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American Put Options

Daniil Kolesnikov

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Daniil Kolesnikov

Student nr: 10604189

Email: danielsidik@gmail.com

Date: April 23, 2015

Supervisor: Prof. Dr. Roger Laeven Second reader: Andrei Lalu

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This document is written by Student Daniil Kolesnikov who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervi-sion of completion of the work, not for the contents.

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Pricing of American Put Options — Daniil Kolesnikov iii

Abstract

This thesis considers the models for pricing American put options. The models applied are the binomial tree model, the Monte-Carlo least squares simulation method and the decomposition method. The first two methods are widely known while the latter one is is not so commonly applied. It was first used by Kim (1990) and it allows to decompose the value of the American put option into a corresponding European put option price and the early exercise premium. The thesis presents the numerical solution of the Kim equation using quadrature for-mulas and involves Newton-Raphson iteration to compute the optimal exercise boundaries at each time step. All three models are fitted to the real data and the results are compared to determine the most efficient method. The sensitivity of the models to particular parameter choices is analyzed and examined.

Keywords American put option, Binomial tree, Black-Scholes, Decomposition method, Early exercise bound-ary, Integral equations, Least-squares method, Monte-Carlo simulation, Volatility

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Preface vii

Introduction 1

1 Numerical Approaches for Pricing American Options 3

1.1 Binomial Tree Model . . . 3

1.2 Monte-Carlo model (LSM) . . . 5

1.3 Decomposition Method . . . 6

2 Practical Implementation 12 2.1 Data Description . . . 12

2.2 Binomial Tree Model . . . 14

2.3 Least-Square Monte-Carlo Model (LSM) . . . 15

2.4 Decomposition method . . . 15 3 Results 16 3.1 Review of Results. . . 16 3.2 Sensitivity analysis . . . 19 Conclusion 22 APPENDICES 23 A Derivation of equation (1.3) 23 B Proof of equation (1.30) 25 C Risk-free interest rates 26 D Code Listings 28 D.1 MATLAB Binomial Tree Method Code . . . 28

D.2 MATLAB Least-Square Method Code . . . 29

D.3 Wolfram Mathematica Decomposition Method Code . . . 31

Bibliography 33

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List of Tables

2.1 Option contract specifications . . . 12

2.2 Option contracts average prices for moneyness/maturity categories (in $) . . . 13

3.1 Option contracts standard errors of estimate in binomial tree model . . . 16

3.2 Errors relative to real option prices in binomial tree model . . . 17

3.3 Option contracts standard errors of estimate in LSM . . . 17

3.4 Errors relative to real option prices in LSM . . . 18

3.5 Option contracts standard errors of estimate in Decomposition method . . . 18

3.6 Errors relative to real option prices in Decomposition method . . . 18

3.7 Overall relative error excluding out-of-the-money options for all methods . . . 19

3.8 Option contracts standard errors of estimate in LSM with a linear regression . . . 20

3.9 Relative change in errors between simple regression and Laguerre polynomials in the LSM pricing approach. . . 20

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2.1 Daily Yahoo! Inc. returns 20/12/13 - 19/12/14. Source: Yahoo! Finance. . . 14 3.1 Critical stock price B(t) for an option with T = 345, K = 55 and S0= 44.05. . . 19

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Preface

I wish to express my sincere gratitude to my supervisors Professor Dr. Roger Laeven and Andrei Lalu. I am thankful to them for sharing their expertise and their valuable guidance.

I am also grateful to my wife and my parents for help, moral support and encour-agement.

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Introduction

An American put option is a derivative contract that gives its holder the right to sell an asset for a certain price prior to and including the maturity date. Most of the options traded on exchanges are American style (Hull, 2012). These types of derivatives can be found in all financial markets: individual equity, equity index, currency, energy, agriculture, precious metals, credit, insurance, etc.

The option contract trading volume has increased by 3.1% from 2013 to 2014 resulting in 9.7 billion contracts traded. Furthermore, individual equity is the largest category with the volume of options traded, around 6.5 billion in 2014.1

While it is never optimal for an American call option contract on non-dividend paying stock to be exercised prior to maturity, hence can be priced as a European call, no closed form solution exists for American put, except for perpetual put on a non-dividend paying stock. Not only the valuation but the optimal exercise of the American options is one of the most puzzling problems in derivatives finance and it continues to be of a great interest to finance theorists.

One of the most famous and influential works on option pricing was written by Fischer Black and Myron Scholes in 1973, which discusses the existence of unique and rational price for European options. The work is based on the assumption of a continuous change of the underlying asset following a geometric Brownian motion with a drift. The work was refined by Merton the same year to show that Black-Scholes equation can be applied to American style options on non-dividend paying stock, but the closed form solution for American put does not exist.

Another common approach to pricing options was proposed by Cox, Ross and Rubinstein in 1978, known as a binomial tree model. The model relies on a discretization of the various price paths for the underlying asset to find the value of the option. It is widely used due to it simplicity and applicability. A paper concerning American put pricing was written by Brennan and Schwartz (1977) which applied a finite difference method to valuate the options. The method approximates the differential equations that describe the development of the option price by a set of difference equations. The price of the American put option is obtained after these equations are solved iteratively.

One of the most recent methods which uses Monte-Carlo simulation for American-style options was developed by Longstaff and Schwartz (2001). The idea is to simulate the underlying asset’s price paths and using backward induction at each discrete tilmestep to determine by using regressions the value of continuation and based on this, decide whether it is optimal to exercise or not by comparing value of continuation and the immediate exercise value.

There are other models available to price American put options, but one is of a particular interest. It was first proposed by Kim (1990). He suggests the decomposition of the American put into a European put and early exercise premium. The European part is calculated using Black-Scholes model and the early exercise premium is computed as an integral containing the critical stock price (the price below

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which the option should be exercised). Only a few papers implemented a numerical solution to evaluate the integral and that is done in this paper. This method is of great interest because unlike many other methods it provides the boundary for optimal exercise during the life of the option.

This thesis provides the solution to two problems: applying the appropriate models for the pricing of American put options and evaluating the performance of these models by analyzing and comparing the pricing results. Hence, answering to the main question of this thesis: “How well do the option pricing models reproduce the American put option market price and which of the methods is the most efficient?”

In order to implement this research the performance of three models was examined using a sample of American put options on Yahoo!Inc. non-dividend paying stocks by fitting the prices of these options during a one-month period. The models applied are the binomial tree model, the least squares Monte-Carlo model and the decomposition model. While the first two are commonly used the latter is infrequently applied which makes this research relevant and up-to-date.

The structure of this thesis is as follows: the first chapter contains the literature review of the existing option pricing models as well as theoretical specifications. The second chapter is dedicated to the practical implementation of the models and justification of the choices made. The third chapter contains the results and their analysis, including the analyses of the models’ sensitivity to the choices made earlier. Finally, in the last chapter the conclusion is provided.

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Chapter 1

Numerical Approaches for Pricing

American Options

This chapter contains a literature review. It includes the detailed overview of the existing pricing models for American put options, different sections of this chapter describe different methods.

1.1

Binomial Tree Model

One of the most popular and useful methods for computing option prices is the binomial option pricing model that was initially proposed by Cox, Ross and Rubinstein (1979).

The basic idea of this model is constructing a binomial tree – that is a graph that represents various possible paths that the stock price can follow during the life of an option. The underlying assumption is that the stock price follows a geometric Brownian motion and that the valuation is done under risk neutrality. Binomial tree is often used for pricing American options that, unlike European options, can be exercised at any time before the maturity date.

The paths that the stock price follows are determined the following way: the stock price S0 can

either go up to a new level S0u or go down to the level S0d. The parameters u and d are determined

by u = eσ √ ∆t (1.1) d = e−σ √ ∆t (1.2)

where σ is a stock volatility and ∆t – the time step.

