Fi−1,k). (4.6.14)

Thus introducing

Σpi−1,j = e−r∆ti

j−1∑

k=0

λi−1,k(Si−1,j − Fi−1,k) (4.6.15)

we have

pji−1,j =

pM(t0, Si−1,j, ti)− Σpi−1,j

λi−1,j. (4.6.16)

Now suppose we know (Si−1,j)i−1j=0 and (λi−1,j)i−1

j=0 at ti−1 and one value of S(ti) at ti, saySi,m, m ∈ N i

0. Then since cmi−1,m is known we compute Si,m+1 using (4.6.5). By repetition,

we obtain Si,m+1, . . . , Si,i. In the same way since pmi−1,m is known we use Si,m in equation

(4.6.7) to compute Si,m−1. By repetition, we obtain Si,m−1, . . . , Si,0

Finally, it remains to show how to choose Si,m at ti−1 in equation (4.6.5) and (4.6.7).Let us go back to Illustration 1 at page 12. If i is an even number, then there will be anodd number of nodes at ti in our tree. We can therefore center our tree at the spot priceS0,0. In other words, we let m = i

2 and thus get

Si,m = Si,i/2 = S0,0.

If on the other hand i is a odd number then we will have an even number of nodes atlevel i in our tree and obviously we can no longer center our tree at the spot price. Butremember that

S(ti) = S(ti−1)eXi

where Xi(ω) ∈ u(S(ti−1), ti−1), d(S(ti−1), ti−1). Thus if S(ti−1) = Si−1,j

Si,j+1 = Si−1,jeu(Si−1,j ,ti−1) (4.6.17)

andSi,j = Si−1,jed(Si−1,j ,ti−1). (4.6.18)

For notational convenience we will from now write i2 = bi/2c, the integer part of i/2. Wenow impose the condition

u(Si−1,j, ti−1) = −d(Si−1,j, ti−1)

forcingSi,i2+1Si,i2 = S2

i−1,i2 . (4.6.19)

35

The equations (4.6.19) and (4.6.5) give a second order polynomial equation for Si,i2+1. Thetwo solutions of this equation are Si−1,i2 and

Si,i2+1 =Si−1,i2(e

r∆ticM(t0, Si−1,i2 , ti) + λi−1,i2Si−1,i2 − er∆tiΣci−1,i2)

λi−1,i2Fi−1,i2 − er∆ticM(t0, Si−1,i2 , ti) + er∆tiΣci−1,i2

(4.6.20)

(se Appendix A.1 for details) where Σci−1,i2 is given by

Σci−1,i2 = e−r∆ti

i−1∑

k=i2+1

λi−1,k(Fi−1,k − Si−1,i2).

We choose (4.6.20) to be our initial value. Knowing, Si,i2+1 we then can resolve Si,i2 using(4.6.19). Hence, if i is a odd number we then use Si,i2+1 as a initial value in (4.6.7) tocompute the upper part of the tree at ti, i.e. Si,k, k = i2 + 1, . . . , i. And, naturally,we use Si,i2 as a initial value to compute the lower part of the tree at ti, i.e. Si,k, k =i2 − 1, . . . , 0. We have now presented the main core in our algorithm for building animplied volatility tree. Unfortunatley, we have to deal with some additional problems thatmay arise during the computations described above. For example, it may happen that atransition probability qi,j 6∈ ]0, 1[.

4.7 Bad transition probabilities

Before we continue with our algorithm at the next time level ti+1, calculating the set(Si+1,j)i+1

j=0 we of course have to know the sequence (λi,j)ij=0 of Arrow-Debreu prices at ti.

We already know, from equation (4.4.14) in Section 4.4 how to compute (λi,j)ij=0 given

the sets (λi−1,j)i−1j=0 and (qi−1,j)i−1

j=0. Furthermore, we also know how to calculate (qi−1,j)i−1j=0

using the following relationship

qi−1,j =Fi−1,j − Si,j

Si,j+1 − Si,j(4.7.1)

for j ∈ N i−10 . So, at first glance it seems to be a straithforward procedure to compute

(λi,j)ij=0. But unfortunatley, it may happen that

qi−1,j ≥ 1 (4.7.2)

orqi−1,j ≤ 1. (4.7.3)

There are several methods to adjust such bad transition probabilites. However, this is avery special topic and we do not go into details here (for more details, see for example[CN], [DK1] or [HA2]). In this context, let us just mention that our implementation seemsto lead to bad probabilities only for nodes (i, j) in the upper or lower part in the tree i.e.for nodes corresponding to stock prices quite far from the spot price S(t0). Fortunately,if we at every time level ti where bad probabilities occour, sum up the probabilities πi,j

for the set of nodes that has been adjusted Bi = j1, j2, . . . , jb, we find that this sum∑

j∈Biπi,j in practice is always very small, typically around 0.01. The misspricings due to

bad probabilities will in such cases be quite negligible.

