On Construction of Multinomial Lattices for Option Pricing

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Mälardalen University Press Licentiate Theses No. 259

ON CONSTRUCTION OF MULTINOMIAL

LATTICES FOR OPTION PRICING

Carolyne Ogutu 2017

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ISSN 1651-9256

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Abstract

The field of option pricing has had quite an explosion in terms of research since the publication of the works by Bachelier and Samuelson. The main idea is to be able to find the fair price of an option while taking into account no-arbitrage opportunities. In the wealth of research in this area, many models have been developed but most often there are no closed form solutions to them. This makes option pricing very difficult. But numerical methods have been developed to help with this difficulty. One on the numerical methods is lattice pricing. This was first introduced by Cox, Ross and Rubinstein when they developed the binomial lattice pricing model. In this thesis, we focus on lattice construction. We seek to develop a multinomial lattice structure that can be used for option pricing. First, we present an overview of research in the area of option pricing and also describe the construction of lattices using moment-matching technique which results in an equation system described by the Vandermonde matrix. Using the properties of inverse of Vandermonde matrix, we come up with expressions for the jump probabilities of the resulting system. Secondly, we extend the lattice building to cases when the underlying asset follows Merton-Bates jump diffusion process. We apply Sturm’s theorem to express the polynomial in α, the jump amplitude, for quadrinomial and pentanomial lattices. Finally, by restricting the lattice to one with symmetrically placed and evenly spaced nodes to prevent exponential growth of the lattice, we develop explicit expressions for probabilities for quadrinomial and pentanomial lattices. We also study their properties when the approximation distribution is lognormal - that is, when the underlying asset follows the geometric Brownian motion. This allows us to come up with conditions that have to be satisfied for α and the probabilities to be real and positive.

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This thesis is dedicated to my dad, Professor G.E.M. Ogutu.

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This work was financially supported by the Swedish International Development Agency and the International Science Programme at Uppsala University -Sweden, in partnership with the East African Universities Mathematics Programme in collaboration with the School of Mathematics, University of Nairobi and Mälardalens University in Västerås, Sweden

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Acknowledgements

I would like to convey special thanks to my supervisors, Professor Sergei Silvestrov, Professor Patrick Weke, and co-supervisors Dr. Ying Ni, Professor Anatoliy Malyarenko and Dr. Ivivi Mwaniki for their continuous and prompt guidance and fruitful cooperation as I worked on my research. I would also like to especially thank Karl Lundengård for his fruitful and inspiring cooperation and for always being helpful in my studies and research. I am also grateful to Professor Dmitirii Silvestrov for useful comments and bibliographical remarks.

I would also like to thank International Science Programme (ISP) in collaboration with East African Universities Mathematics Programme (EAUMP) for awarding me a research fellowship at Mälardalen University. And to the staff at ISP, Leif Abrahamson, Therese Rantakokko, thank you for the your support, guidance and for organizing very fruitful fellow evenings. To Pravina Gajjar, you have been a mother figure to me, making sure I am not alone and encouraging me even in moments of difficulty. I would also like to thank the staff at School of Education, Culture and Communication (UKK), at Mälardalen University for providing a wonderful academic and research environment for Applied Mathematics.

To my past and present PhD colleagues in the ISP and Sida programmes, Betuel Canhanga, Alex Tumwesigye, Jean-Paul Murara, Pitos Biganda, Asaph Muhumuza, Benard Abola, Tin Nwe Aye and John Musonda: thank you, first for the sumptuous dinners you prepared and which I enjoyed, and secondly for your unending encouragement as we braved the cold dark winter days.

To my Vallby family, Erick Mokaya, James Omweno, Shedrack Lutembeka, Polite Mpofu and Katarína Škrabáková your friendship is unmatched. The English men say "All work and no play makes ’Jill’ a dull girl", so i thank

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you for all the family times filled with games that made sure ’Jill’ was never dull.

To my family, words fail me to express my gratitude. To mum and dad, I want to say a big thank you, for your constant and unwavering support and encouragement. Without you, all this would not have been possible. This is in your honor. To my brothers and sisters, Timothy Ogutu, Andrew Ogutu, Sheena Ogutu, Faith Ogutu, Roy Okello, Tonia May, Kevin Okello and Brian Ogutu, thank you for being there for me and being the best aunts and uncles that Trevor Manuel could ever have asked for. It surely does take a village to raise a child. Thank you for being that village. Finally, to the two most important men in my life. To my husband, Paul, I am the luckiest girl in the world to have you in my life. No amount of words can express just how grateful I am. Thank you for taking up the heavy task of raising our son Trevor Manuel and for granting me the opportunity to study. Thank you for being strong for me and standing by me in moments of weakness especially at those times I almost gave up. To Trevor Manuel, of all the gifts I have received from God, you are the best one. Thanks for being the best son mommy could have asked for. Thank you for all the entertaining story times over Skype and for enduring my absences with much grace and love.