After the values of the stock are obtained at the final node the put option price is determined by max(0, K − ST). Then the price of the option at node T − 1 is found using the formulas1:

fT −1= e−r∆t[pfu+ (1 − p)fd] (1.3)

where

p =e

r∆t− d

u − d (1.4)

The binomial tree relies on a discretization of the stock price path at a number of time steps between the evaluation and expiration dates of the option. Due to the fact that the option is American style

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at each node (except the node at time T ) it should be decided whether to exercise early. We compare K − Stwith the value received from (1.3) and if it exceeds the f then the put option price at this node

is replaced by the value of the difference K − St, otherwise leave the value f as an option price at this

node. The valuation of the option price is performed iteratively, working backwards from each of the final nodes towards the first node. The value obtained at each node represents the option price at that moment in time and at that point in the tree. The value of the option at the initial node represents the price of the option at the time of the valuation.

The advantages of this approach are its ease of implementation and its flexibility when applied to different types of options, e.g. Bermudan options, or longer-dated options with dividend payments. The main disadvantage is a low speed of calculations when increasing the amount of time steps in order to achieve a more accurate result. Another disadvantage is that the change in the stock price doesn’t go in discrete time steps, it goes on continuously, while the model assumes the opposite.

Another method used for American put options pricing is the finite difference method intro-duced by Brennan and Schwartz (1977). Unlike the extension of the Black Scholes model implemented by Merton (1973) that allows to price a perpetual American put option on a non-dividend paying stock, the finite difference method solves the problem of pricing a finite maturity put that is subject to dividend payments on the underlying stock (Brennan and Schwartz, 1977).

The method approximates the continuous-time differential equation that characterises the be-haviour of the option price over time by several discrete-time difference equations. After the partial differential equation is expressed in a discrete form, through finite differences approach, the behaviour of the option price is modelled using the grid with y-axis representing the price of the option from 0 to Smax2, while the x-axis representing time from 0 to maturity. Each node of the grid corresponds with

the stock price at a certain time point. Equations containing the stock prices for some of the nodes are derived afterwards and solved simultaneously to obtain a set of option prices one of which is then chosen (Hull, 2012; Wilmott, Howison, Dewynne, 1995).

According to Hull (2012) this approach can be used for the same derivative pricing problems as binomial and trinomial tree approaches are used, however due to its considerable computational costs, that similar to the binomial tree method come from the numerous amount of steps needed in constructing a grid for determining the prices of the option, it is not applied very frequently. Finite difference methods can sometimes require complicated algorithms for solving immense linear systems of equations that also leads to computational cost increase.

Binomial tree and finite difference methods are widely used to price American options. Other pop-ular approaches are the least squares approach involving the Monte-Carlo simulation (Longstaff, Schwartz 2001), the randomisation method by Carr (1998) and the decomposition method by Carr, Jarrow and Myneni (1992) that will be presented below.

Models considered below are known as approximation methods and became popular due to their relative simplicity and applicability.

2S

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Pricing of American Put Options — Daniil Kolesnikov 5

1.2

Monte-Carlo model (LSM)

As mentioned above, American options are commonly priced using an approximation method called least-squares approach proposed by Longstaff and Schwartz (2001). This method involves Monte-Carlo simulations to obtain derivatives prices. One of the advantages of this model is the ability to calculate the value of the option when the option payoff depends on multiple factors.

Assuming an underlying complete probability space (Ω, F , Q) and a finite time horizon [0, T ], where the state space Ω is a set of all possible states of the stochastic economy in the time frame of T with a typical element ω representing a sample path. F is a sigma field of distinguishable events at time T, P is a physical measure which under no-arbitrage paradigm admits the existence of an equivalent martingale measure Q for this economy denoting the risk-neutrality. Define S as the vector of prices of the underlying asset that follows a Markov process3, S ∈ R+.

At the maturity date the optimal strategy for any option is to exercise it if it is in the money4. However, before the maturity date the optimal strategy is the comparison of immediate exercise value with the value of expected cashflows if continuation takes place, and exercise if the immediate value exceeds the value of expected cashflows. C(ω, s; t, T ) – is a path between [t, T ] of expected cashflows generated by the option, based on the condition that the option is not exercised at or earlier than time step t and that the optionholder follows the optimal stopping strategy for all s, t < s ≤ T . The crucial factor of optimally exercising an American option is finding the expected value of continuation. Consider the time grid 1, . . . , k, . . . , K. At time tk the continuation value F (ω; tk) can be defined as:

F (ω; tk) = EQ   K X j=k+1 exp − Z tj tk r(ω, s)ds ! C(ω, tj; tk, T ) | Ftk  . (1.5)

The least square approach approximates the conditional expectation function F (ω; tk) at tK−1, tK−2, . . . , t1.

For example, assume, that at time prior to maturity tK−1the unknown functional form of F (ω; tK−1)

in (1.5) can be presented as a linear combination of a countable set of FtK−1– measurable basis func-tions5 (Longstaff and Schwarz 2001). The value of continuation F (ω; tK−1) can be represented as

F (ω; tK−1) = ∞

X

j=0

ajLj(S), (1.6)

where aj are real constants, Lj(S) – countable set of polynomial functions of underlying asset price,

e.g. weighted Laguerre polynomials.

The Least Squares Method (LSM) algorithm is as follows:

1. Generate a number of sample random price paths the underlying stock may follow. The prices are log-normally distributed ln N (µ, σ2).

2. Split the time from valuation date to maturity in K discrete times 0 < t1 ≤ t2 ≤ t3 ≤ · · · ≤ tK

= T .

3. Working backwards in time from the maturity date get the approximation of F (ω; tK−1) using

first M < ∞, M ∈ N Laguerre basis functions, and denote this approximation as FM(ω; tK−1).

3A Markov process is a type of stochastic process where only current value of the variable is relevant for predicting

the future, while the past history and the way the present value have been achieved are irrelevant (Hull 2012).

4

For a put option this occurs whenever the strike price is above the market price of the underlying.

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4. Estimate the approximation FM(ω; tK−1) by regressing the discounted cash flows C(ω, s; tK−1, T )

onto polynomial functions for the in-the-money paths, and denote the fitted value as bFM(ω; tK−1).

5. Decide whether early exercise at time tK−1 is optimal for an in-the-money ω by comparing the

immediate exercise value with the fitted value bFM(ω; tK−1), if immediate exercise value is greater

than the continuation value, choose to exercise. 6. Do the same for each in-the-money path.

7. Proceed recursively to time step tK−2 and repeat the procedure until the exercise decision is

made at each time step along each path.

8. Start at time 0 and move forward along each path until the first stopping time reached, then discount the cashflows back from stopping time to time 0.

9. Take average of discounted cashflows over all simulated paths ω.

This approach is computationally efficient, easy to implement and accurate. (Longstaff and Schwartz 2001). This method is used for options with complicated features, e.g. exotic options, and for path dependant options, like Asian options and lookback options.

1.3

Decomposition Method

An alternative approximation method for American option prices is the decomposition method pro-posed by Carr, Jarrow and Myneni (1992). The authors suggest to decompose the American option value into the corresponding European style option price and the early exercise premium. As opposed to previous approximations by Barone-Adesi and Whaley (1988) and MacMillan (1986), the paper by Carr et al. provides a precise definition of the early exercise premium. This decomposition was also used in the earlier works of Kim (1990) and Jacka (1991) but it was derived through different approaches. Notably, Carr et al. (1992) show that the early exercise premium is equivalent to the value of an annuity that pays interest at a certain rate whenever the stock price is low enough so that early exercise is optimal.

The model assumes no-arbitrage opportunities, continuos trading, a constant interest rate r > 0 and stock price St with continuous dividends that follows a geometric Brownian motion:

dSt

St

= (µ + δ)dt + σdWt, for all t ∈ [0, T ], (1.7)

µ ∈ R+ σ ∈ R+

where µ denotes the expected return per time unit, and σ is the volatility per time unit, and δ is a constant rate of continuous proportional dividends. The term dWt denotes increments of a standard

Wiener process on the time set [0, T ] and a complete probability space (Ω, F , Q).