36

4.8 Some numerical results

In this section we will present numerical results corresponding to some implied volatilitytrees. Assume for simplicity that a given stock S possesses an implied volatility surfaceindependent of time of maturity and such that

σimp(K, t) = −0.0012K + 0.62. (4.8.1)

Here the implied volatility drops around 1% when the strike price drops 10%. We nowbuild an implied volatility tree that spans one year from today, i.e. t0 = 0 and T = 365days. The total number of time steps n is n = 66 and we let

∆ti =36566

for i = 1, 2, . . . , 66. The spot price of S at t0 is given by S(t0) = 86.0. We assume thatthere are no dividends during the period [t0, T ] corresponding to the stock S and that theinterest rate r is given by r = 0.05. All the figures in this chapter springs from this tree.Figures 2, 3, and 4 show the local volatility σ(S(t), t) for this tree from three differentangles. As usual, σ(S(t), t) is computed using the relation (4.2.22). We have ”pruned” thetree at S = 270 in Figure 2, 3, and 4.

0

100

200

300

400

050

100150

200250

300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time (days)

stock price

loca

l vol

atili

ty

Figure 2: The local volatility surface

37

0

100

200

300

4000

50100

150200

250300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

stock pricetime (days)

loca

l vol

atili

ty

Figure 3: The local volatility surface from another angle

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

time (days)

stoc

k pr

ice

Figure 4: The local volatility surface from above

Figures 5, 6, and 7 show the local volatility and the implied volatility at three different

38

0 50 100 150 200 250 300 3500.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

stock price for local volatility, strike price for implied volatility

local vol implied vol time =205.3125 days from start

Figure 5: Local volatility and implied volatility t = 205 days from start

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9




time points, t = 205, t = 250 and t = 302. A very interesting feature is observed in thesefigures. Let us in each figure approximate the local volatility by a line in the stock price(or equvivalently, strike price) interval [100, 150]. If we calculate the slope of this line,

39

0 50 100 150 200 250 300 350 400 4500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8




kloc and then compare it with the slope of the implied volatility, kimp by computing thequotient kloc

kimpwe get the values 2.0678, 2.0979 and 2.1259 . This result is consistent

with the Rule of Thumb 1 that Goldman & Sachs states at page 13 in the article ”TheLocal Volatility Surface-Unlocking the information in Index Option Prices”(see [DKZ]). This rule states that the local volatility varies about twice as rapid at thecorresponding implied volatility.

40

0

100

200

300

4000 50 100 150 200 250 300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


prob

abili

ty d

ensi

ty

Figure 8: πi,j as a function of Si,j and ti

Once we know the Arrow-Debreu prices λi,j, j = 0, . . . , i, at ti, the risk-neutral proba-bility

πi,j = Q[ S(ti) = Si,j |S(t0) = S0,0 ]

is computed using the relationπi,j = er(ti−t0)λi,j.

Figures 8 and 9 show the function πi,j = Q[ S(ti) = Si,j |S(t0) = S0,0 ] as a funtion of Si,j

and ti. We see that this is a quite smooth function (compared with the corresponding localvolatility surface).

Figures 10, 11 and 12 show πi,j as a function of Si,j at tree different time points t = 205,t = 252 and t = 302.

41

0

100

200

300

400

050

100150

200250

300

0

0.2

0.4

0.6

0.8

1

time (days)

stock price

prob

abili

ty d

ensi

ty


0 50 100 150 200 250 300 3500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

stock price

riskneutral probability distribution time =205.3125 days from start

Figure 10: πi,j as a function of Si,j at ti = 205 (days)

4.9 Valuing barrier options in the presence of the smile

Once we have build our implied volatility tree we can use it to value any exotic contingentclaim secure in the knowledge that our model is calibrated with market prices on European

42

0 50 100 150 200 250 300 350 4000

0.02

0.04

0.06

0.08

0.1

0.12

stock price



0 50 100 150 200 250 300 350 400 4500

0.02

0.04

0.06

0.08

0.1

0.12

stock price



puts and calls. This can be useful if one wants to hedge the exotic claim using static optionreplication with standard puts and calls as underliers (for more details about this see [DEK]and [DB]).

43

0

100

200

300

400

0

50

100

1500

10

20

30

40

50

60

time (days)stock price

barr

ier

optio

n pr

ice

Figure 13: Knock-out call

In this section we will value certain barrier options using the model developed above.These barrier options will be European up-and-out call options. An (European) up-and-out call with strike price K, barrier level B and time of expiry T is a contingent claim, χB

which pays max(S(T ) −K, 0)1[maxt0≤t≤T S(t)<B] at T . Furthermore, if S(t∗) ≥ B for somet∗ ∈ [t0, T ], then the claim will pay some rebate, RB at the hitting time t∗ or at the expiryT .

