**School of Education, Culture and Communication**

**Division of Applied Mathematics**

### Bachelor Thesis in Mathematics / Applied Mathematics

### Option Pricing and Early Exercise Boundary of

### American Options under Markov-Modulated

### Volatility

### By

### Danny Zina

### 26th February, 2020

### Kandidatarbete i matematik / tillämpad matematik

### DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

**School of Education, Culture and Communication**

**Division of Applied Mathematics**

Bachelor thesis in mathematics / applied mathematics

Date:

2020-02-26

Project name:

Option Pricing and Early Exercise Boundary of American Options under Markov-Modulated Volatility

Author:

Danny Zina, Student

Supervisor:

Ying Ni, Senior lecturer

Reviewer:

Milica Ran£i¢, Senior lecturer

Examiner:

Anatoliy Malyarenko, Professor

Comprising:

Abstract

The CRR binomial model is one of the most important models in nancial math-ematics. In this thesis we consider an extension to this model with Markov switching-state volatility. We present a detailed algorithm for obtaining early exercise boundaries for American options, as well as, fair prices for both Ameri-can and European options. To provide extensive numerical results, we experiment with dierent variations of the parameters and analyze the results. In particu-lar, we study properties of the resulting early exercise boundaries. Moreover, we give three approximation methods for the pricing of European options under this model.

Keywords: Pricing American Options, Early Exercise Boundary,

Acknowledgments

I would rst like to thank my thesis supervisor Ying Ni for her consistent guidance and immense support and for introducing me to the topic of this thesis. I would also like to thank Milica Ran£i¢ for her remarks on this thesis, which elevated my mathematical writing in this report. I would also like to thank Marko Dimitrov for his comments that helped me to enhance my report. Finally, I would also like to express my sincere gratitude to my girlfriend for her unwavering love and support throughout my studies from the start.

## Contents

1 Introduction 7

1.1 Background and Literature Review . . . 7

1.2 Contribution of This Thesis . . . 10

1.3 Outline . . . 11

2 A Review of Classical CRR Model 12 3 Model Implementation 17 3.1 Extended CRR Model . . . 17

3.2 Model Assumptions . . . 22

3.3 Algorithm . . . 22

3.4 Technical Aspects . . . 23

4 Numerical Results and EEBs 26 4.1 Correction of Reference Paper . . . 26

4.2 Option Pricing under Extended CRR . . . 27

4.3 Approximations for Extended CRR Prices . . . 30

4.4 Properties of EEB for American Put Option . . . 32

4.5 Properties of EEB for American Call Option with Dividend . . . 35

5 Conclusion 37 5.1 Summary of Results . . . 37

5.2 Further Research . . . 38

A Code 42 A.1 MATLAB Function . . . 42

A.2 Sub-Function . . . 47

B Proofs 48 B.1 Proof of Proposition 3 . . . 48

## List of Figures

3.1 three-steps tree of the extended CRR model with two volatility states i = H, L. . . 20 4.1 EEB of ATM American puts with underlying S(0) = 100, maturity

T = 0.25, and risk-free rate r = 0.05. . . 32 4.2 EEBs of ATM American puts with underlying S(0) = 100, maturity

T = 0.25, and risk-free rate r = 0.05. . . 33 4.3 EEBs of ATM American puts with underlying S(0) = 100, maturity

T = 0.25, and risk-free rate r = 0.05. . . 34 4.4 EEBs of ATM American puts with underlying S(0) = 100, maturity

T = 0.25, and risk-free rate r = 0.05. . . 34 4.5 EEBs of ATM American calls with underlying S(0) = 100, maturity

## List of Tables

4.1 Prices of options with, N = 25, S(0) = 100, r = 0.05, σH = 0.4,

σL= 0.2, pHH = 0.9834 and pLL = 0.9889. . . 27

4.2 Prices of options with, N = 50, S(0) = 100, r = 0.05, σH = 0.4,

σL= 0.2, pHH = 0.9 and pLL = 0.95. . . 28

4.3 Prices of options with, N = 50, S(0) = K = 100, r = 0.05, T = 0.25. 29 4.4 Prices of European call options under the extended CRR, along with

three dierent approximations, with the following xed parameters, N = 50, S(0) = K = 100, r = 0.05, T = 0.25. . . 32 B.1 Two portfolios that include one option each, both options have strike

K and maturity T . . . 48 B.2 Two portfolios that include one option each, both options have strike

### Abbreviations and Acronyms

AT M At-The-Money

BSM Black-Scholes-Merton CRR Cox-Ross-Rubinstein EEB Early Exercise Boundary IT M In-The-Money

OT M Out-of-The-Money

T P M Transition Probability Matrix w.p. with probability

### Symbols

B Amount invested in the money market κ Number of volatility states

P Transition probability matrix σ Volatility or Standard deviation σ2 Variance

τ Time of exercise

d Down-movement factor

∆ Amount invested in the stock ∆t Time increment

H High state K Strike price L Low state

n Intermediate time step N Total number of steps π Stationary distribution S(n) Stock price at time-step n S0 Risk-free bond price

T Maturity

u Up-movement factor V (n) Value at time-step n

## Chapter 1

## Introduction

Financial derivatives are nowadays considered as one of the most important pil-lars of the nancial world. They are traded on many exchanges throughout the world. Whether it is for hedging, arbitrage or speculation, nancial derivatives' transactions constitute a huge proportion of all the transactions in the nancial markets. A nancial derivative is a nancial instrument whose value depends on another underlying variable which is usually more basic (Hull, 2003). It is worth mentioning that some derivatives depend on another derivative as an underlying variable.

Options derivatives have grown more popular in the nancial market since the Chicago Board Options Exchange was established in 1973 (Hull, 2003). Today many option types and many options' trading strategies exist.

### 1.1 Background and Literature Review

Like other nancial derivatives options depend on an underlying asset, for example, an equity stock. All sorts of options exists in pairs, e.g. European options exist in the following pair:

• A European call option gives the holder the right but not the obligation to buy the underlying asset on a certain date in the future, known as the maturity T and for a certain price called the strike price K or the exercise price.

• A European put option gives the right but not the obligation to sell the underlying asset for a strike price, on the maturity date.

The most common sorts of options are European options and American options. Unlike the European options where the owner of an option can choose to exercise

on maturity, American options gives its owner further control over the exercise right, with the ability to exercise at any time up to the maturity date. This broader choice comes with an additional price added to the value of the option (Hull, 2003).

Option-pricing models have been the subject of numerous researches. In 1973,
the revolutionary Black-Scholes-Merton (BSM) (Black and Scholes, 1973; Merton,
1973) model constituted a complete theoretical description for pricing standard
European options1_{. It was not until 1979 that pricing American options became}

practically possible through the work of JC Cox, SA Ross and M Rubinstein on the binomial trees model (Cox et al., 1979), later to be known as the CRR model. Similar to BSM, CRR model assumes that the drift and the volatility are constants and that the stock price process S(t) follows a geometric Brownian motion, i.e.

dS(t) = µS(t)dt + σS(t)dZ(t), (1.1)

where µ is the drift of the stock price and σ is the volatility of the stock price, and dZ(t) is a standard Wiener increment, dZ(t) ∼ N(0, dt). Additionally, the CRR discretizes in time the stochastic process above so that the stock price follows the process,

∆S(t) = µS(t)∆t + S(t)σ√∆t,

where ∼ N(0, 1) and ∆t = T/N is the time increment, where N is the number of the equidistant sub-intervals of the nite time interval [0, T ]. The time epochs in this discrete setting becomes 0, ∆t, 2∆t, . . . , N∆t. For simplicity, we enumerate the time epochs as time-steps 0, 1, 2, . . . , n − 1, n, n + 1, . . . , N.

The solution of the stochastic dierential equation (1.1) which is known from Black and Scholes (1973) and Merton (1973), can also be discretized as follows,

S(n + 1) = S(n) expn∆tµ − σ

2

2

+ σ√∆to, (1.2) where n is the intermediate step. Furthermore, the random variable can be approximated by limiting it, to take only values from the set {−1, 1}. It is proven in Cox et al. (1979), that in this setting and as ∆t approaches zero, Equation (1.2) is reduced to,

S(n + 1) = S(n) expnσ√∆to. (1.3) Equation (1.3) yields the binomial tree process, which can be easily evaluated numerically. The CRR model captures the essence of the stock movement to a highly accurate level and it is computationally possible for a reasonably large

1_{By a standard option we refer to an option with a stock as the underlying variable and a}

number of steps due to its combining property. It can be shown that the number of possibilities after n steps is equal to n+1. The model can be used for evaluating both American and European options. In particular, the CRR binomial tree model is highly convenient for analyzing early exercise decisions for American options.