Based on McKean (1965), Carr et al. (1992) represent the American put value as a function of the exogenous variables K, S0, σ, T and the exercise boundary. This boundary is the time path of critical

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Pricing of American Put Options — Daniil Kolesnikov 7

American put should be exercised early. Let Ptdenote the price of an American put option with strike

price K and maturity date T at time t ∈ [0, T ]. We obtain:

if St≤ B(t), then Pt= max[0, K − St], (1.8)

and if St> B(t), then Pt> max[0, K − St]. (1.9)

The exercise boundary is independent of the current stock price and is a smooth, non-decreasing function of time t ending at the strike price K (Carr et al. 1992). The put option value is also a function defined as P (S, t) with the domain D ≡ (S, t) ∈ [0, ∞) × [0, T ]. B(t) divides the put value function’s domainD into a stopping region S ≡ [0, B(t)]×[0, T ] and continuation region C ≡ (B(t), ∞)×[0, T ]. Equation (1.8) states that the put value function P (S, t) equals its exercise value in the stopping region, while the inequality (1.9) demonstrates that put is worth more “alive” than “dead” in the continuation region.

The Black-Scholes partial differential equation in the continuation region C is: σ2S2 2 ∂2P (S, t) ∂S2 + (r − δ)S ∂P (S, t) ∂S − rP (S, t) + ∂P (S, t) ∂t = 0 (1.10) for (S, t) ∈C . According to McKean (1965), the exercise boundary B(t) and the American put value function P (S, t) solve a free-boundary problem represented by equation (1.10) subject to a number of conditions:

P (S, T ) = max[0, K − S], (1.11) lim S↑∞P (S, t) = 0, (1.12) lim S↓Bt P (S, t) = K − B(t), (1.13) lim S↓Bt ∂P (S, t) ∂S = −1. (1.14)

Equation (1.11) is the payoff at maturity. Equation (1.12) imposes that the value of the American put is zero when stock price tends to infinity. Conditions (1.13) and (1.8) are called value-matching conditions and they indicate that the put price is continuous across the exercise boundary. Moreover, the high contact conditions (1.14) and (1.8) introduced by Samuelson (1965) and proved by McKean (1965) state that a put option function P (S, t) touches the payoff function tangentially. In case this condition is violated there would be a possibility to arbitrage (in case the condition > −1) or the exercise boundary would be below its optimal level (in case the condition < −1) (Seydel, 2009). Equations (1.13) and (1.14) are jointly known as smooth fit conditions.

All of the above leads to the main decomposition theorem of the Carr et al. (1992) paper:

Theorem 1 (Main Decomposition of the American Put, Carr et al.(1992)). On the continuation region C the American put value P0 can be decomposed into the corresponding European put price p0

and the early exercise premium e0:

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The European put price part is calculated using Black-Scholes (1973): p0 = Ke−rTN (−d2(S0, K, T )) − S0e−δTN (−d1(S0, K, T )), (1.16) where d2(x, y, τ ) = ln(x/y)+(r−σ 2/2)τ σ√τ , d1(x, y, τ ) = ln(x/y)+(r+σ2/2)τ

σ√τ . While the early exercise premium

has the following form:

e0 = rK Z T 0 e−rtN −ln(S0/B(t)) + (r − σ 2/2)t σ√t ! dt, (1.17) where N (x) ≡Rx 0 exp(−z2/2)

2π dz is the standard normal distribution function.

When including the dividend payouts the early exercise premium takes a slightly different form (Kim, 1990): e0 = Z T 0 h rKe−rtN (−d2(S0, B(t), t)) − δS0e−δtN (−d1(S0, B(t), t)) i dt (1.18)

When p0 and e0 from (1.16) and (1.18) are combined together as in equation (1.15) that results in the

following equation that was first introduced by Kim (1990):

P0 = Ke−rTN (−d2(S0, K, T )) − S0e−δTN (−d1(S0, K, T ))+ Z T 0 h rKe−rtN (−d2(S0, B(t), t)) − δS0e−δtN (−d1(S0, B(t), t)) i dt. (1.19) The value-matching condition (1.13) can be solved given that the initial boundary value B(0) equals the initial stock price S0:

Ke−rTN (−d2(S0, K, T )) − S0e−δTN (−d1(S0, K, T ))+ Z T 0 h rKe−rtN (−d2(S0, B(t), t)) − δS0e−δtN (−d1(S0, B(t), t)) i dt = K − S0. (1.20)

Because B(0) depends on the future boundary values B(t), t ∈ (0, T ], it should be determined by equalizing the terminal boundary value and the strike price B(T ) = K (or in case with dividends B(T ) = K min(1, r/δ)) and working backwards through time from T to 0. The intuition behind B(T ) = K is that B(T ) represents the critical stock price below which the option should be exercised, and at the maturity date the strike price K is exactly the price level below which the option is exercised.

After the boundary values are found they are inserted in equation (1.19) and the price of the American option can be obtained.

The practical implementation of the method threfore consists of three steps. The first two steps are required to find the values of the early exercise boundary B(t). In order to obtain the early exercise boundary values B(t) at each time step from equation (1.20) a Newton-Raphson iteration procedure should be applied (Kallast, Kivinukk 2003). When these values are obtained, the third step is the numerical integration of (1.19) to get the price of the American put option. This can be done with the use of quadrature formulas, as used in Sullivan (2000), Kallast and Kivinukk (2003).

Assume that the early exercise boundary function B(t) is well-defined, unique and continuous t 7→ B(t), where B(t) ≥ 0 and 0 ≤ t ≤ T . According to Kim (1990) the function t 7→ B(t) is

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Pricing of American Put Options — Daniil Kolesnikov 9

non-decreasing on [0, T ] and

B(T ) = K min(1, r/δ). (1.21)

For what follows denote

f (x, y, τ ) = rye−rτN (−d2(x, y, τ )) − δxe−δτN (−d1(x, y, τ )), (1.22)

p(x, y, τ ) = ye−rτN (−d2(x, y, τ )) − xe−δτN (−d1(x, y, τ )). (1.23)

For numerical approximation we split the integration interval [0, T ] into M subintervals [ti−1, ti]

with length ∆t = ti− ti−1 for i = 1, . . . , M . Define B(ti) = Bi for i = 0, . . . , M and since tM = T we

rewrite the boundary condition (1.21) as

BM = K min(1, r/δ). (1.24)

If there are no dividends to be paid then this equation takes the simple form of BM = K.

Now the Kim’s equations (1.19) and (1.20) look the following way:

P0 = p(S, K, T ) + Z T 0 f (S, B(t), t)dt, (1.25) K − B(t) = p(B(t), K, T − t) + Z T t f (B(t), B(ν), ν − t)dν. (1.26)

Due to (1.24) the value BM is known and putting t = tM −1 in (1.26) we get

K − BM −1= p(BM −1, K, ∆t) +

Z tM

tM −1

f (BM −1, B(ν), ν − t)dν. (1.27)

In order to get the value of BM −1 the integral can be approximated using the trapezoidal rule

(Press,1992). As a consequence, the equation transforms to

K − BM −1= p(BM −1, K, ∆t)+ ∆t 2 h f (BM −1, BM, ∆t) + f (BM −1, BM −1, 0 · ∆t) i , (1.28) and the value BM −1 can be obtained. Similarly, we generalize for values Bi, where i = M − 2, M −

3, . . . , 0 and get the following equation:

K − Bi = p(Bi, K, (M − i)∆t)+ ∆t 2 h f (Bi, BM, (M − i)∆t) + f (Bi, Bi, 0 · ∆t) i + ∆t M −1 X j=i+1 f (Bi, Bj, (j − i)∆t). (1.29)

The f (x, x, 0) value in equations (1.28) and (1.29) is determined by the limit process: f (x, x, 0) = lim τ →0+f (x, x, τ ) = 1 2(rK − δx). 6 (1.30) 6See Appendix B.