Let us consider an up-and-out call with barrier B = 146.2, strike price K = 86.0 anda rebate RB = 4.3 which is paid at expiry T , where T = 365 days from the inception dayt0 = 0 (”today”). Now, suppose that the market provides us with an implied volatilityfor the underlying stock S, given by equation (4.8.1). Furthermore, we assume that thereare no dividends during the period [t0, T ], the interest rate is given by r = 0.05, andthe spot price at t0 is S(t0) = 86.0. We now want to value our barrier option under theabove circumstances. To value this up-and-out call we build an 70 step implied volatilitytree, with ∆ti constant for i = 1, 2, . . . , n. This means that the barrier option will bemonitored 70 times during the period [t0, T ], and every monitoring takes place at the time

44

points t0, t1, . . . , tn. We would like to point out that an up-and-out call obviously is a pathdependent option. Nevertheless, as long as the option has not been knocked out, we treatthe claim as if it is a simple European derivate. Furthermore, since we know the value ofthe option when it has been knocked out it is possible to use a modified variant of equation(4.3.3) to find the option price at time t0. However, we have used a more advanced methodfor valuing the up-and-out call, described in the Goldman & Sachs article ”EnhancedNumerical Methods for Options with Barriers” (see [DKEB]).

0

200

400

050100150

0

10

20

30

40

50

60

time (days)

stock price

barr

ier

optio

n pr

ice

Figure 14: Knock-out call

In Figure 13 we plot the price of the up-and-out call as a function of stock price S andtime t. This figure indicates that, for fixed stock price, the price of the option increasesas T − t decreases. The intuitive explanation of this is of course that that the probabilityof S(t∗) > B for some t∗ ∈ [t, T ] decreases as t → T if S(t) is fixed and less than B.Furthermore, in Figure 14 we see how the price drastically falls as S(t) ↑ B from beingdeep in the money to only be worth the discounted rebate when S(t) ≥ B. This featurethat is characteristic for barrier options is problematic from a hedging strategy point ofview. This is due to the fact that the hedging parameter ∆, ∆ = ∂v

∂s will ”blow-up” whenS(t) ↑ B.

In order to compare our result of valuing a barrier option on an implied volatility treewith some example in the litterature we have computed the value of an up-and-out callunder the same circumstances as described in the Goldman & Sachs article ”The LocalVolatility Surface-Unlocking the information in Index Option Prices”, pp 19-20(see [DKZ]). In that case the strike price is given by K = 100, the barrier B = 130, thespot price at t0 is S(t0) = 100, the time to expiry is 365, days and the interest rate is

45

30 40 50 60 70 80 90 1006.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7

tree size

barr

ier

optio

n va

lue

Standard barrier method Enhanced barrier method Value calculated by Goldman & Sachs

Figure 15: The price of a knock-out call as a function of tree size

constant r = 0.05. Futhermore, σimp(K, t) is given by

σimp(K, t) = −0.001K + 0.25. (4.9.1)

Here the volatility drops 1% as the strike price drops 10%. Goldman & Sachs obtain theoption price 6.46. The result from our calculations is displayed in Figure 15. It shows thevalue of the barrier option as function of the number of steps in the tree. Furthermore,we have used two methods to calculate the price of the the up-and-out call, one is thestandard method and the other is the enhanced method (discussed above). The solid lineshows the Goldman & Sachs option price. We should mention that there are no volatilitythat produces this price using the existing explicit formula for European up-and-out calloptions in the Black-Scholes model. In fact, using this formula we will get a maximumvalue of 5.99 when the volatility is about 9.6 %.

Finally we want to demonstrate how sensitive the price of a barrier option is to localvariations in the volatility. Figure 16 shows the price of an up-and-out call as a functionof different implied volatility surfaces σimp(K, t) given by

σimp(K, t) = αK + m (4.9.2)

46

0 2 4 6 8 10 12 14 16 18 204

5

6

7

8

9

10

11

sigma1−sigma

2

barr

ier

optio

n va

lue

Standard barrier method Enhanced barrier method steps in tree=70

Figure 16: The price of a knock-out call as a function of skew, i.e. σ1 − σ2

whereα =

σ1 − σ2

K1 −K2

and where K1 = 86, K2 = 146.2, σ1 = 0.54 and σ2 varies between 0.34 and 0.54 or, statedequivalently, σ1 − σ2 varies between 0 and 0.2. For each such implied volatility surface webuild an 70 step implied volatility tree and then value an up-and-out call on this tree (onceagain we display both the enhanced method and the standard method). The strike-priceis given by K = 86.0, the barrier B = 146.2, the spot price is S = 86.0, the interest rater = 0.05, and time to maturity T = 365. Furthermore, the barrier option pays a rebate ofthe amount 4.3 at T if it is knocked out some time during the period [t0, T ]. Note that theprice of the option is dubbled as σ1 − σ2 goes from 0% to 18% (the case σ1 − σ2 = 0% isequivalent to a constant volatility of m according to equation (4.9.2)).

47

5 Implied binomial trees

In this section we present an implied binomial tree which is yet another generalized binomialmodel. This model was introduced by Mark Rubinstein (see [RM]), at the University ofCalifornia, Berkeley. In short, the purpose of an implied binomial tree is to build a flexiblerecombining binomial tree such that the stock price has a terminal risk-neutral probabilitydistribution equal to an a priori given probability distribution.