Västerås,

May, 2017 Carolyne

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List of Papers

This thesis is based on the following papers:

Paper I. K. Lundengård, C. Ogutu, S. Silvestrov and P. Weke,

Asian Options, Jump–Diffusion Processes on a Lattice, and Vandermonde Matrices, in Modern Problems in Insurance Mathematics, Springer International Publishing, EEA Series 335 - 363 (2014)

Paper II. C. Ogutu, K. Lundengård, S. Silvestrov and P. Weke,

Pricing Asian Options Using Moment Matching on a Multinomial Lattice, in AIP Conference Proceedings 1637, 759 - 765 (2014)

Paper III. K. Lundengård, C. Ogutu, S. Silvestrov and P. Weke,

Construction of moment–matching multinomial lattices for pricing of Asian Options under Jump–Diffusion Processes, in New Trends in Sochastic Modeling and Data Analysis, ISAST, 211 - 220, (2015).

Paper IV. K. Lundengård, C. Ogutu, J. Österberg, Y. Ni, S. Silvestrov

and P. Weke, Moment–Matching multinomial lattices using Vandermonde Matrices for Option Pricing, in Stochastic and Data Analysis Methods and Applications in Statistics and Demography, ISAST, 15 - 29, (2016).

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Introduction

1 Overview

The study and development of financial models on the evolution of asset prices is not only important to financial engineers but other practitioners too. For instance, investors and speculators are trying to find any opportunities to make money, governments are trying to find a way to derive fair and orderly market, while academicians are trying to find better models for the asset prices and the derivatives that depend on them [16]. These models are developed to explain the stochastic behavior of asset prices, and, the simplest of which is Brownian motion. In Figure 1 we have a plot of the daily asset prices of the Swedish OMX Index, on the left, and simulated Brownian motion on the right. From visual analysis alone, we can safely assume that the evolution of the Swedish OMX Index prices mimics the simplest stochastic process - Brownian motion.

However, further empirical analysis on the daily Swedish OMX Index data, generates results that cause us to shy away from assuming that the evolution of asset prices follow Brownian motion. In Figure 2, we have plots of the histogram and q-q plot of the log return of the data, and we can see that the histogram may be considered Gaussian but the q-q plot shows a marginal deviation from the normal distribution assumption of the Brownian motion which means, the distribution of the log

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1. OVERVIEW

Figure 1: The plot on the left hand side depicts the daily Swedish OMX Index prices from 27th_{October 2000 to 28}th_{October 2016 while the one on}

the right hand side shows a simulated Brownian motion.

Figure 2: Histogram, q-q plot and plot of log-return of Swedish OMX Index

returns has heavy tails. In addition to the heavy tails, the plot of differenced data shows that volatility varies with time, that is, high at some points and low at some points. In summary, we can say that the Swedish OMX Index, as with other asset prices exhibits (a) heavy tails, (b) asymmetry in the rates of return, (c) nonconstant volatility, and (d) clustering of volatility, which are not captured by assumption of Gaussian distribution as underlying distribution of the asset returns.

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On Construction of Multinomial Lattices for Option Pricing

2 The history of Option Pricing

The history of option pricing theory dates back to the beginning of 19th _{century. In finance however, it was the work of Louis} Bachelier in his PhD thesis titled The Theory of Speculation [3], that the first analytical valuation for options was presented. The publication of Bachelier’s thesis did two major things. First, it introduced the world to mathematical finance, and secondly, it provided an agenda for probability theory and stochastic analysis. This agenda was quickly picked up by mathematicians and physists in the 20th century and the theories of probability and stochastic analysis were developed [14]. Bachelier argued that small fluctuations in price seen over short time interval are independent of the current value of the price. He obtained the increments of his stock price process as independent Gaussian random variables, used lack of memory property of the price process to construct the stochastic differential equation [14].

dS(t) = S(0){µdt + σdW (t)} S(0) > 0

For a long while, Bachelier’s work lay dormant. It was not until 50 years later in 1955, when Samuelson got wind of Bachelier’s work, that a spark was ignited [37]. Samuelson quickly realized that Bachelier’s Brownian motion model for asset prices, assumes that asset prices at any one time follow a normal distribution with the implication that the asset prices can assume negative values. He remedied this by assuming the asset prices evolve according to a Geometric Brownian Motion (GBM). After applying Itô formula to his asset prices, he concluded that the asset prices satisfied the stochastic differential equation [38]

dS(t) = µS(t)dt + σS(t)dW (t) S(0) > 0 (1.1) Where µ and σ are some constants, S(t) is the asset price, and W (t) is Brownian motion. A few years later in 1973,

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3. OPTION PRICING BASICS

F. Black and M. Scholes [6] wrote their seminal paper that improved on the theories of Boness [7] and Samuelson [38]. Their model revolutionized the option pricing industry leading to an explosive growth in trading of derivatives in the world

wide financial market. This was mainly because of the

compactness, tractability, robustness and simplicity of the model[20]. However, empirical studies have shown that there exists limitations in the Black–Scholes model. For instance, the normal distribution assumption doesn’t accurately describe observed asset returns properties and therefore other models have been developed to improve on this limitation among others [10], [12].