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Now we convert (1.29) by moving the left-hand side of the formula to the right and denoting this as F (Bi). We get: F (Bi) = Bi− K + p(Bi, K, (M − i)∆t) + ∆t 2 h f (Bi, BM, (M − i)∆t)+ f (Bi, Bi, 0 · ∆t) i + ∆t M −1 X j=i+1 f (Bi, Bj, (j − i)∆t). (1.31)

The approximations of the early exercise boundary values Bi can be obtained by applying the

Newton-Raphson iteration to (1.31). The Newton-Raphson iteration is a common method based on linear approximation for numerical evaluation of the routes of a complicated function. The approxi-mations of Bi can be found by:

Bi(k+1)= B(k)i − F (B (k) i ) F0(B(k) i ) . (1.32)

where Bi(k) are approximations of Bi of order k, where k = 0, 1, . . . and F0(Bi(k)) is the derivative

of F (Bi(k)) with respect to Bi. For the first approximation we apply the rule B(0)i = Bi+1 (i =

M − 1, M − 2, . . . , 0). The approximations are obtained for Bi starting from the boundary value at

the step previous to maturity BM −1, since the value of BM is already known from (1.24). According

to the rule above the approximation BM −1(0) is equal to BM, knowing BM and using (1.32) we get

approximations of BM −1 of several orders. Kallast and Kivinukk (2003) claim that three to four

iterations are enough to give a high accuracy of estimation. After we get the estimation of BM −1 we

move to BM −2 and apply BM −2(0) = BM −1, eventually getting the BM −2 estimation and so on unless

we reach the B0. After all of the boundary values are known we can solve (1.19) using the Simpson’s

rule (Press, 1992). The use of quadrature formulas implies the application of Simpson’s rule because it uses quadratic to approximate the integral while trapezoidal rule approximates the function with a straight line. Assuming that M is an even number, Kim’s equation (1.19) will have the form:

P0 = p(S, K, T ) + ∆t 3 [f (S, B0, 0∆t) + 4f (S, B1, ∆t)+ 2f (S, B2, 2∆t) + 4f (S, B3, 3∆t) + . . . + 2f (S, BM −2, (M − 2)∆t)+ 4f (S, BM −1, (M − 1)∆t) + f (S, BM, M ∆t)]. (1.33)

The value f (S, B0, 0) in the previous formula can not be determined uniquely, because parameter

representing time equals 0, hence leading to undeterminacy of the parameters d1(x, y, τ ) and d2(x, y, τ )

from (1.22). By (1.22) and (1.30) f (S, B0, 0) takes the form:

f (S, B0, 0) =          rK − δS, S < B0, 1 2(rK − δS), S = B0, 0, S > B0. (1.34)

Formula (1.33) is not used when S < B0, taking into account the definition of B(t) and the fact

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Pricing of American Put Options — Daniil Kolesnikov 11

S < B0. When S > B0 considering (1.33) and (1.34) we get:

P0 = p(S, K, T ) + ∆t 3 [4f (S, B1, ∆t)+ 2f (S, B2, 2∆t) + 4f (S, B3, 3∆t) + . . . + 2f (S, BM −2, (M − 2)∆t)+ 4f (S, BM −1, (M − 1)∆t) + f (S, BM, M ∆t)]. (1.35)

Changing the formula (1.33) according to (1.34) we finally approximate the Kim equation to calculate the American put option value.

This method is straightforward and directly solves the equation (1.20), obtaining the values of the early exercise boundary B(t), which is a relevant exercise. Earlier works (e.g., Carr, 1998) use simplified and approximated ways of calculating the early exercise premium and no paper solved the Kim equation as provided in the description above until suggested by Kallast and Kivinukk (2003). Therefore this method is interesting and is applied in this research.

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Practical Implementation

This chapter shows how some of the models described in chapter 1 are implemented. It contains data description, specifications of the models and justification of specific choices made for the implementa-tion. The chapter is divided into four sections. The first section serves as an introduction, consisting of the data review and explanation of the parameters values chosen, and each following section is devoted to one of the three methods implemented in this work.

2.1

Data Description

The testing of the models will be conducted using the historical stock prices of the Yahoo! Inc. (YHOO) company. The company’s shares are traded on the NASDAQ exchange and are a constituent of the NASDAQ 100 Index. The stock does not payout any dividends, that means that the δ parameter equals zero in the models considered. The options written on the company’s stocks are American style and traded on NASDAQ .

The option contracts price evaluation is done during a 30-working-days period from 22nd of De-cember 2014 till the 4th of February 2015 on a daily basis. The data for option prices was obtained

from Datastream software while the quotes for stock prices were taken from Yahoo finance website. Options with three different maturity dates T and several strike prices K on Yahoo! Inc. stocks are chosen. The strike prices were chosen that way to investigate the prediction ability of the models on the in-the-money, out-of-the-money and at-the-money options. At-the-money term implies precise equality of strike price and the stock price and since the are no such options during the evaluation dates, we will consider close-to-the-money options with the strikes closest to the stock price. The specifics of the put option contracts are provided in the Table 2.1.

Option type Evaluation date Maturity date Maturity Moneyness (%) Type 1 22.12.14 - 04.02.13 15.01.16 389 - 345 days 68.3 - 160.1 Type 2 22.12.14 - 04.02.13 17.07.15 207 - 163 days 68.3 - 160.1 Type 3 22.12.14 - 04.02.13 20.03.15 88 - 44 days 68.3 - 160.1

Table 2.1: Option contract specifications

Here the American put options are divided into types according to their maturity date, and all types have similar strike prices for comparison convenience. The type 1 option is a set of options with strikes from 35 to 70 and with similar maturity date. Type 2 and 3 options have the same strikes but different maturity dates.

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Pricing of American Put Options — Daniil Kolesnikov 13

The Table 2.2 contains the average prices of the options for each moneyness and maturity cate-gories. The options with the highest average price are long-term options, the smaller the maturity the lower the price of the option.

Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% 16.94 17.45 18.29 Close-to-the-money 87%- 114% 3 4.28 6.19 Out-of-the-money 73%-83% 0.37 1.11 2.13

Table 2.2: Option contracts average prices for moneyness/maturity categories (in $)

In order to calculate the price of the option several parameters are needed, such as the risk-free rate and the volatility of the stock. To determine the risk free rate for the maturities listed in Table 2.1 daily US Treasury yield curve rates are used1. This curve, which relates the yield on a security to its time to maturity is based on the closing market bid yields on actively traded Treasury securities in the over-the-counter market. These market yields are calculated from composites of quotations obtained by the Federal Reserve Bank of New York. Since there are no rates for the exact maturities of the options given, a linear interpolation of the risk-free rates is applied

r = r1+

d − d1

d2− d1

(r2− r1) (2.1)

where r is the desired rate, r1 and r2 are the rates for the periods preceding and exceeding the

desired rate respectively, d is the number of days for which the rate is calculated, d1 and d2 are the

corresponding maturities of r1 and r2.

The interpolation for the first option type is done depending on the evaluation date, since in some cases time to maturity exceeds one year, the rate is then determined based on 2 year and 1 year rates and in cases when maturity is less than a year – based on 12 month and 9 month rates, for the second option type interpolation is done between 6 month and 3 month rates and for the third type – 1 month and 3 month rates. After applying (2.1) the risk-free interest rates presented in Appendix C were obtained.

As for the volatility, the closing daily stock prices of Yahoo! Inc. for the year prior to valuation date were collected and the volatility was estimated from the historical data. The prices were determined only for working days, and there are 252 price observations. To estimate the volatility we first define: n + 1 to be the number of observations, Si is the stock price at the end of ith interval, i = 1, 2, . . . , n,

and τ – length of time interval in years2.