One of the most controversial assumptions in both CRR and BSM is that the variance rate of its return is constant (Black and Scholes, 1973). Since the pio-neer work by Black and Scholes (1973) many researchers demonstrated that this assumption is not consistent with real-world markets. The problem of volatil-ity smiles suggested that the volatilvolatil-ity of the underlying asset's return is rarely constant (Ederington et al., 2002).

In recent years the demand for stochastic volatility models increased and it is
no more reliable to consider only a deterministic volatility for evaluating options2_{.}

Stochastic volatility models have been examined in Merton (1976) and Wiggins (1987). In Hull and White (1987) it is proposed that the stock prices follow the stochastic volatility model. There are many other stochastic volatility models such as the two factors volatility model that was introduced in Heston (1993) and later in Christoersen et al. (2009), which assumes that the volatility of the risk-neutral stock price process is determined by two factors, where the variance of the stock return is the sum of the two variance factors. However, valuation of American option pricing is a very challenging problem under these types of models. There are some results on both numerical and analytical approximations under these stochastic volatility models, but far too few results in properties of early exercising decisions of American options.

The early exercise boundary (EEB) is a term associated with American option pricing. It can be dened as the set of the underlying asset's prices that corre-spond to the minimum of the possible values of the option, at which it is optimal to exercise, at each time step up to the maturity date. In addition to being the best tool for optimizing early exercise decisions, the EEB is considered an important element of pricing of American options. This was demonstrated by Ramaswamy and Sundaresan (1985) and Brenner et al. (1985), where they used the implicit nite dierence method by Brennan and Schwartz (1978) to nd the EEB and the prices of American options on future contracts simultaneously. Kim (1990) and Jacka (1991) derived the EEB in implicit form and used the resulted integral equation to evaluate the price of American options. All methods mentioned above involving the EEB assumed constant volatility. On the other hand, Tzavalis and Wang (2003) used Chebyshev polynomials to approximate the EEB under stochas-tic volatility model, and integral representation of the American option price in terms of the boundary.

Regime-switching or switching-state models are considered a middle ground

between deterministic models and fully stochastic ones. Such models have been used in dierent methods for option pricing. For example, in Naik (1993) and Guo (2001) the authors assumed a geometric Brownian motion with switching-state volatility. In Elliott et al. (2005) and Fard et al. (2014) the authors argue that due to the incompleteness that is introduced to the market by the uncertain economic state, the risk-free measure is no longer unique and they use Esscher transform in order to nd the minimal entropy martingale measure for pricing American options, which is beyond the scope of this thesis but certainly worth mentioning.

Finally, Aingworth et al. (2006) suggested that discrete-time Markov process with a small number of states, is similar to a probabilistic model known as normal mixture diusion (Alexander, 2004; Brigo and Mercurio, 2002), where the volatility follows a specic discrete distribution. Sato and Sawaki (2014) followed a similar model for pricing non-standard callable American options and even have a plot of EEBs.

### 1.2 Contribution of This Thesis

This thesis further investigates the regime-switching extended CRR model pro-posed in Aingworth et al. (2006) and later implemented by Sato and Sawaki (2014). We abbreviate hereafter this model as the extended CRR model. Our object of study is the EEBs associated with the American option pricing and the fair prices of both American and European options. By successfully constructing a recombin-ing tree, we obtain a convenient settrecombin-ing for in-depth analysis on EEB of American options under this model. We note that the study of EEBs was not carried out in Aingworth et al. (2006) and that our numerical study on EEBs for the standard American options is more extensive than Sato and Sawaki (2014). Furthermore, we note that the implementation of this model is nontrivial since there is a lack of a fully detailed algorithm in the mentioned literature. We believe that our algorithm provides a straightforward description of the model implementation problem. In combination with our MATLAB code in Appendix A, our algorithm can be easily modied to examine other types of payo functions or for further extending the model.

A full description of our contribution is given as follows. We rst give a de-tailed description of the algorithm for the model. Then we provide extensive numerical results for both American and European puts and calls, with dierent volatility states and dierent transition probability matrices (T P M). In partic-ular, we obtain the EEBs for American put options and American call options with dividend-paying underlying stock. In conducting numerical studies on EEBs we vary the parameters and demonstrate the EEBs behavior under our model. Finally, we give three dierent approximations for European call prices under our

studied model, using approximation formulas. As a by-product, we also make a correction for some wrongly computed numerical results in the reference paper (Aingworth et al., 2006), which appears to be new in the literature.

### 1.3 Outline

In the next chapter we review the mathematical background of the classical CRR Binomial model. In Chapter 3 we describe our model, the extended CRR model with a Markov switching-state volatility. We include a proof of its complexity level under that setting. Chapter 3 also explains our algorithm under the two-state version of the model for pricing American options and obtaining the early boundary of exercise.

In Chapter 4 we present our experimental studies. The properties of the early exercise boundary under two-state model with dierent volatility values and dier-ent TPMs, are to be examined. We include pricing tables for both American and European options. We also include approximations to the extended CRR using approximation formulas. Finally, in Chapter 5 we summarize our results and guide the reader to some related further research topics.

## Chapter 2

## A Review of Classical CRR Model

Since this thesis aims to investigate an extended CRR model, it is worth reviewing the framework under which the classical CRR model operates.

Throughout this report we use the notation n to describe the discrete-time-step number, i.e.

n + 1 ≡ t + ∆t, n = 0, 1, . . . , N − 1,

where N is the number of the equidistant sub-intervals of the nite time interval [0, T ]. We also denote w.p. short for with probability. Advancing from the stochastic process given by Equation (1.3), we identify the following up- and down-factors,

u = expnσ√∆to, d = expn− σ√∆to= 1 u.

Throughout this section we denote V (n) be the value of the portfolio at time-step n.

Denition 1 (Kijima (2016)). A portfolio process {θ(n); n = 0, 1, . . . , N} is called self-nancing if for the value process {V (n)} the following holds,

V (n) =

m

X

i=0

θi(n + 1)Si(n) n = 0, 1, . . . , N − 1,

where θi is the ith asset in the portfolio.

Denition 2 (Kijima (2016)). A contingent claim X is said to be attainable if there exists a replicating portfolio {θ(n); n = 0, 1, . . . , N}, that is a self-nancing portfolio such that V (N) = X.

In the paper by Cox et al. (1979), the authors used the no-arbitrage pricing method, using risk-neutral probability measure. Here we state necessary deni-tions and theorems, in order to build the pricing argument.

Denition 3 (Kijima (2016)). An arbitrage opportunity is the existence of some self-nancing trading strategy or portfolio {θ(n); n = 0, 1, . . . , N} such that,

(a) the portfolio costs nothing, V (0) = 0,

(b) the portfolio gives prot with no risk with positive probability, i.e. V (N) ≥ 0 with probability 1 (a.s.), and V (N) > 0 with probability p > 0.

Theorem 1 (No-Arbitrage Pricing, Kijima (2016)). For a given contingent claim X, if there exists a replicating portfolio {θ(n)} and if there are no arbitrage oppor-tunities then the initial value of the portfolio is the correct price of the contingent claim, i.e. V (0) = X.

Denition 4 (Kijima (2016)). A stochastic process {X(n); n = 0, 1, . . . , N} that is dened on the probability space (Ω, P, F), with ltration {Fn} is called a

Mar-tingale if,

(a) the stochastic process is integrable, that is, E[| X(n) |] < ∞ for every n, (b) E[X(n + 1) | Fn] = X(n) n = 0, 1, . . . , N − 1.