3 Option Pricing Basics

A financial option, or just option, is a financial asset whose price depends on another financial asset. A financial option contract involves two parties (i) The writer, is the party who issues the option. Their responsibility is to fulfill the contract if the option is exercised by the holder, and (ii) The holder, is the party who holds the option and have the right to exercise the option, but he can choose not to exercise the right depending on his preferences.

At inception of the option there are two key components that have to be included, that is, (a) the strike price which is the price agreed today specifying the price at which the underlying asset will be bought or sold when the option is exercised in the future, and (b) the expiration date which determines the period during or after which the option can be exercised by its holder. The expiration date is dependent on the type of option; European options are exercised only at maturity while American options can be exercised at anytime within the life of the option.

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Options are divided into two depending on the type of right they give to the holder. A call option is one which gives the holder right to buy while a put option gives the holder the right to sell an underlying asset at the strike price at or before expiration. Most often, analysis of options begins with European-style options since they are easier to handle given that the holder cannot exercise till maturity.

The payoff function of a European-style call option is (S(T ) − K)+ = max(S(T ) − K, 0), where K is the strike price and S(T ) is the asset price at maturity. From the payoff function, see figure 3, we notice that the payoff can be 0 especially when the strike is greater than the asset price at maturity. In this case, the holder of the option has no reason to exercise the option. On the other hand, when the asset price is greater than the strike price, the option holder will exercise his right and makes a profit of S(T ) − K with no remaining obligations.

Figure 3: Call Option Payoff Plot

For a European-style put option, the payoff function is (K − S(T ))+ = max(K − S(T ), 0). When K > S(T ) then the option holder will exercise his right and make profit of K − S(T ) with no further obligations. When K < S(T ), the option holder can choose not to exercise this right, see figure 4.

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3. OPTION PRICING BASICS

Figure 4: Put Option Payoff Plot

options which are similar to European-style except for the fact that an American option holder can exercise their right at anytime within the life of the option.

Transactions involving buying and selling of options, the maturity period and strike price are often agreed upon at inception without debate. The biggest contention is what is the right or fair price of the option, that is, what is the minimum value the writer is willing to receive for the option and what is the maximum the buyer is willing to pay. A key principle that determines this price is the no-arbitrage principle which further leads to the Theorem 3.1.

Definition 3.1. Arbitrage Opportunity An arbitrage opportunity is a trading strategy that never costs you anything today or in the future but has strictly positive probability of having strictly positive cashflow in future.

Theorem 3.1. One Price Theorem Identical assets, those

that yield similar cash flows, should have the same price. Otherwise, one could exploit the gap between the prices by buying the cheaper asset and selling it at the higher price and this results

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in arbitrage

4 Pure Diffusion Option Pricing

4.1 Black Scholes Model

Consider a market with two assets in a finite time interval, a risky asset S = {S(t), t ∈ [0, T ]} and a risk-free asset saving account B = {B(t), t ∈ [0, T ]}, paying constant interest rate r > 0 with continuous compounding.

Let Ω, F , (Ft)t∈[0,T ], P

be a filtered probability space. The price process S is assumed to follow geometric Brownian motion, Equation 1.1 and by applying Itô formula, we can show that S satisfies Equation 1.1 if and only if

S(t) = S(0)exp    µ − σ2 2  t + σW (t)   (1.2)

where µ is constant drift, σ is constant volatility and W (t) is a Wiener process. Also, the risk-free savings account B is assumed to evolve as

dB(t) = rB(t)dt t ∈ [0, T ] (1.3)

with initial value B₀= 1. To obtain the fair price of a European call option, Black and Scholes constructed a self-financing portfolio containing units of the two assets, made sure this portfolio was free of arbitrage and showed that there exists a replicating portfolio for the European call option. In the end, the result was a Black–Scholes call option pricing formula

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4. PURE DIFFUSION OPTION PRICING where d1 = ln_KS+ r +σ₂2 T σ√T d₂ = d₁− σ√T .

The Black–Scholes call option price does not depend on µ because the price is calculated under the risk-neutral probability measure Q. Black–Scholes model assumes continuity in the asset price process, which is not the case in real markets since trading takes place at discrete times. To remedy this, Cox, Ross and Rubinstein developed a simple discrete time model for option valuations - the binomial lattice model.

4.2 Recombining Binomial Lattice Model

In the binomial lattice model, trading takes place at discrete times and the asset can only take two possible values. The assumptions in the binomial model are (a) there are no trading costs, (b) there’s no maximum or minimum units of trading, (c) assets and bonds can only be bought and sold at discrete times, and (d) the principle of no arbitrage holds.