5.1 Introduction

The implied binomial tree uses a probability distribution (Qn,j)nj=0, as an input and then

calibrates a generalized binomial model that spans from t = t0 to t = T = tn, so that therisk-neutral probability distribution of the stock price equals (Qn,j)n

j=0 at time tn. Moreprecisely, we have

Definition 5.1.1 Implied binomial tree. Let (Qn,j)nj=0 be a given probability distribu-

tion. An implied binomial tree is a generalized binomial model, (S(ti))ni=0 that spans

from t = t0 to t = T = tn such that the risk-neutral probability distribution

πi,j = Q[ S(ti) = Si,j |S(t0) = S0,0 ], j ∈ N i0, i ∈ Nn

0 (5.1.1)

satisfies (πn,j)nj=0 = (Qn,j)n

j=0 or, stated equivalently,

Qn,j = Q[ S(tn) = Sn,j |S(t0) = S0,0 ], j ∈ Nn0 (5.1.2)

In the next section we will present a way to determine the sequence (Qn,j)nj=0 from market

data on European calls proposed by M. Rubinstein (see [RM]). Moreover, in Section 5.5the Rubinstein method is extended to incorporate subjective opinons regarding the futurerisk-neutral probability distribution corresponding to S.

Building the implied binomial tree will be far more easier than constructing the impliedvolatility tree. Furthermore, the backward algorithm for constructing the tree is extremelysimple and it will quarantee that qi−1,j ∈ ]0, 1[ for every node. Thus, unlike the impliedvolatility tree, we will not have to worry about the unpleasent problem of bad transitionprobabilities. However, unlike the implied volatility tree, this tree will not at all quaranteethat the distributions at ti, t0 ≤ ti < tn, will be consistent with the smile. Nevertheless,due to its simplicity and flexibility regarding how to choose (Qn,j)n

j=0 this tree will be avery intresting complement to the implied volatility tree. To construct an implied binomialtree we proceed in two steps.

First step. Estimate the probability distribution (Qn,j)nj=0.

Second step. Find a generalized binomial model (S(ti))ni=0 and the martingale measure

Q so that (5.1.2) holds.

By abuse of language, from now on we will call (Qn,j)nj=0 a risk-neutral distribution corre-

sponding to the stock price S(tn).

48

5.2 Extracting the risk-neutral probability distribution

The main goal of this section is to describe a method that will provide us with the sets(Sn,j)n

j=0 and (Qn,j)nj=0. In this context Rubinstein discusses two methods, the Longstaff

Amended method and the Shimko method (see [RM]). In addition, Rubinstein presents anQP-optimization method which we now will describe.

Let (Q′n,j)

nj=0 be the distribution of the stock price at t = tn with respect to the risk-

neutral measure of a standard tree. Thus, in view of (3.1.9) and (3.1.10) we let Sn,j begiven by

Sn,j = S0,0eju+(n−j)d (5.2.1)

for every j ∈ Nn0 , since in a standard tree we have

S(tn) = S0,0eX1+...+Xn (5.2.2)

where, Xi ∈ u, d, u > d and u, d are the same for every i ∈ Nn1 . We choose u and d

according tou = σ

√h (5.2.3)

andd = −u. (5.2.4)

Here, σ is choosen as the average of an appropriate class of implied volatilities. Further-more, h is choosen as h =

√∆t where ∆t is the step size in the tree. From Section 3.1 we

know that the risk-neutral transition probability q between Si,j at ti and Si+1,j+1 at ti+1

in a standard binomial tree is given by the constant expression

q =er∆t − ed

eu − ed (5.2.5)

for every i ∈ Nn1 and j ∈ N i

0. Here r and ∆t are constants (as usual r is the risk-freerate). Furthermore, since (Xi)n

i=1 is an i.i.d. with respect to the measure Q′ induced bythe standard tree,

(

nj

)

qj(1− q)n−j = Q′[ S(tn) = Sn,j |S(t0) = S0,0 ]

that is,

Q′n,j =

(

nj

)

qj(1− q)n−j (5.2.6)

for j ∈ Nn0 .

Now, assume that there are m European puts and calls listed on the market with thesame exercise day T , (T = tn) and that has S as an underlier but with different strikeprices Ki, i = 1, 2, . . . , m. Let Cb

i , Cai denote the market bid and ask prices for each such

European call with strike price Ki, i = 1, 2, . . . , n. Finally, let Sb, Sa denote the bid and

49

ask price of the underlying stock S at time t0. Now, Rubinstein proposes (Qn,j)nj=0 to be

a solution of the following QP-minimization problem

minn

∑

j=0

(Qn,j −Q′n,j)

2 (5.2.7)

subject to the following constraints

n∑

j=0

Qn,j = 1 (5.2.8)

Qn,j ≥ 0 j = 1, 2, . . . , n (5.2.9)Cb

i ≤ Ci ≤ Cai i = 1, 2, . . . , m (5.2.10)

Sb ≤ S0 ≤ Sb (5.2.11)

where

Ci = e−r(T−t0)n

∑

j=0

Qn,j max(Sn,j −Ki, 0) i = 1, 2, . . . ,m (5.2.12)

S0 = e−r(T−t0)n

∑

j=0

Qn,jSn,j. (5.2.13)

All the constraints above are obvious and need no comment.