Then the returns of the stocks were calculated the following way:

ui = ln

Si

Si−1

!

for i = 1, 2, . . . , n (2.2)

The daily log returns of the Yahoo! Inc. are shown in the Figure 2.1. The common estimate, ˆs, of the 1

The quotes for US Treasury yield curve were downloaded from U.S. Department of the Treasury website.

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-­‐10.00%   -­‐8.00%   -­‐6.00%   -­‐4.00%   -­‐2.00%   0.00%   2.00%   4.00%   6.00%   8.00%   20-­‐Dec-­‐13   7-­‐ Jan-­‐1 4   22-­‐Jan -­‐14   5-­‐Feb -­‐14   20-­‐F eb-­‐14   6-­‐Mar -­‐14   20-­‐Mar -­‐14   3-­‐Ap r-­‐14   17-­‐Ap r-­‐14   2-­‐May -­‐14   16-­‐May -­‐14   2-­‐Ju n-­‐14   16-­‐Ju n-­‐14   30-­‐Ju n-­‐14   15-­‐Ju l-­‐14   29-­‐Ju l-­‐14   12-­‐Au g-­‐14   26-­‐Au g-­‐14   10-­‐S ep-­‐14   24-­‐S ep-­‐14   8-­‐O ct-­‐14   22-­‐O ct-­‐14   5-­‐No v-­‐14   19-­‐No v-­‐14   4-­‐Dec-­‐14   18-­‐Dec-­‐14  

Yahoo!Inc.  daily  log  returns  

Figure 2.1: Daily Yahoo! Inc. returns 20/12/13 - 19/12/14. Source: Yahoo! Finance.

ui standard deviation is provided by (Hull 2012)

ˆ s = v u u t 1 n − 1 n X i=1 u2i − 1 n(n − 1) n X i=1 ui !2 (2.3)

In the case of our underlying asset we get ˆs = 0.01999. The standard deviation σ can be estimated as ˆσ, where

ˆ σ = √ˆs

τ =

0.01999

p1/252 = 0.3173 (2.4)

That means that the annualized volatility of the Yahoo! Inc. stock is estimated at 31.73%. Assuming normally distributed returns, the standard error of this estimate is approximately ˆσ/√2n which will give us

0.3173 √

2 × 251 = 0.01413, (2.5)

i.e., the standard error of 1.41% per annum.

All the necessary parameters are now known and we can go to the first model applied.

2.2

Binomial Tree Model

For every type of option different amount of steps in the tree was chosen. They were selected that way so that the valuation is done every day during the life of the option. That makes the amount of steps in the tree vary from 389 to 345 for the first type of options, from 207 to 163 for the second type, and 88 – 44 for the third. Since it is nearly impossible to do the number of steps needed to obtain high accuracy in valuation of the option manually, the Binomial tree method was realized in MATLAB. The code of the program is presented in Appendix D.1.

The computational speed of this method is the best among the three methods. The results were obtained instantly in a fraction of a second.

The results produced by this code will be presented, discussed and compared with other models in Chapter 3.

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Pricing of American Put Options — Daniil Kolesnikov 15

2.3

Least-Square Monte-Carlo Model (LSM)

The Monte-Carlo simulation was performed by the steps described in Section 1.2. The data was simulated assuming normal distribution of the log-returns. As basis functions for estimating the con-tinuation value F (ω; tk) a set of three Laguerre polynomials suggested in the paper by Longstaff and

Schwartz were used. The polynomials have the following forms:

L0(X) = e−X/2, (2.6)

L1(X) = e−X/2(1 − X), (2.7)

L2(X) = e−X/2(1 − 2X + X2/2) (2.8)

To implement the Monte-Carlo simulation using simple least-square approach, the MATLAB nu-merical computing environment was used. To determine the American put price 10,000 stock price path simulations were made and the number of steps for each option type differed according to days to maturity, so that the valuation and decision whether to exercise or not took place every day. The MATLAB code for computing the price of the option is given in the Appendix D.2.

Apart from the Laguerre polynomials used to regress the discounted cashflows to obtain the con-tinuation value at each time step a simple linear regression of the form Y = a0 + a1X + a2X2 was

applied for comparison reasons. The results with a simple regression were slightly better for some of the options. Longstaff and Schwartz in their paper mentioned the possibility of getting better results for certain options using a simple regression instead of polynomials, especially for ordinary (options on non-dividend paying stock) types of options. We can conclude that the choice of polynomials for regression does effect the results, but not significantly. As a further research the implementation of these method with different polynomials on more complex options can be done.

The computational speed of this method is significantly worse than the one of the binomial tree method. And since this method and the decomposition method were performed in different computing environments it is hard to compare the speed of calculation, but the calculations with Monte-Carlo simulation were much faster than decomposition method, especially for the options with longer time to maturity. For the options with short maturity LSM showed worse speed results than other methods.

2.4

Decomposition method

The decomposition method was implemented the way described in Section 1.2.2. The results and graphs depicting the critical stock prices B(t) during the life of the option contracts are given in Chapter 3.

The computation of the prices was done on the basis of daily evaluation, that means that amount of steps were exactly the same as the number of days to maturity. The code describing the method was written in Wolfram Mathematica computational environment and is available in Appendix D.3. The computational speed of decomposition method is the worst amongst all applied methods for long-term options, while the prices for options with short life were obtained in fractions of a second. The slow speed of calculations for long-term options maybe explained by the fact that the numerous critical stock prices had to be computed to get the price of the put option contract which increases the amount of calculations for this method.

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Results

This chapter contains the results of all the models fitted to the real data, as well as the analysis of these results and their comparison between the models.

3.1

Review of Results.

Out of the three methods tested the binomial tree model is the simplest one, but it provides satisfactory fit for the Yahoo! Inc. American put option contracts. The results of the model predictions for the options with different maturities and moneyness levels are presented in the Table 3.1. In order to estimate the error of the option values obtained by the model the standard error of the estimate is used. This method is often used in statistics to estimate the accuracy of predictions. The standard error of the estimate is defined as

σest =

s PN

i=1(pi− ˜pi)2

N , (3.1)

where pi - is the actual American put option price, ˜pi - the predicted price and N is the number of

observations.

From the Table 3.1 close-to-the-money options have the biggest standard deviation of prediction errors comparing to other moneyness levels. That means that the errors for close-to-the-money options are spread out over a wider range of values around the mean because of the greater amount of strikes involved and different price levels. To better understand the prediction power of the binomial tree model let’s consider option pricing errors relative to the actual option contract prices expressed in percentage points. Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% 0.3592 0.2979 0.3026 Close-to-the-money 87%- 114% 0.5104 0.4036 0.3251 Out-of-the-money 73%-83% 0.2811 0.3827 0.4244 Overall error 0.4269 0.3745 0.3506

Table 3.1: Option contracts standard errors of estimate in binomial tree model 16

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Pricing of American Put Options — Daniil Kolesnikov 17

Error = pi− ˜pi pi

(3.2) The values in the Table 3.2 are the averages of the relative errors in various moneyness and maturity groups. The binomial tree model as the other applied models has a low prediction ability for

out-of-Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% -1.172% -0.417% 0.705% Close-to-the-money 87%- 114% -28.221% -12.459% -6.011% Out-of-the-money 73%-83% -72.205% -38.278% -21.527% Overall error -32.455% -15.903% -7.460%

Table 3.2: Errors relative to real option prices in binomial tree model

the-money options. The prices for these options are slightly bigger than 0 while all the models give the results very close to zero. That makes the error values reach nearly a 80%. The same applies to close-to-the-money options with short maturity, the prices are too small and even if the residuals are of the same scale as for in-the-money options the relative error will still be of a significant size. Due to immense errors in these categories the overall error rises dramatically. Note the fact that all except one errors are negative which indicates the underestimation of the prices.