Denition 5 (Risk-Neutral Probability, Kijima (2016)). Given a probability space (Ω, P, F) with ltration {Fn}. The probability measure Q is a risk-neutral

proba-bility if and only if,

(a) Q is equivalent to P, that is, P(A) > 0 if and only if Q(A) > 0, for all A ∈ F,
(b) the martingale condition E∗_{[S}∗

i(n + 1) + d∗i(n + 1) | Fn] = Si∗(n) n =

0, 1, . . . , N − 1 holds for all i and n, where E∗ is the expectation under the risk-neutral probability measure, {d∗(n)} is the dividend process that

is adapted to the ltration {Fn} and S∗(n) is the denominated stock price

with the money market account for one unit of money S0 as the numéraire,

where S0(0) = 1.

Now, we build the replicating trading strategy from Cox et al. (1979). We take in our portfolio a ∆ amount of the stock S, along with a correspondent B amount invested in the risk-free account. Then we equate the value of the portfolio to the value of the option V , so that the portfolio is V (n) = S(n)∆ + BS0(0), where

S0(n) = enr∆t, such that the risk-free rate r has the following relation to the

up-and down-factors u > er∆t _{> d}, so that after one step in time, we get,

V (n + 1) =
(
∆uS(n) + Ber∆t _{= V}
u, w.p. p,
∆dS(n) + Ber∆t _{= V}
d, w.p. 1 − p.
(2.1)

where u and d denote the up- and down-factor, respectively. Solving the above system yields that,

∆ = Vu − Vd

uS(n) − dS(n), B =

uVd− dVu

(u − d)er∆t.

With that choice of ∆ and B our portfolio is replicating the option value and it is self-nancing. Substitute back in our portfolio, we get

V (n) = Vu− Vd
u − d
!
+ uVd− dVu
(u − d)er∆t
!
,
=e−r∆t
"
er∆t_{V}
u− er∆tVd− uVu+ uVd+ uVu− dVu
u − d
#
,
=e−r∆t
"
Vu
er∆t_{− d}
u − d
!
+ Vd
u − er∆t
u − d
!#
,
or
V (n) = e−r∆t[Vup∗+ Vd(1 − p∗)] , (2.2)
where
p∗ = e
r∆t_{− d}
u − d . (2.3)

If there are no arbitrage opportunities, then a European option value must be equal to the replicating portfolio value otherwise, we can build an arbitrage strategy by combining the option and its replicating portfolio.

Note that we can dene Q = {p∗_{, (1 − p}∗_{)}} _{as a probability measure, since}

it p∗ _{is always positive and has values between 0 and 1. Moreover, Q is a }

risk-neutral probability measure since it is the result of a portfolio that does not take the risk aversion of the investors in consideration. More importantly, it fullls the conditions in Denition 5; rstly Q(A) is zero if and only if r = σ = 0 which is a reasonable assumption for the real-world measure P(A). Furthermore, it is readily seen that the martingale condition under Q holds for all A, i.e.

E∗S(n + 1)/er∆t | Fn = S∗(n).

Now, without a rigorous proof, we state a version of the fundamental theorem of asset pricing (Harrison and Kreps, 1979; Harrison and Pliska, 1981).

Theorem 2 (Kijima (2016)). There are no arbitrage opportunities in a securities market if and only if there exists a risk-neutral probability measure. If this is the case then, the price of an attainable contingent claim X is,

V (0) = E∗ X S0(T ) , (2.4)

for every replicating strategy, where S0 is the money market account.

It should be mentioned that the CRR setting coincides even with the stronger version of the fundamental theorem of asset pricing by Harrison and Pliska (1981) and fullls the so-called complete market assumption.

Denition 6 (Complete Market, Kijima (2016)). A security market is said to be complete if and only if every contingent claim is attainable.

Theorem 3 (Kijima (2016)). Suppose that a securities market admits no arbitrage opportunities, then that market is complete if and only if there exists a unique risk-neutral probability measure.

We mentioned earlier about Equation (2.2), that if the assumption of no ar-bitrage opportunities in the market holds, then the value of the European option must match the replicating portfolio value. Hence by Theorem 1 the value of the European option is given by the following recursion,

V (n) = e−r∆t_{E}∗[V (n + 1)],

where n = 0, 1, 2, . . . , N − 1 denotes the time step number. At time T we have to calculate all possible values of the option, that is, for a call option, {S(T ) − K}+,

where

S(T ) = S(N ) = uWNdN −WNS(0), (2.5)

where {Wn} is a random walk with the Bernoulli underlying random variables

Xi ∼ B(p∗), that is Wn ≡ X1+ X2+ · · · + Xn, n = 1, 2, . . . and Wn takes the

values from the set {0, 1, 2, . . . , n}, so that S(T ) will have N + 1 possible values. Evaluating non-contingent claims like the American option is more compli-cated, since the stopping time can be at any time up to the maturity date.

Denition 7 (Kijima (2016)). For a given probability space(Ω, F, P) with l-tration {Fn} a stopping time is a random variable τ taking values in the set

{0, 1, . . . , N, ∞}, such that, the event {τ = n} belongs to Fn for each n ≤ N.

Theorem 4 (Option Sampling Theorem, Kijima (2016)). Consider a stochastic process {X(n)} dened on a probability space(Ω, F, P) with ltration {Fn}. If the

process is a martingale, then for any stopping time τ ∈

### τ0

, where### τn

denotes the set of stopping times that take values in the set {n, n + 1, . . . , N}, we have,The same no-arbitrage argument can be made for American options. However, the value is no more completely described by Equation (2.2), rather we have to account for the freedom of early exercise and take the highest of the two values so that the value of an American option Y with a payo function h(x) is given by,

Y (n) = max
n
h(S(n)), e−r∆t_{E}∗[Y (n + 1)]
o
, (2.7)
where S(n) is given by (2.5).

Theorem 5 (Kijima (2016)). Suppose that there exist a risk-neutral probability measure Q that the adapted process {Z(n); n = 0, 1, . . . , N} that is dened such that,

Z(n) = max

τ ∈

### τ

nE∗[Y∗(τ ) | Fn], n = 0, 1, . . . , T,

exists and suppose that the American option is attainable. Then the value of the American option is equal to Z(0) and the optimal exercise strategy is given by the equation,

τ (n) ≡ min{s ≥ n : Z(s) = Y∗(s)}. (2.8) The algorithm for American options under the CRR starts by evaluating the stock and the option at maturity. Then we cover all the time steps excluding the maturity, with a loop. Inside the time level loop, we have another loop, that will cover all the nodes at that current level. At each node the algorithm will evaluate both the intrinsic value of the option, if we are to exercise, and the payo, if we are to continue. Then we compare the two values and assign the larger one to the value of the option. Once all nodes at the current time level are covered, we can recover the price of the option, along with the stock prices that we can create the EEB from.

Our model is the extended CRR model. Since our volatility is a stochastic process, as proven by Romano and Touzi (1997), we have an incomplete market. When the market is incomplete, there is more than one equivalent martingale measure. Hence we face a problem of selecting a martingale measure for option pricing. In practice, this is not a problem as one can calibrate the model to option data to obtain the parameters under the market martingale measure. However, as our focus is on the properties of EEBs, we do not address this issue in this study. Instead, we assume that the market martingale measure is already given to us.

## Chapter 3

## Model Implementation

In this chapter, we describe the extended CRR model and its implementation. Our presentation is based on Aingworth et al. (2006) but we provide a substantial amount of important details.

### 3.1 Extended CRR Model

The CRR can be easily extended to regime-switching volatility, so that under the same market assumptions and for κ states of volatility, it follows from Equation (1.3) the value of the underlying stock after one step and is given by,

S(n) = u1S(n − 1), d1S(n − 1), ... uκS(n − 1), dκS(n − 1), where, ui = exp n σi √ ∆to, di = exp n − σi √ ∆to,

for i = 1, 2, . . . , κ, so that we have 2κ possible movements of the underlying asset when advancing one step in time. The transition probability matrix (TPM ) of the

switching-state is given by,
P =
1 . . . κ
1 p_{11} . . . p_{1κ}
... ... ... ...
κ pκ1 . . . pκκ
.