5.3 Building an implied binomial tree

From now on we will take for granted that we are equipped with a risk-neutral probabilitydistribution (Qn,j)n

j=0 with support Sn,0, . . . , Sn,n, where the stock prices Sn,0, . . . , Sn,n

are given by (5.2.1). The main goal in this section is to calibrate a generalized binomialmodel so that S(tn) satisfies the condition (5.1.2) at t = T = tn where Q is the risk-neutralmeasure. According to Definition 5.1.1 we will refer to this generalized binomial model asan implied binomial tree.

It is not hard to realize that there are infinite many solutions, (S(ti))ni=0 to this problem.

So in order to find one of these solutions we have to make an additional assumption that willprovide us with a uniqe implied binomial tree. We therefore make the following assumptionin our flexible recombining binomial tree.

Let (sk)ik=1 and (sk)i

k=1 be two different realisations of the process (S(tk))ik=1 that satisfy

si = si.

ThenQ[ (S(tk))i

k=1 = (sk)ik=1 ] = Q[ (S(tk))i

k=1 = (sk)ik=1 ].

50

Furthermore, if si = Si,j set

qri,j = Q[ (S(tk))i

k=1 = (sk)ik=1 ]. (5.3.1)

Now, since the stochastic process (S(ti))ni=0 represents a recombining binomial tree there

will be(i

j

)

different realisations of (S(tk))ik=1 such that

S(ti) = Si,j. (5.3.2)

Hence, in view of (5.3.1),(

ij

)

qri,j = Q[ S(ti) = Si,j for some realization (S(tk))i

k=1 ] (5.3.3)

or, stated equivalently,

πi,j =(

ij

)

qri,j (5.3.4)


1 since

πi,j = Q[ S(ti) = Si,j |S(t0) = S0,0 ]= Q[ S(ti) = Si,j for some realization (S(tk))i

k=1 ]. (5.3.5)

Graphically, this means that the risk-neutral probability to reach the node (i, j) at timeti is the same for every path or every realisation that leads from node (0, 0) to node (i, j)in the tree. Thus, qr

i,j is a path-probability which explains why we use the subscript r (ras in realisation) to distinguish it from qi,j which is the risk-neutral transition probabilitybetween the node (i, j) and (i + 1, j + 1), i.e.

qi,j = Q[ S(ti+1) = Si+1,j+1 |S(ti) = Si,j ]. (5.3.6)

In view of the smile, (5.3.1) is probably a bit unrealistic. Nevertheless, it will make thealgorithm for building our tree extremely simple.

We now present a backwards recursive algorithm that will provide us with an impliedbinomial tree. Assume that we know the sequences (Si,j)i

j=0 and (qri,j)

ij=0 at ti. We now

want to compute (Si−1,j)i−1j=0 and (qr

i−1,j)i−1j=0 at ti−1. This will be done in three steps, as

follows (see Illustration 3).

51

qri,j+1

qri,j

qri−1,j

qi−1,j

ti−1 ti

Illustration 3

Step 1:qri−1,j = qr

i,j+1 + qri,j (5.3.7)

Step 2:

qi−1,j =qri,j+1

qri−1,j

(5.3.8)

Step 3:Si−1,j = e−r∆ti [qi−1,jSi,j+1 + (1− qi−1,j)Si,j] (5.3.9)

Now, let us explain these steps. We start with (5.3.8) in step 2. Intuitively, in view of(5.3.1) this step says that the path-probability of reaching node (i, j + 1) must be equal tothe path probability of reaching node (i− 1, j) and then make an up move between node(i− 1, j) and(i, j + 1) (of course all probabilities are with respect to Q). But not to leaveany doubts let us show this. Let (sk)i

k=1 be a realisation of (S(tk))ik=1 such that

si = Si,j+1 (5.3.10)si−1 = Si−1,j. (5.3.11)

Then, in view of (5.3.1),

qri,j+1 = Q[ (S(tk))i

k=1 = (sk)ik=1 ]

= Q[ (S(tk))i−1k=1 = (sk)i−1

k=1 , S(ti) = si ]= Q[ (S(tk))i−1

k=1 = (sk)i−1k=1 ]

· Q[S(ti) = si | (S(tk))i−1k=1 = (sk)i−1

k=1]= qr

i−1,jQ[S(ti) = si |S(ti−1) = si−1]= qr

i−1,jQ[S(ti) = Si,j+1 |S(ti−1) = Si−1,j]= qr

i−1,jqi−1,j (5.3.12)

which is exactly what we wanted to show. Here, in the fourth equality, we have used theMarkov property of the process (S(tk))i

k=1 with respect to Q and (5.3.1). From (5.3.8) weimmediately get

qri,j+1 = qr

i−1,jqi−1,j. (5.3.13)

52

In a similar way,qri,j = qr

i−1,j(1− qi−1,j) (5.3.14)

and summing (5.3.13) and (5.3.14) we get

qri−1,j = qr

i,j+1 + qri,j

which is (5.3.7). Finally, equation (5.3.9) is just a rewritten version of the familiar rela-tionship


where, Fi−1,j denotes the forward price of S contracted at ti−1 with delivery date ti, giventhat S(ti−1) = Si−1,j.