To compare this model’s results, examine the least squares method (with Laguerre polynomials) presented in the Table 3.3. The standard deviation for in-the-money and close-to-the-money options is greater for all maturities for least squares method comparing to binomial tree which means binomial tree has its error values situated closer to the mean in these categories. The overall standard deviation is smaller for the binomial tree model than for the LSM, that leads to a conclusion of a better prediction power of the model.

Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% 0.3803 0.3219 0.3205 Close-to-the-money 87%- 114% 0.5179 0.4214 0.3535 Out-of-the-money 73%-83% 0.2777 0.3818 0.4151 Overall error 0.4354 0.3888 0.3635

Table 3.3: Option contracts standard errors of estimate in LSM

At the same time, Table 3.4 shows that LSM has smaller average relative errors than binomial tree in five groups out of nine. The errors are very close between the methods but LSM shows slightly better results. The total error determines the advantage of the binomial tree method over LSM for medium-term and long-term option contracts.

The biggest inaccuracies are again observed for out-of-the-money options which worsens the overall results.

The final method applied to forecast the American put option prices is the decomposition method by Carr et al. In their paper the method is described as a very fast method producing accurate results. The speed appeared to be the worst of all the methods, while the results were of a high precision. The

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Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% -1.114% -0.654% 0.348% Close-to-the-money 87%- 114% -28.291% -12.892% -6.311% Out-of-the-money 73%-83% -70.862% -37.974% -21.032% Overall error -32.139% -16.103% -7.537%

Table 3.4: Errors relative to real option prices in LSM

Table 3.5 contains the standard deviations of the estimate. This method performs the best results for

Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% 0.3587 0.2951 0.3053 Close-to-the-money 87%- 114% 0.5110 0.4052 0.3294 Out-of-the-money 73%-83% 0.2791 0.3815 0.4221 Overall error 0.4268 0.3745 0.3522

Table 3.5: Option contracts standard errors of estimate in Decomposition method

in-the-money options with short and medium maturity and the second result for long-term options in the same moneyness category. It also has the smallest standard deviation for medium-term out-of-the-money options. In all other categories decomposition method shows second smallest standard deviations loosing to one of the other methods. Decomposition method has best overall standard error for short-term options, has the same result as binomial tree for medium-term options and has second best result in long-term maturity category.

The relative prediction errors in the Table 3.6 indicate that decomposition method best evalu-ates the prices of close-to-the-money options for all maturities and out-of-the-money short-term and medium-term maturity options. Again it has the second best result in each of the other categories. The overall error points out the best performance of this method in predicting put contracts.

Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% -1.163% -0.446% 0.700% Close-to-the-money 87%- 114% -28.116% -12.372% -5.961% Out-of-the-money 73%-83% -70.746% -37.953% -21.310% Overall error -32.035% -15.786% -7.388%

Table 3.6: Errors relative to real option prices in Decomposition method

Another benefit of this method is providing a critical stock price curve beyond which it is optimal to exercise. For example, consider the Figure 3.1. It shows that at the evaluation date (time 0) it is optimal to exercise the put option when the stock price goes below $24.6, and at the maturity date B(t) = K and it is optimal to exercise if S < $55 which is true. Thus, this method provides valuable information about the exercise boundary which can be helpful for other research problems.

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Pricing of American Put Options — Daniil Kolesnikov 19

Figure 3.1: Critical stock price B(t) for an option with T = 345, K = 55 and S0 = 44.05.

All of the overall results were distorted by substantial errors of out-of-the-money options, which can be caused by the assumption of constant stock volatility for 30-day period covered. In practice volatility tends to vary between out-of-the-money, at-the-money and in-the-money options. This phenomenon is known as “volatility skew” or “volatility smile”. According to the results in our case volatility skew affects the out-of-the-money options’ prices the most, which leads to considerable errors in predictions. Consider another table with overall relative errors for all of the methods excluding out-of-the-money options. We can exclude these options because they do not have any intrinsic value and it is unlikely that stock price will go beyond that level and these options become in-the-money. Examine Table 3.7 that contains the overall error excluding the out-of-the-money category.

Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) Relative error Binomial

tree -19.205% -8.445% -3.325%

Relative error LSM -19.232% -8.813% -3.647%

Relative error

Decom-position method -19.132% -8.396% -3.296%

Table 3.7: Overall relative error excluding out-of-the-money options for all methods

The error for short-term options dropped on average by 40%, for medium-term options dropped by 46% and for long-term options by 53.5%. The inaccuracy for short-term options is still substantial due to the possible difference in volatility for different moneyness levels. Having excluded the out-of-the-money options we can remark that the decomposition method is indeed the most accurate one for the 30-day period covered and option types considered, with a slight advantage over the other methods.

3.2

Sensitivity analysis

Due to some of the choices that had to be made when working with the models, this section describes the sensitivity of some of these models to these choices and shows the results provided by alternatives. The least-squares method is implemented with the use of Laguerre polynomials described in Chapter 2. As discussed earlier this method was also tested using a simple linear regression Y =

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a0+ a1X + a2X2 in order to check the influence of the polynomial choice on the model’s outcome.

Recall the Table 3.3 and compare the results with the Table 3.8 below. The change in the regression

Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% 0.3814 0.3176 0.3242 Close-to-the-money 87%- 114% 0.5211 0.4081 0.3388 Out-of-the-money 73%-83% 0.2816 0.3814 0.4210 Overall error 0.4381 0.3806 0.3604

Table 3.8: Option contracts standard errors of estimate in LSM with a linear regression type didn’t crucially effect the standard deviation of the estimate performing better for all medium-term options and long-medium-term close-to-the-money options and worse for all others categories. Taking the overall standard deviation into consideration we can not conclude the application of what regression type resulted in a better output.

The error relative to the real option price is presented as a change in this error between the simple regression and Laguerre polynomials (Table 3.9), i.e. the cell representing short-term in-the-money options shows that the relative error decreased by 7.388% when using simple regression comparing to Laguerre polynomials. The majority of cells have negative values, hence the LSM with a simple regression produces better results than LSM with Laguerre polynomials. The overall error doesn’t indicate a significant improvement in the results, which corresponds with the results by Longstaff and Schwartz (2001). Moneyness Moneyness range (K/ ¯S) Maturity Short-term (44-88 days) Medium-term (163-207 days) Long-term (345-389 days) In-the-money 124%-145% -7.338% -16.214% -18.869% Close-to-the-money 87%- 114% 3.369% -1.790% -0.604% Out-of-the-money 73%-83% 2.911% -0.645% 0.143% Overall error including out-of-the-money options 3.024% -1.261% 0.128%

Overall error ex-cluding out-of-the-money options

3.163% -2.147% 0.094%

Table 3.9: Relative change in errors between simple regression and Laguerre polynomials in the LSM pricing approach.

Another parameter that could be changed in the models is the number of steps when the option price was evaluated. By default, the daily evaluation was chosen but some of the options were also evaluated more frequently. That didn’t lead to any notable changes in the outcome but changing this parameter downwards for LSM and especially for short-term options resulted in declining in the accuracy of the predictions. The accuracy of the binomial tree predictions also suffered a significant decrease when reducing the amount of steps. While dramatical decrease in steps (by 2-3 times) for decomposition method didn’t lead to any changes.

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Pricing of American Put Options — Daniil Kolesnikov 21

with the highest accuracy among the models tested, the only drawback is the time consumption. On the other hand the binomial tree provides surprisingly good predictions with the highest calculation speed.

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This thesis provides an overview of the methods for American put option contract pricing, applies them using the real data to predict option prices and compares the results to determine which of the methods is the most efficient and accurate. It also analyses the sensitivity of the models to the choices made when applying these models.