4.2.1 One Period Binomial Model

Consider a market with two assets (as in the Black–Scholes model) in a finite time interval [0, T ] divided into N periods, 0 = t₀ < t₁ < · · · , t_{N −1} < t_N = T . Recall that the price process for the bank account is deterministic and earns interest at a constant risk-free continuously compounding rate of r per annum, therefore, at time t, the bank account value is ert _for

r > 0 [1] , [5]. The price process for the stock is stochastic, starting at time t0 with stock price equal to S(0), in one time step, say one month, the price S(1) is assumed to go up by

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a factor u to uS(0) with probability p or down by factor d to dS(0) with probability 1 − p. Where u and d represent the rate of return on the asset price, 0 < d < u and p ∈ (0, 1). To rid the market of arbitrage, d < er < u, because if er < d, then no one would have any reason to invest in the savings account and if u < er then no one would have any reasons to buy assets, therefore, the rate of return of the savings accounts has to be sandwiched between the best and worst rates of return of the asset [45].

Now, consider a European-style call option with strike price uS(0)

S(0)

dS(0)

Figure 5: Stock Dynamics

Cu

C0

Cd

Figure 6: Option Dynamics

K > 0 and maturity t₁ with price at t₀ given by C₀. At time t₁, we have two payoffs corresponding to the various states of the world, that is, Cuwhen the asset price goes up to uS(0) and Cd

when the asset price goes down to dS(0), as shown in figure 6. On first impulse, suppose an investor believes that the price of the asset will indeed go up with probability p and down with probability 1 − p. He then goes ahead and evaluates the price of the call option at time t = 0 as

C₀ = e−r(pC_u+ (1 − p)C_d)

where p is considered to be the real world (physical) probability. Then, this option price would be incorrect since the investor has

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4. PURE DIFFUSION OPTION PRICING

chosen the probability p from his point of view and not from the markets.

To calculate the true call option price at t = 0 we use the method of replication. We construct a self-portfolio V containing β units of risky asset and α units of the bank account that exactly replicates the value (payoff) of the option.

Definition 4.1. Self-financing A portfolio strategy V is said

to be self-financing if

αter+ βtS(t) = αt+1+ βt+1S(t) for all t = 0, 1, . . . , T − 1

Definition 4.2. Replicating Portfolio A portfolio V is said

to be a replicating portfolio, if it is self-financing and its value is equal to the value of the contingent claim (payoff), that is,

V_T = C with probability 1

Note that the values α, β can be both positive or negative since we can hold both long and short positions in the market [5]. Our portfolio V = (α, β) is deterministic at time t₀ and stochastic at time t1, V0 = αB(0) + βS(0) V1 =     

αB(1) + βuS(0) = αer+ βuS(0) αB(1) + βdS(0) = αer+ βdS(0)

To calculate the option price, we follow these steps (a) find α and β such that the payoff of the option - max(S(T ) − K, 0) for call options, at time t₁ is the same as the value of the portfolio at time t1. and (b) find the option price at time t0 by using the One Price Theorem 3.1.

So, at time t1, we know that Cu is the payoff if the stock price

goes to uS(0) and C_d is the payoff if the stock price goes to

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dS(0) and this leads to the following system of equations C_u = αer+ βuS(0)

Cd = αer+ βdS(0),

which we solve to get

β = Cu− Cd

(u − d) S(0) (1.4)

α = uCd− dCu

er_{(u − d)} (1.5)

now by the one price theorem, we have α + βS = V₀, and thus V₀ = 1 er " er− d u − d ! C_u+ u − e r u − d ! C_d # , (1.6) and by defining q = e r _{− d}

u − d we have equation (1.6) simplifies to V0 =

1

er [qCu− (1 − q) Cd] . (1.7)

Therefore the European-style call option price after one time step is given by

C₀ = 1

er[qCu− (1 − q) Cd] (1.8)

4.2.2 Multi-period Binomial Model

The above pricing method can be extended to the multi-period case. First consider the case with 2 periods. The assumptions are the same as in the one period case, that is, at any given time step the price can only assume two values in the next

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time step. Now adding the additional assumption that ud = 1 [13] (recombination condition), this means that an up movement followed by a down movement is the same as a down movement followed by an up movement, then at time step two the asset price will either go to uuS(0), udS(0) or ddS(0). Deriving the values of Cu and Cd with one period to expiration if the asset

prices are at uS(0) and dS(0), we have

Cu = [qCuu+ (1 − q)Cud] e−r

C_d = [qC_du+ (1 − q)C_dd] e−r. Replacing this into Equation 1.8 results in

C₀ = 1 e2r

h

q2C_uu+ 2q(1 − q)C_ud+ (1 − q)2C_ddi. (1.9) In addition, extending to 3 periods, the asset prices at time step 2 will now move to uuuS(0), uudS(0), uddS(0) and dddS(0) and the option values are derived as

Cuu = [qCuuu+ (1 − q)Cuud] e−r

Cdd = [qCddu+ (1 − q)Cddd] e−r

Cud = [qCudu+ (1 − q)Cudd] e−r.

Substituting this in equation 1.9, we have C₀= 1 e3r [q 3_C uuu+ 3q2(1 − q)Cuud+ 3q(1 − q)2Cudd+ (1 − q)3Cddd i . (1.10)

In general, assuming that the time increment tn − tn−1 is the

same for any n = 1, 2, · · · N , then at any time tn(n = 0, · · · , N )

the asset price process take values in the set {uk_dn−k_S

0}k=0,...,n.