Remember that we already know (Sn,j)nj=0 from (5.2.1) and, moreover,

qrn,j =

Qn,j(n

j

)

for j ∈ Nn0 . By iterating Steps 1-3, we get a generalized binomial model for the stock.

Once we know (qri,j)

ij=0 at ti, for every i ∈ Nn−1

1 we can compute (πi,j)ij=0 since by

(5.3.1),

πi,j =(

ij

)

qri,j. (5.3.16)

At a first glance it is far from obvious that the essential relationship

i∑

j=0

πi,j = 1 (5.3.17)

will hold for every i ∈ Nn−11 when we have computed our implied binomial tree using the

algorithm described above. Obviously (5.3.17) will hold for i = n due to the constraint(5.2.8) and the fact that (πn,j)n

j=0 = (Qn,j)nj=0. So let us prove (5.3.17) for all i. Assume

that (5.3.17) will hold for (πi,j)ij=0. We then want to show that (5.3.17) also will hold for

(πi−1,j)i−1j=0. In view of (5.3.1) have that

πi−1,j =(

i− 1j

)

qri−1,j

=(

i− 1j

)

(

qri,j+1 + qr

i,j

)

=(

i− 1j

)

(

πi,j+1( i

j+1

) +πi,j(i

j

)

)

=j + 1

iπi,j+1 +

i− ji

πi,j . (5.3.18)

53

The second equlality uses (5.3.7), the third equality uses (5.3.4) again and in the finalequality we use the definition of

(nk

)

= n!k!(n−k)! to simplify the expression just above. Let

us now compute∑i−1

j=0 πi−1,j using (5.3.18).

i−1∑

j=0

πi−1,j =i−1∑

j=0

(

j + 1i

πi,j+1 +i− j

iπi,j

)

=1i

(

i−1∑

j=0

(j + 1)πi,j+1 +i−1∑

j=0

(i− j)πi,j

)

=1i

(

i−1∑

j=0

(j + 1)πi,j+1 −i−1∑

j=0

jπi,j + ii−1∑

j=0

πi,j)

)

=1i

(

iπi,i + ii−1∑

j=0

πi,j)

)

= πi,i +i−1∑

j=0

πi,j =i

∑

j=0

πi,j = 1 (5.3.19)

By induction we conclude that (5.3.19) will hold for all i ∈ Nn−10 .

Note that if qri,j > 0 for all nodes (i, j) in the tree, this will, in view of (5.3.7) guarantee

that the conditionqi,j ∈ ]0, 1[

will always be satisfied.

5.4 Some numerical results

In this section we will present some numerical results concerning an implied binomialtree. In order to compare our results with the litterature we use the same data as in theexample at pp 784, in the Rubinstein article ”Implied binomial trees” [RM]. This datawas sampled January 2, 1990 at 10:00 A.M and consists of S&P 500 index call optionsmaturing in June 1990. Table 1 shows the relevant data. We here, for simplicity, assumethat there are no dividends on the index during this period and that the interest rate isr = 9%. The bid ask spread on the index is given by [349.16, 349.26]. The implied volatilityof the two nearest-the-money options (strikes 350 and 355) equals 17% and the time tomaturity is 164 days.

We use 100 nodes at the expiry day and then solve the QP-minimization problem withthese data using MatLab’s QP-solver ”quadprog”. The result is shown in Figure 17 and issimilar to [RM], although this article used 200 nodes at the expiry day.

We now want to build an implied binomial tree using this solution as the terminaldistribution. Since the solution will be zero for S ≤ 186 and S ≥ 441 we remove thezero-elements in the distribution corresponding to these intervalls and build a smaller treecorresponding to the size of the non-zero elements in the distribution (which is 37). Thestep length in the tree is then 164/37 ≈ 4.43 days.

In Figure 18 we show πi,j as a function of Si,j and ti and in Figure 19 the correspondinglocal volatility surface induced by the implied binomial tree.

54

Table 1: Bid and ask prices ($) on S&P 500 calls for different strikes K with 164 days tomaturity.

K bid ask250 109.47 109.71275 85.66 86.71300 63.00 64.04325 42.00 42.75330 37.97 38.60335 34.01 34.64340 30.10 30.73345 26.45 27.13350 22.79 23.48355 19.32 19.88360 15.98 16.54365 13.13 13.75370 10.52 11.15375 8.37 9.12380 6.65 7.40385 4.91 5.60

100 200 300 400 500 600 7000

0.02

0.04

0.06

0.08

0.1

0.12

stock price

Figure 17: Solution to Rubinsteins QP-problem

55

0

50

100200 250 300 350 400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


prob

abili

ty d

ensi

ty


0

50

100

150

200150 200 250 300 350 400 450

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


loca

l vol

atili

ty

Figure 19: Local volatility deduced from the implied binomial tree

56

5.5 Using investor’s biases in an implied binomial tree

In Section 5.2 we presented Rubinstein’s quadratic optimization method to extract a risk-neutral probability distribution for a stock S at time tn using market data on Europeancalls that mature at tn and that have S as an underlier. Note, that all the constraints(5.2.8)-(5.2.11) in this minimization-problem make perfect sense and we do not have tomotivate them further. On the other hand, there are no special reasons that motivate aquadratic objective function of the type

n∑

j=0

(Qn,j −Q′n,j)