Apart from the common methods like binomial tree and least squares method tested in this thesis a not so broadly used decomposition model by Kim (1990) and Carr et al. (1992) was studied. Unlike most of the papers that use this method applying different approximations on the decomposition equation this thesis is based on the numerical solution of this equation using the Newton-Raphson iteration procedure (Kallast, Kivinukk 2003).

Among the three methods performed on the stock prices the decomposition method appears to be the most accurate. Furthermore, it provides additional information about the critical stock prices below which the put option should be exercised before its maturity. These can be useful not only for those who are interested in the price of the option but for analytical purposes as well. The results obtained are correspondent with the results in the previous literature. The only drawback revealed is the considerable computational costs of this method due to the use of the large number of evaluation time steps, but as determined later on the convergence of the results of this method happens on a less amount of time steps compared to the other methods.

Based on the results the binomial tree method justifies its popularity, it provides very accurate results in a very short period of time. The least squares Monte-Carlo simulation method produces satisfactory results as well.

In conclusion, the decomposition method proves to be the most efficient method from those con-sidered in this thesis with the data used for model testing for a certain time period.

For a further research these methods can be tested on either a different option contract type or the put options with a different kind of underlying asset or on the stock with dividend payments. Another possible extension of this thesis is incorporating the volatility skew in the option valuation.

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Appendix A

Derivation of equation

(1.3)

Consider a stock with a price S0 and an option on this stock with the current price f . During the life

of the option the stock price can either go up to S0u, u > 1 or go down to the value S0d, d < 1,

the maturity time of the option is T . The percentage increase when an up movement takes place is u − 1 and when going down 1 − d. The payoff from the option is fu when the stock price is S0u and fd

when the stock price is S0d. Suppose we have a portfolio consisting of a long position in ∆ shares and

a short position in one option. ∆ makes the portfolio riskless. If the upward movement in the stock price takes place, the value of the portfolio at time T is S0u∆ − fu, if there is a downward movement

then the value is S0d∆ − fd. The two are equal when

S0u∆ − fu = S0d∆ − fd

∆ = fu− fd S0u − S0d

(A.1)

The portfolio is riskless and to take no-arbitrage paradigm into account it should earn a risk-free rate. Equation (A.1) indicates that ∆ is the change in the option price to the change in the stock price. If risk-free rate is r then the present value of the portfolio is

e−rT(S0u∆ − fu)

The cost of making the portfolio is S0∆ − f . We can equalize these two

S0∆ − f = e−rT(S0u∆ − fu)

Deriving the f

f = S0∆(1 − ue−rT) + fue−rT

Substituting ∆ from (A.1), we get

f = S0 fu− fd S0u − S0d ! (1 − ue−rT) + fue−rT or f = fu(1 − de −rT) + f d(ue−rT − 1) u − d or f = e−rT[pfu+ (1 − p)fd] (A.2)

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where p = erTu−d−d. If we split the evaluation in time steps of length ∆t instead of evaluating in one step with the length T the equation (A.2) takes the form of (1.3).

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Appendix B

Proof of equation

(1.30)

The equation is f (x, x, 0) = lim τ →0+f (x, x, τ ) = 1 2(rK − δx). (B.1)

Proof. First consider:

f (x, x, τ ) = rKe−rτN (−d2(x, x, τ )) − δxe−δτN (−d1(x, x, τ )) (B.2) where −d2(x, x, τ ) = −ln(x/x) + (r − σ 2/2)τ σ√τ = (σ2/2 − r)τ σ√τ (B.3) −d1(x, x, τ ) = −ln(x/x) + (r + σ 2/2)τ σ√τ = − (σ2/2 + r)τ σ√τ (B.4) Now lim τ →0+f (x, x, τ ) = limτ →0+ " rKe−rτN ((σ 2/2 − r)τ σ√τ ) − δxe −δτN (−(σ2/2 + r)τ σ√τ ) # = rKe0N (0) − δxe0N (0) = 1 2rK − 1 2δx = 1 2(rK − δx) (B.5) The limit limτ →0+N ((σ

2/2−r)τ

σ√τ ) according to L’Hopital’s rule is equal to the limit of the derivative of

the numerator function divided by the derivative of the denominator function with respect to σ, we get limτ →0+N (√σττ) = N (0). The values of N(0) can be found in the cumulative normal distribution

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Risk-free interest rates

Date Risk-free rates (%)

Option type 1 Option type 2 Option type 3

22/12/2014 0,04803278 0,17582417 0,30827397 23/12/2014 0,02934426 0,15516483 0,28961643 24/12/2014 0,01 0,15450549 0,28832876 26/12/2014 0,01 0,11758241 0,28575342 29/12/2014 0,02672131 0,13214285 0,27189041 30/12/2014 0,03 0,12967032 0,25016438 31/12/2014 0,03803278 0,13071428 0,26726027 02/01/2015 0,02 0,12 0,26460273 05/01/2015 0,02721311 0,1087912 0,27150684 06/01/2015 0,02704918 0,10741758 0,25986301 07/01/2015 0,02688524 0,09703296 0,25810958 08/01/2015 0,02344262 0,08576923 0,23747945 09/01/2015 0,02 0,08461538 0,22608219 12/01/2015 0,02606557 0,09164835 0,19304109 13/01/2015 0,02590163 0,08131868 0,20186301 14/01/2015 0,0314754 0,0904945 0,1809041 15/01/2015 0,03 0,08 0,16 16/01/2015 0,02540983 0,06956521 0,16945054 20/01/2015 0,01950819 0,0772826 0,16752747 21/01/2015 0,01918032 0,07673913 0,16703296 22/01/2015 0,02442622 0,07619565 0,16653846 23/01/2015 0,02 0,06565217 0,16560439 26/01/2015 0,02377049 0,07402173 0,17395604 27/01/2015 0,02 0,07217391 0,16406593 28/01/2015 0,02 0,07152173 0,16357142 29/01/2015 0,01655737 0,06391304 0,16230769 30/01/2015 0,01311475 0,06184782 0,17093406 02/02/2015 0,01262295 0,06021739 0,16010989 03/02/2015 0,02 0,05967391 0,19538461 26

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CONCLUSION 27

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Code Listings

D.1

MATLAB Binomial Tree Method Code

function put_price=american_put(S, K, r, sigma, t, steps); S=44.7; K=20; r=0.00195385; sigma=0.3173; T=0.94794521; steps=346; R = exp(r*(T/steps)); Rinv = 1/R; u = exp(sigma*sqrt(T/steps)); d = 1/u; p_up = (R-d)/(u-d); p_down = 1-p_up; price = zeros(steps+1,1); price(1) = S*(d^steps); uu = u*u; for i=2:steps+1 price(i) = uu*price(i-1); end

val = max(0, (K-price));

for step=steps:-1:1

val = Rinv * ( p_up*val(2:step+1) + p_down*val(1:step) ); price = u*price(1:step);

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CONCLUSION 29

val = max(val,K-price); end

put_price=val(1);