The same no-arbitrage pricing scheme that we showed in Chapter 2 can be applied here, and the replicating portfolio yields the following system,

Vi(n + 1) = ( ∆u1S(n) + Ber∆t = Vu1, w.p. p1, ∆d1S(n) + Ber∆t = Vd1, w.p. 1 − p1. w.p. pi1, n ... n ... n ... ( ∆uκS(n) + Ber∆t = Vuκ, w.p. pκ, ∆dκS(n) + Ber∆t = Vdκ, w.p. 1 − pκ. w.p. piκ,

where Vi _{is the value of the option (or the replicating portfolio). Notice that}

the resulting system is an extended version of the system in Equation (2.1). Fur-thermore, since we are assuming that the switching probabilities of the volatility states and the stock movement probabilities are independent, solving the above subsystem separately yields κ dierent amounts of ∆ and B. By substituting the resulted amounts weighted by their transition probabilities and simplifying in the same manner that we did earlier in Chapter 2 we get,

Vi(n) =e−r∆t[V_{u}1p1 + V_{d}1(1 − p1)]pi1+ [Vu2p
2_{+ V}2
d(1 − p
2_{)]p}
i2+ . . .
+ [Vuκp
κ_{+ V}
dκ(1 − p
κ_{)]p}
iκ ,

after expanding and reordering the terms, we get the formula for the expected values of the stock at time t and under any volatility state i is,

Vi(n) =e−r∆tV_{u}1p1pi1+ Vd1(1 − p
1
)pi1+ Vu2p
2
pi2+ Vd2(1 − p
2
)pi2+ . . .
+ Vuκp
κ_{p}
iκ+ Vdκ(1 − p
κ_{)p}
iκ .
(3.1)
Assuming that the TPM that we are using, has risk-neutral probabilities, we
get that the risk-neutral probability measure vector that corresponds to the initial
states vector is as follows,

Q =
p1_{p}
i1,
q1_{p}
i1,
... i = 1, 2, . . . , κ,
pκpiκ,
qκpiκ,

where pi _{is the same as p}∗ _{in Equation (2.3), that is,}

pi = e
r∆t_{− d}
i
ui− di
,
qi = 1 − pi.

We can see that Q fullls the martingale condition since it is a weighted average
of the martingale measures pi_{.}

In Figure 3.1 we can see a lattice representation of the model for two stats, after we apply the combining eect that we explained earlier. The TPM in this case is of the form,

P =

H L

H pHH pHL L pLH pLL

where, i =H,L are the volatility states. Since there are only two states, we refer to the higher value state as High and the lower value state as Low.

The resulting tree recombines in a more complex manner than it does in the original CRR, nevertheless, it does not have exponential complexity but rather polynomial one. To analyze the complexity of this model, we nd the count system of the nodes of the lattice generated by the model.

Proposition 1. In the regime-switching extended CRR model the number of dis-tinct possible nodes at time step n is n+2κ−1

2κ−1

(Aingworth et al., 2006).

Proof. To determine the number of all distinct1 _{possible nodes after n time steps,}

we look at the underlying stock value after this time,

S(n) = S(0)(u1)U1(d1)D1(u2)U2(d2)D2. . . (uκ)Uκ(dκ)Dκ, (3.2)

where, U1, D1, U2, D2, . . . , Uκ, Dκ are the powers of the movement factors i.e. the

number of steps taken in each movement factor.

1_{Here we are not accounting for the further combination that is allowed due to the fact that}

u3
H
u2
H u2HuL
u2_{H}dH
uHuL u2HdL
uH uHu2L
uHdH uHuLdH
uHuLdL
uHdL uHd2H
uL uHdHdL
u2
L uHd2L
H, L
uLdH u3L
dH u2LdH
uLdL u2LdL
uLdHdL
d2
H uLd2H
dL uLd2L
dHdL d3H
d2_{H}dL
d2
L dHd2L
d3
L

Figure 3.1: three-steps tree of the extended CRR model with two volatility states i = H, L.

It is given that the sum of all the powers is equal to n, keeping in consideration the powers can take values 0, 1, . . . , n. We also know that there are 2κ dierent possible movements, but we are only choosing 2κ − 1, since the last one is de-termined by the last property. Finally, we have n to choose from, in addition to 2κ − 1 since in this setting we can repeat zeros 2κ − 1 since, it is required that at least one of the powers is greater than zero. Hence, we can conclude that we have in total n + 2κ − 1 to choose from.

Proposition 2. The number of distinct possible nodes after n steps is n+2κ 2κ

(Aingworth et al., 2006).

Proof. Using Proposition 1 we sum up the nodes at each time level, i.e.

n X j=0 j + 2κ − 1 2κ − 1 =2κ − 1 2κ − 1 + 2κ 2κ − 1 + n X j=2 j + 2κ − 1 2κ − 1 , expanding the rst two terms using the denition of combination, we get,

n X j=0 j + 2κ − 1 2κ − 1 =(2κ − 1)! (2κ − 1)! + (2κ)! (2κ − 1)! + n X j=2 j + 2κ − 1 2κ − 1 =(2κ)! (2κ)! + 2κ(2κ)! (2κ)! + n X j=2 j + 2κ − 1 2κ − 1 =(2κ)!(1 + 2κ) (2κ)! + n X j=2 j + 2κ − 1 2κ − 1 =(2κ + 1)! (2κ)! + n X j=2 j + 2κ − 1 2κ − 1 =(2κ + 1)! (2κ)! + 2κ(2κ + 1)! (2κ)!(2!) + n X j=3 j + 2κ − 1 2κ − 1 =2(2κ + 1)! + 2κ(2κ + 1)! (2κ)!(2!) + n X j=3 j + 2κ − 1 2κ − 1 =(2κ + 2)! (2κ)!(2!) + n X j=3 j + 2κ − 1 2κ − 1 ... =(2κ + n)! (2κ)!n!

Proposition 2 tells us that the tree under the extended CRR has a polynomial growth, since n + 2κ 2κ =(2κ + n)! (2κ)!n! = (2κ + n)(2κ + n − 1)(2κ + n − 2) . . . (n + 1)(n)(n − 1) . . . 1 (2κ)!n! = (2κ + n)(2κ + n − 1)(2κ + n − 2) . . . (n + 1)n! (2κ)!n! = (2κ + n)(2κ + n − 1)(2κ + n − 2) . . . (n + 1) (2κ)! = Q2κ−1 i=0 (2κ + n − i) (2κ)! .

which is a polynomial of degree 2κ.

### 3.2 Model Assumptions

The assumptions of our model are similar to the assumptions of CRR and BSM. We assume that all investor's actions in the market have insignicant eect on the probability distribution of the existing securities. The nancial market is frictionless, that is, there are neither transaction fees nor taxation. The securities that we are using are assumed to be innitesimally divisible, short selling is allowed, with no restrictions. We also assume that the rates of the risk-free securities are the same for both lending and borrowing.

### 3.3 Algorithm

The algorithm we implement in Chapter 4 evaluates one step at the time. Start-ing from the maturity and loopStart-ing backward toward the present value, takStart-ing advantage of the combining eect we mentioned earlier. We start by dening zero matrices for the stock values S and option values fκ, and zero vectors for the early

exercise values.

The algorithm starts using Equation (3.2) and the payo function to evaluate the stock and the option values at the last time level, i.e. at the maturity. Then we start the main time loop that covers all the time steps excluding the maturity. Inside the time level loop, we have the node loop, that covers all the nodes at that current level. At each node the algorithm evaluates both the payo value of the option through Equation (3.1) if we continue and its intrinsic value if we

exercise, then compare the two values and hold the larger as the value of the option. Furthermore, if the intrinsic value is larger than the payo then we take a copy of the value at current node and put it in a vector Eκ. After that all nodes

at the current time level are covered, the function takes the minimum value that we collected at the possible exercises and assign it in the early exercise vectors Bκ.

At the end of the algorithm we are left with the EEBs along with a simulated path for the same stock.

See Appendix A, for MATLAB code of the algorithm under two volatility states.

### 3.4 Technical Aspects

We used MATLAB to apply the algorithm above with two initial volatility states. MATLAB's toolboxes and specialized functions are very useful in nance and applied mathematics in general. For an inexperienced programmer, MATLAB requires a relatively short time to achieve tasks with relatively high level of com-plexity.