2. (5.5.1)

As Rubinstein proposes (see [RM]), this ”least-square” type of minimization function isjust one of several alternatives for an objective function in our optimization problem. Thusa goodness of fit function of the type

n∑

j=0

(Qn,j −Q′n,j)

2

Q′n,j

(5.5.2)

or the maximal-entropy function,

n∑

j=0

Qn,j lnQn,j

Q′n,j

(5.5.3)

or an absolute difference type of objective function,

n∑

j=0

|Qn,j −Q′n,j| (5.5.4)

are all natural alternatives. Needless to say, the Rubinstein minimization problem forfinding the risk-neutral distribution (Qn,j)n

j=0 has some subjectivity in the choice of theobjective function for the optimization problem.

Now, let us go one step further in this ”subjective” thinking. Remember that thealgorithm for building an implied binomial tree presented in Section 5.3 is completelyindependent of how we determined the risk-neutral distribution (Qn,j)n

j=0 and the corre-sponding stock prices Sn,0, . . . , Sn,n at tn. The algorithm just takes the values (Qn,j)n

j=0and Sn,0, . . . , Sn,n as an input and then produces a flexible recombining binomial tree or,equivalently, a generalized binomial model as an output, where

Qn,j = Q[ S(tn) = Sn,j |S(t0) = S0,0 ]

for j ∈ Nn0 at tn.

The fact that the algorithm is independent of the method for extracting (Qn,j)nj=0 will

turn out to be very useful. Consider for example the following very common scenario.

57

An investor has a strong opinion regarding the risk-neutral probability distribution corre-sponding to a stock S at some future time point T . He/she then wants to find a reasonableprice today for a certain contingent claim YI given that this subjective distribution holdsat T . To this end, it is natural to base the pricing on the implied binomial tree above.

Finally, we will here give another example of how to determine a risk-neutral probabilitydisribution corresponding to beliefs of an investor. We will still use Rubinstein’s QP-minimization problem but we will add some constraints that naturally force the solution(Qn,j)n

j=0 to fulfil some conditions imposed by the investor (for details, see [HA1]). InFigure 20 we show an example of a subjective distribution. This distribution was achievedby adding the constraint

a∑

j=0

Qn,j = 0.08 (5.5.5)

to the previous example that we presented in Section 5.4. Here Sn,a = 300 and this togetherwith the constraint (5.5.5) imply that the risk-neutral probability that the S&P 500 indexwill be lower or equal 300, is equal to 0.08 or 8%. If we solve this modified QP-minimizationproblem we arrive at a solution as in Figure 20.

100 200 300 400 500 600 7000

0.02

0.04

0.06

0.08

0.1

0.12

stock price

Figure 20: A subjective distribution

58

References

[BS] Black, F., Scholes, M. , The pricing of options and corporate liabilities, Journal ofPolitical Economy 81, 637-659, 1973.

[BI] Bilting, U., Skansholm, J., Vagen till C, Studentlitteratur, 1999.

[BT] Bjork, T., Arbitrage Theory in Continuous Time, Oxford University Press, 1998.

[BC1] Borell, Ch., Optioner och matematik, kompendium, Chalmers Tekniska Hogskola.

[BC2] Borell, Ch., Finansiella derivat och stokastisk analys, kompendium, ChalmersTekniska Hogskola.

[CN] Chriss, N.A, Black-Scholes and beyond, McGraw-Hill, 1997.

[CS] Clewlow, L., Strickland, C., Implementing Derivatives models, John Wiley & Sons,2000.

[CW] Copeland, T.E, Weston, J.F, Financial Theory and Corporate Policy, Addison-Wesley, Publishing Company, 1992.

[DK1] Derman, E., Kani, I., The Volatility Smile and Its Implied Tree, Goldman & SachsQuantitative Strategies Research Notes, 1994.

[DEK] Derman, E., Ergener, D., Kani, I., Static Options Replication, Goldman & SachsQuantitative Strategies Research Notes, 1994.

[DKC] Derman, E., Kani, I., Chriss, N., Implied Trinomial Trees of the Volatility Smile,Goldman & Sachs Quantitative Strategies Research Notes, 1996.

[DK2] Derman, E., Kani, I., Stochastic Implied Trees: Arbitrage Pricing With StochasticTerm and Strike Structure of Volatility, Goldman & Sachs Quantitative StrategiesResearch Notes, 1998.

[DKZ] Derman, E., Kani, I., Zou, J.C, The Local Volatility Surface. Unlocking the Infor-mation in Index Option Prices, Goldman & Sachs Quantitative Strategies ResearchNotes, 1996.

[DKEB] Derman, E., Kani, I., Ergener, D., Bardhan, I., Enhanced Numerical Methodsfor Options with Barriers, Goldman & Sachs Quantitative Strategies Research Notes,1995.

[DFW] Dumas, B., Fleming, J., Whaley, R.E, Implied Volatility Functions: EmpiricalTests, The Journal of Finance, December 1998.