D.2

MATLAB Least-Square Method Code

S = 4 4 . 0 5 ; %S t o c k p r i c e on t h e v a l u a t i o n d a t e K=70; %S t r i k e p r i c e T= 0 . 9 4 5 2 0 5 4 8 ; %Time t o m a t u r i t y i n y e a r s r = 0 . 0 0 1 8 4 6 1 5 3 8 ; %Risk−f r e e i n t e r e s t r a t e d i v =0; %d i v i d e n d s sigma = 0 . 3 1 7 3 ; %v o l a t i l i t y n s i m u l =10000; %number o f s i m u l a t i o n s n s t =345; %number o f t i m e s t e p s r=r−d i v ; z =−1; Smat=z e r o s ( n simul , n s t ) ; CC=z e r o s ( nsim ul , n s t ) ; %c a s h f l o w from c o n t i n u a t i o n CE=z e r o s ( n simul , n s t ) ; %c a s h f l o w from e x e r c i s e EF=z e r o s ( nsim ul , n s t ) ; %E x c e r c i s e f l a g dt=T/ ( n s t − 1 ) ; Smat ( : , 1 ) = S ; d r i f t =(r−sigma ˆ 2 / 2 ) ∗ dt ; s i g m a s q r d t=sigma ∗ s q r t ( dt ) ; f o r i =1: nsim ul , s t=S ; c u r t i m e =0; f o r k =2: n s t , c u r t i m e=c u r t i m e+dt ; s t=s t ∗ exp ( d r i f t +s i g m a s q r d t ∗ randn ) ; Smat ( i , k)= s t ; end end CC=Smat ∗ 0 ; %c a s h f l o w from c o n t i n u a t i o n CE=Smat ∗ 0 ; EF=Smat ∗ 0 ; %E x c e r c i s e f l a g s t=Smat ( : , n s t ) ; CE ( : , n s t )=max ( z ∗ ( s t −K) , 0 ) ;

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CC ( : , n s t )=CE ( : , n s t ) ; EF ( : , n s t )=(CE ( : , n s t ) > 0 ) ; par=z e r o s ( 3 , n s t ) ; %c o e f f o f b a s i s f u n c t i o n s f o r k=n s t −1: −1:2 , s t=Smat ( : , k ) ; CE ( : , k)=max ( z ∗ ( s t −K) , 0 ) ; %o n l y t h e p o s i t i v e p a y o f f p o i n t s a r e i n p u t f o r r e g r e s s i o n i d x=f i n d (CE ( : , k ) > 0 ) ; X=Smat ( idx , k ) ;

Y=CC( idx , k+1)∗ exp(− r ∗ dt ) ;

%R e g r e s s d i s c o u n t e d c o n t i n u a t i o n v a l u e a t n e x t t i m e s t e p % t o S v a r i a b l e s a t c u r r e n t t i m e s t e p %Using L a g u e r r e P o l y n o m i a l s L0 = @(X) o n e s ( s i z e (X ) ) ; L1 = @(X) (1−X ) ; L2 = @(X) 1/2∗(2 −4∗X−X . ˆ 2 ) ; %Handle t o f u n c t i o n g e n e r a t i n g t h e r e g r e s s i o n m a t r i x L = @(X) [ L0 (X) exp(−X/ 2 ) . ∗ L1 (X) exp(−X/ 2 ) . ∗ L2 (X ) ] ; R = L (X/S ) ; % R e g r e s s i o n m a t r i x a=R\Y; % L i n e a r r e g r e s s i o n s t e p

CC( idx , k)=R∗ a ; % Cash f l o w s a s p r e d i c t e d by t h e model %I f e x e r c i s e v a l u e i s more than c o n t i n u a t i o n v a l u e %c h o o s e t o e x e r c i s e

EF( idx , k)=CE( idx , k ) > CC( idx , k ) ; EF( f i n d (EF ( : , k ) ) , k +1: n s t ) = 0 ;

par ( : , k)=a ;

i d x=f i n d (EF ( : , k ) == 0 ) ;

%don ’ t keep t h e r e g r e s s e d v a l u e o f CC f o r n e x t u s e CC( idx , k)=CC( idx , k+1)∗ exp(− r ∗ dt ) ;

i d x=f i n d (EF ( : , k ) == 1 ) ; CC( idx , k)=CE( idx , k ) ; end p a y o f f s u m =0; f o r i =1: n s t , i d x=f i n d (EF ( : , i ) == 1 ) ; s t=Smat ( idx , i ) ; p a y o f f v e c=exp(− r ∗ ( i −1)∗ dt ) ∗max ( z ∗ ( s t −K) , 0 ) ; p a y o f f s u m=p a y o f f s u m+sum ( p a y o f f v e c ) ; end

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CONCLUSION 31

AmericanPP=p a y o f f s u m / n s i m u l ; d i s p l a y ( AmericanPP )

D.3

Wolfram Mathematica Decomposition Method Code

M = 345; r = 0.184615; δ = 0.0; S = 44.05; K = 70; t = T /M ; σ = 0.3173; T = 0.945205; d1[x , y , τ ] = τ  −δ+r+σ2 2  +log[x/y] σ√τ ; d2[x , y , τ ] = τ−δ+r−σ22 +log[x/y] σ√τ ;

n[a ] = CDF[NormalDistribution[], a]; Array[B, M + 1, 0]; B[M ] = K; f [x , y , τ ] = Kr exp[−rτ ]CDF [NormalDistribution[], −d2[x, y, τ ]] − δx exp(−δτ )CDF [NormalDistribution[], −d1(x, y, τ )] ; p[x , y , τ ] = y exp[−rτ ]CDF [NormalDistribution[], −d2[x, y, τ ]] − x exp[−δτ ]CDF [NormalDistribution[], −d1[x, y, τ ]] ; f [x , x , 0] = 12(Kr − δx); n2[x ] = PDF[NormalDistribution[0, 1], x]; dp[x , y , τ ] = − exp[−δτ ]n[−d1[x, y, τ ]]; df0 = −δ/2; df1[x , y , τ ] = exp[−δτ ] n2[d1[x,y,τ ]] σ√τ  δ − Kry − δn[−d1[x, y, τ ]] ! ; For " l = M − 1, l ≥ 0, l − −, B[l] = B[l + 1]; For  k = 0, k < 15, k++, B[l]= B[l] −  B[l] − K + p[B[l], K, t(M − l)t] +2t(f [B[l], B[M ], (M − l)t] + f [B[l], B[l], 0]) +

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t × Sum[f [B[l], B[j], (j − l)t], {j, l + 1, M − 1}]   1.0 + dp[B[l], K, (M − l)t] + t × Sum[df1[B[l], B[j], (j − l)t], {j, l + 1, M − 1}] + t 2(df1[B[l], B[M ], (M − l)t] + df0)  ; Print[B[l]] # P0 = p[S, K, T ]; P0 = P0 +3.0t f [S, B[M ], M × t] + 4 × Mod[M − 1, 2]f [S, B[M − 1], (M − 1) × t]; For  i = 0, i < M/2 − 1, i++, P0 = P0 +3.0t 4 × f [S, B[2 × i + 1], (2 × i + 1) × t] + 2 × f [S, B[2 × i + 2], (2 × i + 2) × t]); Print[P0] 

tlist = Table[B[i], {i, 0, M }]; ListLinePlot[tlist]

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Bibliography

Acworth, W. (2015, March). FIA ANNUAL VOLUME SURVEY. Retrieved from

https://fimag.fia.org/articles/2014-fia-annual-global-futures-and-options-volume-gains-north-america-and-europe-offset

Barone-Adesi, G., & Whaley, R. E. (1988). On the valuation of American put options on dividend-paying stocks. Advances in Futures and Options Research, 3(1): 1-13.

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The journal of political economy, 81(3): 637-654.

Carr, P. (1998). Randomization and the American put. Review of Financial Studies, 11(3): 597-626. Carr, P., Jarrow, R., & Myneni, R. (1992). Alternative characterizations of American put options.

Mathematical Finance, 2(2): 87-106.

Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of financial Economics, 7(3): 229-263.

Hull, J. C. (2012). Options, Futures, and Other Derivatives. Prentice Hall

Jacka, S. D. (1991). Optimal stopping and the American put. Mathematical Finance, 1(2): 1-14. Kallast, S., & Kivinukk, A. (2003). Pricing and hedging American options using approximations by

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least-squares approach. Review of Financial studies, 14(1): 113-147.

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Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (1992). Numerical Recipes in C: The Art of Scientific Computing (2nd ed.), New York, NY: Cambridge University Press.

Samuelson, P. A. (1965). Rational theory of warrant pricing. Industrial management review 6(2): 13-32. Seydel, U.R. (2009). Tools for Computational Finance (4th ed.). Berlin: Springer.

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