The function we provide has the advantage of a good level of computation speed. Nonetheless, there are still some parts that could be improved by a pro-fessional programmer. The run time is 2.12 minutes for 50 steps, with MATLAB (version R2019a) on a PC with Intel i7-9700 processor and 16 GB of RAM. The largest number of steps we experimented with is 120 which took about 20 hours. Given the complexity of the tree, our function's eciency is satisfactory.

The most challenging part of the algorithm is in line 18, where it is required to identify the parent nodes in the n + 1 time-step. In order to achieve this in an ecient manner and without the need of tracing complex indices, we used a simple fact about the change in the power vector by which we mean the set of the exponents that corresponds to the set of movement factor. For example, in the basic case of two states, as in Figure 3.1, we have that the price of the stock at any time n is,

S(n) = S(0)(uH)UH(dH)DH(uL)UL(dL)DL,

here the power vector is [UH, UL, DH, DL]. Since a one-step advancement in time

corresponds to increase in one of the values in this vector, subtracting the current node's power vector from all the power vectors in the next time step identies all the parent nodes.

For example, in Figure 3.1, the rst node in the rst time step has the power vector [1,0,0,0]. The power vectors of all nodes in the next time step, can be represented in the following matrix,

2 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 2 0 0 0 1 1 0 0 1 0 1 0 0 2 0 0 0 1 1 0 0 0 2 .

Subtracting the power vector of the current node from each power vector in the next time step we obtain a matrix with only 2κ or in this case 4 non-negative vectors. In our example we get the following matrix,

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 −1 2 0 0 −1 1 1 0 −1 1 0 1 −1 0 2 0 −1 0 1 1 −1 0 0 2 ,

where the non-negative vectors correspond to the parent nodes. There is a variety of methods one can use to obtain the required indices from the matrix above. With this method we do not need to have a specic order to the nodes. We can order the obtained nodes with the help of the resulting non-negative vectors. The position of the digit 1 in the non-negative vector corresponds to the type of the parent node, i.e. the type of the movement towards this parent node.

The MATLAB function provided by Isaac (2019) is an ecient method that our MATLAB function utilizes for obtaining power matrices, i.e. sets of power vectors.

Algorithm 1: Pricing Options and EEBs under Extended CRR. Input : S0 - current stock price, K - strike,

T - expiry time, r - interest rate,

σ - volatility values, P - TPM corresponding to the volatility states, h(S) - the payo function, Nsteps - number of time-steps,

Output:

price1 - option price given that the rst volatility is the current state,

...

priceκ - option price given that the last volatility is the current state,

K1 - the early exercise boundary vector for the rst state,

...

Kκ - the early exercise boundary vector for the last state. 1 Define ∆t = T/Nsteps;

2 Define u and d;

3 Define the risk-free measures Q; 4 Define κ = length (σ);

5 Define NLeaves = N steps+2κ−1_{2κ−1} ;

6 Define matrices S, V1, . . . , Vκ size (NLeaves, Nsteps); 7 Define vectors B1, . . . , Bκ size (Nsteps);

8 for h ← 1 to NLeaves do 9 for j ← 1 to Nsteps do

10 Evaluate S;

11 end

12 end

13 V1(all, N steps) = · · · = Vκ = h(S(all, N steps)); 14 for n ← (UNsteps − 1) to 1 do

15 Define NLeaves = n+2κ−1_{2κ−1} ;
16 Define nLeaves = n−1+2κ−1_{2κ−1} ;
17 for h ← 1 to NLeaves do
18 Identify all 2κ parent nodes;
19 Evaluate V1, . . . , Vκ;
20 V1, . . . , Vκ = max((V1, . . . , Vκ), h(S));
21 if V1, . . . , Vκ < h(S) then
22 E1, . . . , Eκ = V1, . . . , Vκ;
23 end
24 end
25 I1, . . . , Iκ = Index(min(E1, . . . , Eκ);
26 B1, . . . , Bκ = (S(I1, . . . , Iκ));
27 end
28 price1, . . . , priceκ = V1, . . . , Vκ(0);

## Chapter 4

## Numerical Results and EEBs

We test our model in the setting of dierent parameters. We tested the algorithm thoroughly and decided to x the number of steps in this chapter to be N = 50, since it makes the computational time reasonable and sucient amount of details on the boundary. Also, it was demonstrated in Aingworth et al. (2006) that the total number of steps n = 50 yields a similar result to n = 200 in terms of a produced forward put option prices. A forward option is an option with a forward contract as the underlying asset, see Hull (2003). We also choose to x the underlying stock initial price and the risk-free interest rate to be S(0) = 100 and r = 0.05, respectively, throughout this chapter.

Here we mention that in the case of no dividend and non-negative risk-free rate, American calls are never optimally exercised before maturity. Since given the no-arbitrage assumption, the call value must always be greater than S(t) − K, otherwise there would be an easy arbitrage strategy; simply buy the call and short the stock and invest K units of money in a risk-free investment (Hull, 2003; Cox et al., 1979). That means, in the case of no dividend and non-negative risk-free rate, American calls are worth the same as European calls. Throughout this chapter we assume that there is no dividend on our underlying stock, with the exception of Section 4.5.

### 4.1 Correction of Reference Paper

Here we demonstrate that at least some of the numerical results for American puts' prices stated in Table 2 of Aingworth et al. (2006) are incorrect, since they violate the put-call parity for American options (Hull, 2003).

Proposition 3. Given no dividend, for the prices of American options the follow-ing inequality holds,

where P and C are the prices of the American call and put, respectively, both with the same underlying S(0) and same strike K. Otherwise, arbitrage opportunities can be produced from those options (Hull, 2003).

Given the proposition above and the fact that American and European calls have the same values when there is no dividend and the risk-free rate is non-negative, it is readily seen that we can build an arbitrage opportunity using some of the prices in Aingworth et al. (2006). For example, we take the call price in the last entry of their Table 1, Call(H)= 10.642 and the American put price in the last entry of their Table 2, Put(H)= 37.019. The prices correspond to the same parameters K = 115.64, T = 1, r = 0.05 and S(0) = 100. Substitute in the left part of the relation in Proposition 3, we get that,

100 − 115.64 10.642 − 37.019,

which violates the relation. It is also worth to mention, that the example we
show here, is not a unique case. In Table 4.11 _{we redo the last three rows from}

the mentioned tables in Aingworth et al. (2006) and we get signicantly dierent prices.

Table 4.1: Prices of options with, N = 25, S(0) = 100, r = 0.05, σH = 0.4,

σL= 0.2, pHH = 0.9834 and pLL = 0.9889.

Maturity Strike Call(H) Call(L) _{Put(H) Put(L) Put(H) Put(L)}European Americans

1 94.61 19.126 12.062 9.122 2.058 9.607 2.186

1 105.13 14.229 5.693 14.231 5.696 15.070 6.478 1 115.64 10.563 2.459 20.559 12.454 21.944 15.640

### 4.2 Option Pricing under Extended CRR

In this section, before we start presenting our results, we give some useful termi-nology. Denote K as the strike price and S as the underlying stock price. We say that an option can exist in three states at any time during its life (Hull, 2003):

• at-the-money (AT M), when K = S,

• in-the-money (IT M), when S > K for calls and S < K for puts, • out-of-the-money (OT M), when K > S for calls and S > K for puts.

1_{the staying probabilities for this table are obtained using the generator matrix given in}

We say that the option is deeper in-the-money or deeper out-of-the-money as the values of S and K gets further apart. This relation between K and S, is usually referred to as the moneyness (Hull, 2003). We also need to denote CallH and PutH

as call and put option prices given that the initial state is the High state, and CallL

and PutL as call and put option prices given that the initial state is Low state.

Now we build price tables for both American and European options and reect on the results. In Table 4.2, we vary the moneyness and the maturity and x pHH = 0.9 the probability of staying at the High state, pLL = 0.95the probability

of staying at the Low state. We evaluate ATM, ITM and OTM puts and calls, with dierent values for the volatility.

Table 4.2: Prices of options with, N = 50, S(0) = 100, r = 0.05, σH = 0.4,

σL= 0.2, pHH = 0.9 and pLL = 0.95.