Therefore the only paths that lead to uk_dn−k_S

0 are

n! k!(n − k)! [15]. Further, the probability of moving to node uk_dn−k_S

0 is n!

k!(n − k)!q

k_{(1 − q)}n−k_.

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With this in mind, we can rewrite Equation (1.10) as C0= 1 e3r 3 X k=0 3! k!(3 − k)!q k_{(1 − q)}3−k_maxh 0, ukd3−kS0− K i , where K is the strike price. This can be extended to n-periods and results in the general equation [13, 21]

V0=_e1nr    n X k=0 n! k!(n − k)!q k_{(1 − q)}n−k_maxh 0, uk_dn−k_{S(0) − K}i    . (1.11)

4.3 Moment Matching: Solving for u and d

In the previous section we constructed a risk-neutral

multi-period binomial model but did not derive the expressions for u and d. Let the time increment defined in section 4.2.2 be denoted by ∆t. In risk neutral world, the asset price evolution in discrete time is given by

St+∆t= Stexp r − 1 2σ 2 ! ∆t + σ √ ∆tW ! (1.12) where W is a Gaussian random variable with zero mean and unit variance. The Equation (1.12) is similar to Equation (1.2) except we have replaced µ with r. In addition, we know that the Black–Scholes model follows a lognormal distribution and so the first and second moments are given as

Et(St+∆t) = Ster∆t

Et(St+∆t2 ) = S

2

te

(2r+σ2_)∆t

Now, to solve for u and d, we need to construct the binomial lattice in such a way that the limit of the process as ∆t approaches zero corresponds to the diffusion process of the

Black–Scholes model. We do this via moment-matching

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to those of the diffusion process of the Black–Scholes model. This leads to the following system of equations

qu + (1 − q)d = er∆t (1.13)

qu2+ (1 − q)d2 = e(2r+σ2)∆t (1.14) The above system has two equations and three parameters and therefore to solve for the parameters u, d, q we have to impose an additional condition. In their model, Cox et al [13] assumed that ud = 1 and this results in the following values for u and d

u = eσ √

∆t _{u = e}−σ√∆t

In 1993, Jarrow and Rudd developed their binomial lattice model by assuming that p = 1/2 and later in the same year Tian further improved on the binomial lattice model by no only matching the first two moments but matching the first three. He argued that since the binomial distribution is skewed, adding the condition that the third moment of the discrete-time process and the continuous-time process be equal is sensible. The results in the Table 1.1 show a comparison of the At-The-Money European call option prices from a multi-period Binomial model and the Black–Scholes model.

Table 1.1: Comparison of binomial lattice option prices (Call prices, C. P.) with Black–Scholes (B–S) option price

N. St. 10 20 40 60 80 · · · 500 B–S

C. P. 9.4278 9.5306 9.5826 9.6000 9.6087 · · · 9.6307 9.6349

From the Black–Scholes model and binomial lattice model discussed thus far, two points clearly emerge. First, when the asset price process follows geometric Brownian motion, there exists only one arbitrage free price and lastly, the writer of the option can hedge his position at maturity by just constructing

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Figure 7: Convergence of multi-period Binomial Model At-The-Money European Call option price to the Black Scholes Model Counterpart

a self-financing replicating portfolio. These points can only be achieved because the market in question is complete. In lattice modeling, when the number of lattices L is greater than 2, then the market becomes incomplete.

4.4 Trinomial Lattice Model

The simplest form of trinomial lattice model is constructed by thinking of it as a two period CRR binomial model. In this case the up, down and middle parameters become u = eσ

√ 2∆t and d = e−σ

√

2∆t _{and m = 1. The resultant prices and plot are} shown below

Table 1.2: Comparison of CRR Trinomial lattice option prices (Call prices, C. P.) with Black–Scholes (B–S) option price

N. St. 10 20 40 60 80 · · · 500 B–S

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Figure 8: Convergence of multi-period CRR Trinomial Lattice Model At-The-Money European Call option price to the Black–Scholes Model Counterpart

4.4.1 Boyle Trinomial Lattice

This model was first introduced by Boyle [8]. Consider an asset, say stock, S with lognormal distribution of returns. At an interval of time ∆t, the Boyle’s lattice structure assumes that the price of stock can either move up to uS(∆t), down to dS(∆t) or remain the same mS(∆t), as shown in Figure 9. To calculate the

u2_S 0 uS0 umS0 S0 mS0 m2S0 dS0 dmS0 d2_S 0

Figure 9: Trinomial tree asset price evolution 16

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jump magnitudes u and d and the probabilities pu, pm, pd, Boyle

employed the method of moment-matching, where he equated his system of equations to the first two moments of the lognormal distribution. The system of equations is given by

pu+ pm+ pd = 1

puSu + pmS + pdSd = SM

pu(S2u2− S2M2) + pm(S2− S2M2) + pd(S2d2− S2M2) = S2V

where M = er∆t _{and V = M}2_(eσ2_∆t

− 1), the first and second moments of the lognormal distribution. In addition, he let ud = 1 still hold in this case. Then the system of equations now becomes pu+ pm+ pd = 1 (1.15) puSu + pmS + pdS1u = SM (1.16) pu(S2u2− S2M2) + pm(S2− S2M2) + pd(S2 1_u 2 − S2_M2₎ ₌ _S2_V _(1.17) From Equation (1.15) we have p_m = 1 − p_u− p_d. Substituting in Equations (1.16) and (1.17) we have

pu(u − 1) + pd 1 u− 1 ! = M − 1 (1.18) pu(u2− 1) + pd 1 u2 − 1 ! = V + M2− 1 (1.19)