[DB] Dupire, B., Pricing With a Smile, Risk, 7, No.1 1994, pp 18-20.

59

[EK] Elliot. R.J, Kopp, P.E, Mathematics of Financial Markets, Springer-Verlag 1999.

[HE] Haug, E.G., The Complete Guide to options pricing formulas, McGraw-Hill, 1997.

[HA1] Herbertsson, A.G., Implicita binomial trad, working paper at Handelsbanken In-vestment Banking, 2001.

[HA2] Herbertsson, A.G., Implicita volatilitets trad, working paper at Handelsbanken In-vestment Banking, 2001.

[HC] Hull, J.C, Options, futures & other derivatives, Prentice-Hall International, Inc.,2000.

[HL] Haggstrom, O., Lindvall, T., Sannolikhetsteorins grunder, kompendium, ChalmersTekniska Hogskola.

[JJ1] Jackwerth, J.C., Generalized Binomial Trees, Journal of Derivatives 5, No. 2, 7-17,1997.

[JJ2] Jackwerth, J.C., Option-Implied Risk-Neutral Distributions and Implied BinomialTrees: A Literature Review, Journal of Derivatives 7, No. 2, 66-82, 1999.

[JJ3] Jackwerth, J.C., The Price of a Smile: Hedging and Spanning in Option Markets,Review of Financial Studies 14, No. 2, 495-527, 2001.

[KP] Kelley, A., Pohl, I., A Book on C. Programming in C, Addison-Wesley, 1998.

[KF] Klebaner, F., Introduction to Stochastic Calculus with Applications, Imperial CollegePress, 1999.

[RM] Rubinstein, M., Implied Binomial Trees, Journal of Finance, July 1994.

[SH] Shreve, S., Steven Shreve’s Lecturers on Stochastic Calculus and Finance, 2001.

[VRD] van Rossum, G., Drake, F.L, Python Tutorial. Release 2.0b1, 2001.

[WD] Williams, D., Probability with Martingales, Cambridge University Press, 2000.

[WP] Wilmott, P., Derivatives. The Theory and Practice of Financial Engineering, JohnWiles & Sons, 2000.

[WHD] Wilmott, P., Howison, S., Dewynne, J., The Mathematics of Financial Derivatives.A student introduction, Cambridge University Press, 1999.

60

A Appendix A

A.1 Centering condition

In this section we will show how to determine the equation

Si,i2+1 =Si−1,i2(e

r∆ticM(t0, Si−1,i2 , ti) + λi−1,i2Si−1,i2 − er∆tiΣci−1,i2)

λi−1,i2Fi−1,i2 − er∆ticM(t0, Si−1,i2 , ti) + er∆tiΣci−1,i2

(A.1.1)

that we presented in Section 4.6 (equation (4.6.20)). Note that i2 = bi/2c. Using equation(A.1.1) it is possible to find the two stock prices that are used as initial values in thealgorithm for finding the stock prices at odd time levels in an implied volatility tree.

Now, let us derive equation (A.1.1). We will use the equations (4.6.12) and (4.6.5) forthe case j = i2 = bi/2c,

Si,i2+1 =ci2i−1,i2Si,i2 + e−r∆ti(Si,i2 − Fi−1,i2)Si−1,i2

ci2i−1,i2 + e−r∆ti(Si,i2 − Fi−1,i2)

(A.1.2)

and

ci2i−1,i2 =

cM(t0, Si−1,i2 , ti)− Σci−1,i2

λi−1,i2, (A.1.3)

respectively. We will also use equation (4.6.19)

Si,i2+1Si,i2 = S2i−1,i2 . (A.1.4)

For notational convenience we will from now on use the following notation

Si+1 = Si,i2+1 (A.1.5)Si = Si,i2 (A.1.6)S = Si−1,i2 (A.1.7)c = er∆ticM(t0, Si−1,i2 , ti) (A.1.8)Σ = er∆tiΣc

i−1,i2 (A.1.9)F = Fi−1,i2 (A.1.10)λ = λi−1,i2 . (A.1.11)

Now, if we combine (A.1.3) and (A.1.2) and multiply the denominator and numerator in(A.1.2) by the term er∆tiλi−1,i2 , we get

Si+1 =Si (c− Σ)− λS(F − Si)

c− Σ− λ(F − Si). (A.1.12)

Moreover, (A.1.4) yields Si = S2

Si+1and by using (A.1.12) we obtain the following expression

S2i+1 (c− Σ− λF ) + Si+1

(

λS2 + λSF)

= S2 (c− Σ) + λS3. (A.1.13)

61

Therefore, introducing

a = c− Σ− λF (A.1.14)b = λS2 + λSF (A.1.15)d = −

(

S2 (c− Σ) + λS3) (A.1.16)

we obtainaS2

i+1 + bSi+1 + d = 0. (A.1.17)

By inspection we see that Si+1 = S is a root in (A.1.17). The other root is then given by

Si+1 =daS

= − 1S

S2 (c− Σ) + λS3

c− Σ− λF. (A.1.18)

Thus, if we simplify (A.1.18), we finally obtain (A.1.1).

62

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Implied Binomial Models