Maturity Strike CallH CallL _{Put}European Americans
H PutL PutH PutL

0.25 80 21.34 21.23 0.34 0.24 0.35 0.24 0.25 90 12.83 12.51 1.71 1.39 1.74 1.41 0.25 100 6.49 6.02 5.25 4.78 5.36 4.87 0.25 110 2.78 2.37 11.42 11.00 11.74 11.29 0.25 120 1.04 0.79 19.55 19.30 20.27 20.04 0.5 80 23.09 22.82 1.11 0.84 1.14 0.86 0.5 90 15.42 14.91 3.20 2.69 3.29 2.76 0.5 100 9.51 8.84 7.04 6.38 7.29 6.58 0.5 110 5.48 4.82 12.76 12.10 13.33 12.59 0.5 120 2.97 2.45 20.01 19.48 21.11 20.52 1 80 26.42 25.91 2.52 2.01 2.63 2.08 1 90 19.59 18.84 5.20 4.45 5.47 4.65 1 100 14.06 13.14 9.18 8.27 9.76 8.72 1 110 9.85 8.89 14.48 13.52 15.55 14.42 1 120 6.76 5.85 20.90 19.99 22.70 21.61

The prices produced in Table 4.2 are consistent and follow the standard theory of option pricing, e.g. the deeper in the money the option is the higher its value and the longer the maturity of the option, the higher its value. Most importantly, for the European options, our prices are not a direct source of arbitrage opportunities and the put-call parity for European options holds at this setting. Moreover, extended put-call parity also holds for the American options.

Next we vary the volatility states, which are in the form [σH, σL] and the

Table 4.3: Prices of options with, N = 50, S(0) = K = 100, r = 0.05, T = 0.25.
σH σL pHH pLL CallH CallL _{Put}European Americans

H PutL PutH PutL

0.3 0.25 0.99 0.90 6.49 6.34 5.17 4.68 5.36 5.22 0.3 0.25 0.90 0.85 6.23 6.18 4.99 4.93 5.10 5.04 0.3 0.25 0.90 0.80 6.29 6.25 5.05 5.00 5.15 5.11 0.3 0.25 0.90 0.70 6.36 6.33 5.12 5.09 5.22 5.20 0.3 0.25 0.80 0.70 6.22 6.20 4.98 4.96 5.08 5.07 0.3 0.25 0.70 0.80 6.03 6.01 4.79 4.77 4.89 4.87 0.3 0.25 0.70 0.90 5.89 5.86 4.64 4.61 4.75 4.72 0.3 0.25 0.60 0.95 5.74 5.71 4.50 0.47 4.61 4.58 0.3 0.25 0.40 0.95 5.69 5.67 4.45 4.43 4.55 4.54 0.3 0.25 0.30 0.90 5.74 5.73 4.49 4.49 4.60 4.60 0.3 0.25 0.10 0.15 6.10 6.11 4.86 4.87 4.96 4.97 0.3 0.25 0.80 0.15 6.41 6.41 5.17 5.17 5.27 5.27 0.5 0.15 0.99 0.90 9.59 6.30 8.35 5.06 8.99 8.16 0.5 0.15 0.95 0.80 9.62 9.32 8.38 8.07 8.48 8.17 0.5 0.15 0.65 0.60 8.16 8.12 6.92 6.88 7.02 6.98 0.5 0.15 0.52 0.65 7.47 7.44 6.22 6.20 6.32 6.29 0.5 0.15 0.52 0.55 7.85 7.84 6.61 6.60 6.71 6.70 0.5 0.15 0.50 0.98 7.79 7.78 6.55 6.54 3.22 2.97 0.5 0.15 0.15 0.95 4.38 4.36 3.14 3.11 3.25 3.22 0.5 0.15 0.15 0.90 4.92 4.91 3.68 3.67 3.78 3.77 0.5 0.15 0.15 0.85 5.35 5.35 4.12 4.12 4.22 4.22 0.5 0.15 0.15 0.80 5.72 5.73 4.48 4.49 4.58 4.59 0.5 0.15 0.15 0.70 6.30 6.32 5.06 5.08 5.16 5.18 0.5 0.15 0.15 0.60 6.74 6.77 5.50 5.53 5.60 5.63 0.5 0.15 0.15 0.20 7.84 7.89 6.60 6.65 6.71 6.76 0.5 0.15 0.15 0.10 8.02 8.07 6.78 6.83 6.89 6.94 0.5 0.15 0.05 0.10 7.85 7.91 6.61 6.66 6.71 6.77 0.5 0.15 0.01 0.02 7.92 7.97 6.68 6.73 6.79 6.84 0.75 0.1 0.9 0.05 14.72 14.73 13.48 13.49 13.59 13.59 0.75 0.1 0.9 0.3 14.50 14.46 13.26 13.22 13.36 13.32 0.75 0.1 0.9 0.5 14.18 14.08 12.94 12.84 13.04 12.94 0.75 0.1 0.9 0.7 13.53 13.28 12.29 12.04 12.39 12.13 0.75 0.1 0.9 0.9 11.43 10.56 10.19 9.32 10.28 9.40 0.75 0.1 0.7 0.9 8.42 7.98 7.18 6.74 7.26 6.81 0.75 0.1 0.5 0.9 7.06 6.83 5.82 5.58 5.90 5.66 0.75 0.1 0.3 0.9 6.24 6.14 5.00 4.90 5.09 4.99 0.75 0.1 0.05 0.9 5.57 5.59 4.33 4.35 4.42 4.44

The prices in Table 4.3 holds for the put-call parity and reect mostly intu-itive results. The gap between the prices of the same option with dierent initial volatility state, is larger with higher probabilities of staying and smaller with lower probabilities of staying. The gap becomes smaller as the TPM approaches a sta-tionary distribution and it disappears completely when the TPM is at a stasta-tionary distribution. Here we mention that in the case of two states the TPM is at sta-tionary distribution when the sum of the staying probabilities is equal to one. We give a further explanation for stationary distributions in Section 4.3.

However, there are some counter-intuitive results in Table 4.3. When the prob-ability of staying at the High state is less than the probprob-ability of transiting to High, the prices at High are lower than the prices at Low. It is mentioned in Sato and Sawaki (2014) in Assumption 2.1, that the TPM is assumed to be stochastic increasing matrix, that is, for each i ≤ κ, Pκ

λ=iphλ is non-decreasing in h. This

is to conciliate with the following sucient condition for option prices: With a high volatility state the option prices should be greater than with a low volatility state. Our results show the importance of the stochastic increasing TPM assump-tion since validaassump-tion of this assumpassump-tion results in an over-weighted Low state that leads to prices that violate the overly mentioned sucient condition for option pricing.

### 4.3 Approximations for Extended CRR Prices

In this section, we give three approximations for the prices under our model.
First, we look at the case where the current state of volatility is unknown. In
fact, it is within the properties of Markov chains, that for a large number of steps
the initial state becomes less important as the chain settles into its stationary
distribution. The stationary distribution (limiting distribution) π is calculated by
solving the system of equations π> _{= π}>_{P} _{under the condition P}

f or all iπi = 1.

The stationary distribution always exists in the case of two states, as long the
probabilities of staying are not equal to 1 or 02_{. Also, in Fuh et al. (2012) the}

authors had a similar switching-state model, where they mention that in the case of the unknown current state, a weighted average of the resulted prices can be used as an approximation, with the π as the weights. In our model for the two-state case, the stationary distribution can be denoted as π = [πH, πL]. We denote

the weighted average for a European call price by Callavg and our approximation

becomes,

2_{Classication of Markov chains and TPMs are beyond the scope of this thesis, therefore}

we chose to omit the theoretical ground for the limiting distribution and its existence, see Ross (2014) for details.

Callavg = πHCallH + πLCallL. (4.1)

Next, we approximate our extended CRR model with the xed volatility
mod-els, BSM and classical CRR. Here we need to represent κ volatility states in a
xed volatility value. It was mentioned in Hull and White (1987) and later in
Ball and Roma (1994), that the average variance can be used as an approximation
of a stochastic variance in the case of European options if the variance σ2 _{is not}

correlated with the underlying asset. In our case, we can take the square root of the average of the volatility states weighted by the limiting distribution. We denote ¯σ as our approximation for two volatility states and we get,

¯

σ =pπH(σH)2+ πL(σL)2. (4.2)

We can use the approximated volatility from Equation (4.2) to nd approxi-mations using constant volatility models. We denote CallBSM and CallCRR to be

the approximations of a European call, resulted from using ¯σ in the BSM formula (Hull, 2003) and the algorithm for the classical CRR model, respectively.