From Equations (1.18) and (1.19) we have pu = (V + M2_{− M )u − (M − 1)} (u − 1)(u2_{− 1)} (1.20) pd = (V + M2_{− M )u}2_{− u}3_{(M − 1)} (u − 1)(u2_{− 1)} (1.21) pm = 1 − pu− pd (1.22)

After solving this system of equations he realized that by assuming the same up and down parameters as in the CRR

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lattice, that is, eσ∆t _{would result in negative probabilities for}

the second probability. To rectify this, he added a dispersion parameter λ > 1 and the resultant up and down parameters are given by u = eλσ√∆t _{and d = e}−λσ√∆t_{. The results in the} Table 1.3 show a comparison of the At-The-Money European call option prices from a multi-period Boyle trinomial model and the Black–Scholes model. The results in the table are better illustrated in the Figure 10.

Table 1.3: Comparison of Boyle Trinomial lattice option prices (Call prices, C. P.) with Black–Scholes (B–S) option price

N. St. 10 20 40 60 80 · · · 500 B–S

C. P. 9.4270 9.5303 9.5827 9.6002 9.6090 · · · 9.6311 9.6349

Figure 10: Convergence of multi-period Boyle Trinomial Lattice Model At-The-Money European Call option price to the Black–Scholes Model Counterpart

Up until now, the models discussed have all assumed that the asset price process follows geometric Brownian motion. Notwithstanding its simplicity, geometric Brownian motion is unrealistic in practice. As mentioned earlier, actual asset prices are most likely going to stray beyond the ±3σ from the mean.

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5 Jump Diffusion Option Pricing

The Black–Scholes option pricing model does not make room for large market fluctuations or jumps that can occur such as market crashes. Research also shows that including jumps in option valuation can explain some of the biases observed in the their option pricing model [2], [32], [33]. For example, real data shows that markets have skewed log-return distributions which is contrary to no skewness in the normal distribution. In addition, real market log-return distributions are leptokurtic while the normal distribution is mesokurtic. To capture some of these features, the pure diffusion process of Black and Scholes can be improved upon by adding a jump parameter. In option pricing, jumps are introduced via the Poisson distribution and to build a jump process one starts from the building block which is the Poisson process [39].

Definition 5.1 (Poisson Process). A stochastic process

{Jt, t ≥ 0} is called a Poisson process if the following conditions

hold

(a) J₀= 0

(b) Jt− Js are integer-valued for 0 ≤ s < t < ∞ and

P(J_t− J_s= k) = λ

k_{(t − s)}k

k! e

−λ(t−s) _{f or} _{k = 0, 1, 2...}

(c) The increments Jt2− Jt1 and Jt4− Jt3 are independent for

all 0 ≤ t₁< t2 < t3< t4

5.1 Merton Jump Diffusion Model

The paper by Merton [33] pioneered the pricing of options governed by jump-diffusion processes. He proposed log-normally sized jumps that arrive according to a Poisson process and

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5. JUMP DIFFUSION OPTION PRICING

developed a closed-form formula for pricing European options in continuous time. Let τ_j be the instances when the jump occurs and τ1 < τ2 < τ3 < . . . and let the random variable S(t) (asset price) jump at τj, where τ+ is the moment after a

particular jump and τ− is the moment before a particular jump. The absolute size of the jump therefore becomes

∆S = S(τ+) − S(τ−) (1.23)

Suppose q is the proportion of the jump, we can then say that S(τ+_{) = qS(τ}−_{) and replacing it back into Equation (1.23) we} have ∆S = qS(τ−) − S(τ−) = (q − 1)S(τ−).With this the jump process becomes

dS(t) = (qt− 1)S(t)dJ(t) (1.24)

Jumps provide a natural framework to model prices as aforementioned and can be modeled separately or can be

superimposed into the diffusion model [39]. When we

superimposed the jump process in equation (1.24) into the diffusion process we have

dS(t) = µS(t)dt + σS(t)dWt+ (q − 1)S(t)dJ (t)

In general, the underlying asset S(t) under jump-diffusion models follow the stochastic differential equation

dS(t)

S(t) = (µ − λk) dt + σdW (t) + (q − 1) dJ (t) (1.25) where J (t) is a Poisson process, k is the expected relative jump size, λ is the rate of the Poisson process, and q − 1 is a random variable producing a jump from St to S(t)q.