Finally, since the rst approximation is the closest to our model, we can use it as a benchmark for the last two. We denote RECRR and REBSM to be the relative

errors for the approximations resulted from classical CRR and BSM, respectively, for the approximation in Equation (4.1), that is,

RECRR =

CallCRR−Callavg

Callavg , REBSM =

CallBSM −Callavg

Callavg

.

The results in Table 4.4 give a hint of the performance of the three approxi-mations.

Table 4.4: Prices of European call options under the extended CRR, along with three dierent approximations, with the following xed parameters, N = 50,

S(0) = K = 100, r = 0.05, T = 0.25.

σH σL pHH pLL CallH CallL Callavg RECRR REBSM

0.3 0.25 0.99 0.90 6.494 6.359 6.482 1.7 × 10−3 2.8 × 10−3
0.3 0.25 0.90 0.85 6.235 6.177 6.212 4.8 × 10−3 3.2 × 10−4
0.3 0.25 0.70 0.90 5.887 5.856 5.864 4.8 × 10−3 3.4 × 10−4
0.3 0.25 0.60 0.95 5.738 5.711 5.714 3.8 × 10−3 5.2 × 10−4
0.3 0.25 0.40 0.95 5.688 5.675 5.676 3.5 × 10−3 8.8 × 10−4
0.3 0.25 0.30 0.90 5.736 5.731 5.731 4.4 × 10−3 1.7 × 10−4
0.5 0.15 0.99 0.90 10.121 9.321 10.048 8.9 × 10−4 _{5.6 × 10}−3
0.5 0.15 0.95 0.80 9.619 9.316 9.559 2.5 × 10−3 2.2 × 10−3
0.5 0.15 0.15 0.85 5.358 5.358 5.358 3.7 × 10−3 5.6 × 10−4
0.5 0.15 0.15 0.80 5.721 5.730 5.728 4.4 × 10−3 5.2 × 10−5
0.5 0.15 0.15 0.70 6.299 6.320 6.315 5.4 × 10−3 9.5 × 10−4
0.5 0.15 0.01 0.02 7.924 7.971 7.948 6.9 × 10−3 2.4 × 10−3

### 4.4 Properties of EEB for American Put Option

Now we examine the behavior of the EEB with some dierent parameters. We have chosen a handful of EEB plots to include in our report.

(a) σH = 0.4, σL= 0.2and pHH = 0.9, pLL = 0.95. 0.05 0.1 0.15 0.2 0.25 t 82 84 86 88 90 92 94 96 98

Underlying Asset price

Continuation Region

Exercise Region

EEB when

H is the active state

EEB when

L is the active state

(b) σH = 0.3, σL= 0.25and

pHH = 0.9, pLL= 0.95.

Figure 4.1: EEB of ATM American puts with underlying S(0) = 100, maturity T = 0.25, and risk-free rate r = 0.05.

In general, the early exercise of American put options becomes more attractive as S(0) becomes less with respect to the strike, as r increases and as the volatility decreases (Hull, 2003). The results we obtain do indeed coincide with the general theory on American options. In Figures 4.1-4.3 we see that the boundary in the Low state is higher than the boundary in the High state. We also see that as the volatility values get more spread out, the EEBs follow.

Here, it is important to mention that the exercise region for American puts is always under the EEB, a consequence of the direction of the moneyness, i.e. the lower the price of the stock with respect to the strike is, the deeper in the money the option is. The the opposite applies for American calls and the exercise region lies above the EEB.

0.05 0.1 0.15 0.2 0.25 t 65 70 75 80 85 90 95 100

Underlying Asset price

Continuation Region

Exercise Region

EEB when

H is the active state

EEB when

L is the active state

(a) σH = 0.5, σL= 0.15and

pHH = 0.9, pLL= 0.4.

(b) σH = 0.5, σL= 0.15and

pHH = 0.99, pLL = 0.98.

Figure 4.2: EEBs of ATM American puts with underlying S(0) = 100, maturity T = 0.25, and risk-free rate r = 0.05.

We notice in Figure 4.2 that the spread of the boundaries is also being aected by the values of the TPM. The higher the staying probabilities, the more spread the boundaries are.

In Figure 4.3 we see two examples where the boundaries of the two stats coincide when we use a stationary distribution as a TPM.

0 0.05 0.1 0.15 0.2 0.25 t 88 90 92 94 96 98 100

Underlying Asset price

Continuation Region

Exercise Region

EEB when

H is the active state

EEB when

L is the active state

(a) σH = 0.5, σL= 0.15and pHH = 0.01, pLL= 0.99. 0 0.05 0.1 0.15 0.2 0.25 t 75 80 85 90 95 100

Underlying Asset price

Continuation Region

Exercise Region

EEB when

H is the active state

EEB when

L is the active state

(b) σH = 0.5, σL= 0.15and

pHH = 0.3, pLL = 0.7.

Figure 4.3: EEBs of ATM American puts with underlying S(0) = 100, maturity T = 0.25, and risk-free rate r = 0.05.

(a) σH = 0.5, σL= 0.15and pHH = 0.01, pLL= 0.02. 0 0.05 0.1 0.15 0.2 0.25 t 75 80 85 90 95 100

Underlying Asset price

Continuation Region

Exercise Region

EEB when

H is the active state

EEB when

L is the active state

(b) σH = 0.5, σL= 0.15and

pHH = 0.2, pLL = 0.6.

Figure 4.4: EEBs of ATM American puts with underlying S(0) = 100, maturity T = 0.25, and risk-free rate r = 0.05.

In Figure 4.4 we notice again the counter eect that occurs when we have a probability of staying at the High state is less than the probability of transiting to High. That is, when we violate the assumption of a stochastic increasing TPM

mentioned in Section 4.2. Therefore, we see that the boundary in the High state is above the one in the Low state, as opposed to the standard case.

### 4.5 Properties of EEB for American Call Option

### with Dividend

In order to illustrate the EEB in American calls, we took the simplest case of dividend, that is, where the underlying assets pays a continuous dividend yield. Like the put examples, we take ATM option on a stock with initial price equal to 100 and 3−months maturity. As expected, the obtained boundary has an asymptotic behavior where it approaches the maturity from above, as opposed to the asymptotic behavior in the put case where it approaches the maturity from below, which coincides with results obtained in Jönsson (2001) and in Sato and Sawaki (2014).

Adding a dividend yield larger than the risk-free rate acts as a negative rate, since it represents a return. The rate of return becomes r − q, where q is the continuous dividend yield rate (Hull, 2003; Kijima, 2016).

We are free to choose any positive value for dividend yield and the risk-free rate, as long as the following relation is preserved, see Hull (2003),

ui < e(r−q)∆t < di, for all i.

(a) σH = 0.5, σL= 0.15and

pHH = 0.7, pLL= 0.8.

(b) σH = 0.5, σL= 0.15and

pHH = 0.98, pLL = 0.99.

Figure 4.5: EEBs of ATM American calls with underlying S(0) = 100, maturity T = 0.25, risk-free rate r = 0.05 and dividend yield y = 0.07.

In the gure above we see that the EEBs for calls have a similar behavior with respect to the volatility values and the TPM and its spread.

In this section, we chose to have a xed moneyness in all plots, in order to demonstrate other important variations. Nevertheless, it is worth mentioning that varying the moneyness with the other parameters xed results in shifts in the stock price axis. As expected, the deeper in-the-money the option is initially the larger the exercise region is.

## Chapter 5

## Conclusion

In this report, we conclude that the switching-regime volatility extended CRR model is a simple and eective way of obtaining the early exercise boundaries along with the fair prices of options. We demonstrate that the model is suitable for pricing both American and European options. We also show how the boundary changes as the parameter of the underlying and we illustrated how the changes coincide with the theoretical ground of American options. We analyze numerically the model through pricing tables and give examples of the EEBs.