Suppose there’s no jump in a small time interval τ , then Equation (1.25) simplifies to

dS(t)

S(t) = (µ − λk) dt + σdW (t)

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By applying the Itô lemma for pure diffusion processes, we have S(t) = S(0)exp     µ − λk − σ2 2  t + σW (t)    (1.26) Now suppose that jumps occur at time τ₁, τ₂, · · · , τ_k, then

S(t) = S(0)exp     µ − λk − σ2 2  t + σW (t)    · q1 = S(0)exp     µ − λk − σ2 2  t + σW (t)    · q₁· q₂ = ... = S(0)exp     µ − λk − σ2 2  t + σW (t)    k Y i=1 q1 (1.27) Let X(t) = ln S(t) − ln S(0), γ = µ − λk − σ₂2 and ln qt = Qt

then equation (1.27) simplifies to X(t) = γt + σW (t) +

J (t)

X

i=1

Qi (1.28)

5.2 Characteristic Function of Merton’s model

In order to calculate the moments of Merton’s model, we first need to find the characteristic function of Merton’s jump-diffusion model.

Definition 5.2. Characteristic Function The characteristic

function of a process S(t) is defined to be

E[exp(izS(t))] z is real or complex

The characteristic functions of Brownian motion and Poisson process are

Brownian motion = exp{−1

2σ 2_z2_t} Poisson process = exp{λteiz− 1

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6. PROBLEM FORMULATION

Since the jump-diffusion model (1.28) is a sum of Brownian motion and compound Poisson process, the characteristic function is Et0[exp(iz(S(t)))] =  exp  iz  γt + σWt+ J (t) X k=1 Q       = exp{izγt −1 2σ 2 z2t}Et0  E_t₀  exp  iz   J (t) X k=1 Q     Q = x, J (t) = k     = exp izγt −1 2σ 2_z2_t ∞ X k=0 exp (−λt) (λt)k k! k Y n=1 Z∞ −∞e izxl(dx) λ = exp izγt −1 2σ 2 z2t exp (−λt) exp λt Z∞ −∞ eizxl(dx) λ = exp t izγ −1 2σ 2_z2 + Z∞ −∞ eizx− 1l(dx) (1.29)

As aforementioned Merton assumes that the log asset price jump size follows a normal distribution and thus

l(dx) = √λ 2πδ2exp    (dx − µ)2 2δ2   

Replacing this into equation (1.29) we get that the integral part of the expression becomes

Z∞ −∞ eizx− 1 l(dx) = Z∞ −∞ n eizx− 1o λ √ 2πδ2exp    (dx − µ)2 2δ2    = λ " eizµ−δ2z22 − 1 # (1.30) Now replacing Equation (1.30) into (1.29) we have

E [exp(iz(X(t)))] = exp t izγ −1₂σ2_z2_{+ λ}_eizµ−δ2z2 2 − 1 (1.31) which is the characteristic function of Merton’s jump diffusion model.

6 Problem Formulation

The first multinomial lattice was in principle first introduced by Boyle in 1988 [8]. From the theory discussed in Section 4.4,

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he extended it to a two assets case by considering two assets with bivariate lognormal distribution and obtaining the mean, variance and covariances of the two assets from the lognormal distribution. He constructed a pentanomial lattice by using the moment matching method and setting u_i = exp(λ_iσ_i√∆t) for i = 1, 2, 3, 4, 5. Boyle et al [9] went on to improve on this methodology while maintaining the two assets, and reducing the number of lattices to 4. They did this by assuming that u_id_i= 1 for i = 1, 2 and ui = eσi

√

∆t_{. In 1991, Kamrad and Ritchken} [22] went ahead to generalize the methods by Boyle and Boyle et al. In addition, a comprehensive survey of stochastic approximation methods for pricing American options is given in the fundamental work by Silvestrov [42, 43]. In particular, optimal stopping domains for American knock out options in discrete time were considered in [24, 25, 26, 27, 28, 29, 30].

In this thesis, we adopt the work of Primbs et al in constructing our multinomial lattice. This method of construction helps us to determine the objective probabilities of our framework. We work with jump-diffusion process - particularly, that assumed by Merton [33], to construct our multinomial lattice because we are employing higher order moments. Unlike Primbs et al, we match the first L moments of the continuous-time distribution to those of our approximating distribution - the constructed lattice. Let the approximating discrete random variable be denoted by Z where

Z = m1+ (2l − L − 1) α, l = 1, . . . , L with probabilities pl (1.32)

where m1is the mean of X, the continuous random variable, and α > 0 is a free parameter. So, the equations for the moments of Z are [see [44]]

L

X

l=1

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6. PROBLEM FORMULATION

where µj are the jth central moments of X. In matrix form, the

general case is given as

A~p = ~µ (1.34)

where ~p is the vector of probabilities, ~µ is the vector of moments and A is the general matrix given by equation (1.35). Note that when L is odd, the central column of A will have all zeros except for the first element

A =                      1 · · · 1 · · · 1 (1 − L)α · · · (2n − L − 1)α · · · (L − 1)α ((1 − L)α)2 _{· · ·} _{((2n − L − 1)α)}2 _{· · ·} _{((L − 1)α)}2 ... ... ... ... ... ((1 − L)α)m _{· · ·} _{((2n − L − 1)α)}m _{· · ·} _{((L − 1)α)}m ... ... ... ... ... ((1 − L)α)L−1 _{· · · ((2n − L − 1)α)}L−1 _{· · · ((L − 1)α)}L−1 ((1 − L)α)L _{· · ·} _{((2n − L − 1)α)}L _{· · ·} _{((L − 1)α)}L                      (1.35) 24

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7 Solutions and Chapter Summaries

Chapter Two

This paper we present a general introduction into option pricing theory by giving general definitions for European-style options with concentration on the path-dependent options Asian options. We also give an extensive review of literature on option pricing on the lattice for European-style options and for Asian options. In addition to the classical option pricing models -Black–Scholes model and binomial option pricing model, we introduce a discussion on jump diffusion model by Merton and discuss two methods that have employed this model in pricing options on a lattice. Further, we show explicit expressions that lead to the development of matrix A in Equation (1.35) and build on the premise of its similarity to the Vandermonde matrix, when the last row is omitted, to express its inverse as

V_L−1 ij = (−1)j−1 (−1)i−1 · αL−j αL−1· ˜ σL−j,i 2L−1_{(i − 1)!(L − i)!} = (−1) j−i 2L−1_αj−1 · ˜ σL−j,i (i − 1)!(L − i)! .

In addition, we show that the solution for the probabilities is given as pi = L X j=1 V−1 ijµj−1 = L X j=1 (−1)j−i 2L−1_αj−1 · ˜ σL−j,i (i − 1)!(L − i)!µj−1

We then use the omitted last row to express a polynomial of degree L with coefficients

L X j=1 αL−j+1µj−1   L X i=1 (2i − L − 1)Lσ˜L−j,i 2L−1_{(i − 1)!(L − i)!}  − µ_j = 0 . (1.36)

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7. SOLUTIONS AND CHAPTER SUMMARIES

Chapter Three

In this chapter we extend the lattice building when the underlying asset follows Merton-Bates jump diffusion process. We use the characteristic function (1.31) to calculate the first six moments. We then simplify the expression for the polynomial (1.36) into P (α) =                    µL− bL 2c X j=1 c(2j − 1) µL−2j αL−2j if L even, µL− bL 2c X j=1 c(2j) µL−2j+1 αL−2j+1 if L odd where c(j) is defined by c(j) = L X i=1 (−1)L−j(2i − L − 1)Lσ˜L−j,i 2L−1_{(i − 1)!(L − i)!} .

In addition, we use Sturm’s theorem to examine whether there exists any positive roots for the polynomial. We give particular cases when L = 4, L = 5 and L = 6. The Sturm’s chain for L = 4 and L = 5 are given below

S₄(α) =    − 9α4_{+ 10µ} 2α2− µ4, − 36α3+ 20µ2α, 36 5 µ4 µ₂ − 20µ2 ! α, − µ4    , S5(α) =    − 64µ1α4+ 20µ3α2− µ5, − 256µ1α3+ 40µ3α, − 10µ3α2+ µ5, 256 10 µ1 µ3 µ5− 40µ3 ! α, µ5    . Finally we find the range of λ - the jump intensity, that would guarantee positive real roots to the polynomial. Concentrating

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on quadrinomial L = 4 and we find that the range of λ is 0 < λ < 36

100

3 + 6η2+ η4 (1 + η2₎2 , where η is the mean jump.

Chapter Four

This chapter extends the methods in chapter three to pentanomial lattices L = 5. Here we realize that the number of positive real roots vary according to the Table 1.4 below.

Conditions on moments Number of real positive roots

µ1> 0 , µ3> 0 , µ5< 0 : q =    1 if 16µ1µ5> 25µ23, 0 otherwise. µ1> 0 , µ3< 0 , µ5> 0 : q =    1 if 16µ1µ5< 25µ23, 0 otherwise. µ1< 0 , µ3> 0 , µ5< 0 : q =    1 if 16µ1µ5> 25µ23, 0 otherwise. µ1> 0 , µ3< 0 , µ5< 0 : q = 1 µ1< 0 , µ3> 0 , µ5> 0 : q = 1 µ1= 0 , µ3> 0 , µ5> 0 : q = 1 µ1= 0 , µ3< 0 , µ5< 0 : q = 1 otherwise : q = 0

Table 1.4: The number of real positive roots, q, of the polynomial P (α) for

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BIBLIOGRAPHY

Chapter Five

In this chapter, we restrict the lattice to one with symmetrically

placed and evenly spaced nodes. This allows us to have

a lattice whose number of nodes at each time step don’t grow in exponential time. We develop explicit expressions for probabilities for quadrinomial and pentanomial lattices. We also study their properties when the approximation distribution is lognormal - that is, when the underlying asset follows the geometric Brownian motion. This allows us to come up with conditions that have to be satisfied for α and the probabilities to be real and positive.

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