### 5.1 Summary of Results

The prices of both American and European options under our extended CRR model are consistent and they have a positive relationship with the moneyness, the maturity duration, and the volatility values. The gap between two prices that correspond to two initial volatility states, has a positive correlation with the staying probabilities in the TPM (Table 4.3). In Section 4.3 we demonstrate how can we use a simple approximation in the case of the unknown initial state. We also demonstrate approximations that use xed volatility value, by approximating the volatility states into a xed value. In Table 4.4 we give a brief comparison between these approximations.

The EEBs plots we obtained show an asymptotic behavior as it approaches maturity. Our results demonstrate how the spread of the two EEBs is eected by the volatility values and the staying probabilities. We even show how validating the assumption of a stochastic increasing TPM, could result in an inverse position of the EEBs.

### 5.2 Further Research

One of the further research questions is to attempt to approximate EEBs with the switching-regime in a closed form, similar to Lauko and ev£ovi£ (2010). It is also interesting to compare numerically, the development in the results as the number of volatility states increases.

One can also do numerical analysis and parameterizations to match the TPM of the volatility states with the market. Furthermore, numerical comparisons with other options pricing models can be addressed.

Finally one can examine the behavior of the EEB of non-standard payo func-tions under the extended CRR. Such payos can be step-wise, logarithmic or square of a standard payo, as studied in Jönsson (2001). The algorithm and the corresponding MATLAB function that we provide can be modied to accompany most payos and it is of interest to demonstrate the limitation of this model, in terms of non-standard payos.

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## Appendix A

## Code

### A.1 MATLAB Function

Here is the function that we made in order to implement our extended CRR model under two volatility states. The function outputs the prices that correspond to each initial volatility state, along with the corresponding EEBs.

1 f u n c t i o n [ price1 , p r i c e 2 ] = . . .

2 Bintree_M ( S0 , K, r , y , T, sigma , P, Nsteps , opttype )

3

4 % A Function that return the p r i c e o f Both

5 % American and European Option

6 % With Markov Switching two State v a r i a b l e s .

7 % 8 % 9 %Input 10 % S0 − current stock p r i c e 11 % K − s t r i k e 12 % T − expiry time 13 % r − i n t e r e s t r a t e 14 % y − dividend y i e l d 15 % sigma − v o l a t i l i t y values [H, L ]

16 % P − The t r a n s i t i o n matrix corresponding to the v o l a t i l i t y s t a t e s

17 % opttype − 0 f o r a c a l l , otherwise a put

18 % Nsteps − number o f timesteps

19 %

20 %Output

21 % p r i c e 1 : option p r i c e in the f i r s t v o l a t i l i t y s t a t e 22 % p r i c e 2 : option p r i c e in the second v o l a t i l i t y s t a t e

23 % K1 : the e a r l y e x e r c i s e boundary vector f o r the f i r s t s t a t e

24 % K2 : the e a r l y e x e r c i s e boundary vector f o r the second s t a t e

26 %

27 % Account f o r the dividend r a t e

28 R=r−y ;

29 %time−step s i z e ( Delta t ) and t r e e parameters

30 d e l t = T/ Nsteps ;

31

32 % compute the up and down movements

33 u = exp( sigma ∗ s q r t( d e l t ) ) ;

34 d = 1 . / u ;

35 % the discount f a c t o r

36 a = exp( (R) ∗ d e l t ) ;

37 % the p r o b a b i l i t y o f the up and down movements a c c o r d i n g l y

38 p = ( a − d) . / ( u − d) ;

39 q = 1−p ;

40 % number o f v o l a t i l i t y values

41 Nsigma = length( sigma ) ;

42 % the complexity o f the t r e e as given in the model

43 NLeaves= nchoosek ( Nsteps −1+(2∗Nsigma ) , (2∗ Nsigma ) −1) ;

44

45 % i n i t i a t i n g p a y o f f s and Stock p r i c e s in the t r e e in a d d i t i o n to 46 % e a r l y e x e r c i s e v e c t o r s . 47 f 1 = z e r o s( NLeaves , Nsteps+1) ; 48 f 2 = z e r o s( NLeaves , Nsteps+1) ; 49 S = z e r o s( NLeaves , Nsteps+1) ; 50 B = ones ( NLeaves , 1 ) ; 51 E1 = z e r o s( Nsteps +1 ,1) ; 52 E2 = z e r o s( Nsteps +1 ,1) ; 53

54 % here we match the order o f the p r o b a b i l i t y vector and

55 % the movement vector

56 % The P r o b a b i l i t y v e c t o r s f o r both s t a t e s . 57 P1 = double ( [ p ( 1 ) . ∗P( 1 , 1 ) , p ( 2 ) . ∗P( 1 , 2 ) , q ( 1 ) . ∗P( 1 , 1 ) , q ( 2 ) . ∗P( 1 , 2 ) ] ) ; 58 P2 = double ( [ p ( 1 ) . ∗P( 2 , 1 ) , p ( 2 ) . ∗P( 2 , 2 ) , q ( 1 ) . ∗P( 2 , 1 ) , q ( 2 ) . ∗P( 2 , 2 ) ] ) ; 59

60 % the movements vector

61 ud = [ u d ] ;

62

63 % Obtaining the powers o f the movements value f o r the l e a v e s

64 [ Nmatrix ] = multinomial_powers_recursive ( Nsteps , Nsigma ∗2) ;

65

66 % Compute the stock and the options values in the l e a v e nodes at the

67 % maturity 68 f o r h=1:NLeaves 69 f o r j =1: 4 70 B(h) = ud ( j ) ^( Nmatrix (h,5− j ) ) ∗B(h) ; 71 end 72 S(h , 1 )= S0 ∗ B(h) ;

73 end

74 f 1 ( : , 1 ) = opttype ∗max(K−S ( : , 1 ) ,0) + (1− opttype ) ∗max(S ( : , 1 )−K, 0 ) ;

75

76 f 2 ( : , 1 ) = f 1 ( : , 1 ) ; 77

78 % the l a s t value o f the EEB i s always equal to the s t r i k e

79 E1( Nsteps+1)=K;

80 E2( Nsteps+1)=K;

81

82 % v a r i a b l e s f o r holding the e a r l y e x e r c i s e values at each l e v e l

83 e1=z e r o s( NLeaves +1 ,1) ;

84 e2=z e r o s( NLeaves +1 ,1) ;

85

86 % Start the backward r e c u r s i o n

87

88 f o r n=1: Nsteps

89 % the number o f nodes at the current time step

90 Nnodes= nchoosek ( Nsteps−1−n+(2∗Nsigma ) , (2∗ Nsigma ) −1) ;

91 % the number o f nodes at the time step a f t e r the current one

92 Nleaves= nchoosek ( Nsteps−n+(2∗Nsigma ) , (2∗ Nsigma ) −1) ;

93 % power matrix that corresponds to the time s t e p s mentioned above

94 [ Nmatrix ] = multinomial_powers_recursive ( Nsteps+1−n , Nsigma ∗2) ;

95 [ Nmatri ] = multinomial_powers_recursive ( Nsteps−n , Nsigma ∗2) ;

96

97 B = ones ( Nnodes , 1 ) ;

98 f o r h=1:Nnodes

99

100 % B i s the movement f a c t o r s accumulation at the current node .

101 f o r j =1:4

102 B(h) = ud ( ( j ) ) ^( Nmatri (h,5− j ) ) ∗B(h) ;

103 end

104 % S i s the expected value o f the a s s e t at the current node

105 S(h , n+1)= S0 ∗ B(h) ;

106 % phi i s the i n t r i n s i c value o f the option at the current

node

107 phi1 = opttype ∗(K−S(h , n+1) ) + (1− opttype ) ∗(S(h , n+1)−K) ;

108

109 % To i d e n t i f y the t r a n s i t i o n d i r e c t i o n to the previous step ,

110 % s u b t r a c t the powers o f the current node

111 % from the powers o f a l l o f the nodes in the next time step ,

112 % then i d e n t i f y the i n d i c e s o f the only 4 p o s i t i v e valued

113 % rows .

114 Trans = bsxfun (@ (A,B) A−B , Nmatrix , Nmatri (h , : ) ) ;

115 [Row, ~ ] = f i n d( Trans <0) ;

116 Trans (Row , : ) = 0 ;

117 [ cc , ~ ] = f i n d( Trans==1) ;

118

119 % here we have an a l t e r n a t i v e ( l e s s e f f i c i e n t ) method