option pricing and early exercise boundary of american...

School of Education, Culture and CommunicationDivision 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 UNIVERSITYSE-721 23 VÄSTERÅS, SWEDEN

School of Education, Culture and CommunicationDivision 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 :

15 ECTS credits

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 Markovswitching-state volatility. We present a detailed algorithm for obtaining earlyexercise boundaries for American options, as well as, fair prices for both Ameri-can and European options. To provide extensive numerical results, we experimentwith di�erent variations of the parameters and analyze the results. In particu-lar, we study properties of the resulting early exercise boundaries. Moreover, wegive three approximation methods for the pricing of European options under thismodel.

Keywords: Pricing American Options, Early Exercise Boundary,Markov-Modulated Volatility, Switching-State Volatility, Extended CRR Model.

Acknowledgments

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

Contents

1 Introduction 7

1.1 Background and Literature Review . . . . . . . . . . . . . . . . . . 71.2 Contribution of This Thesis . . . . . . . . . . . . . . . . . . . . . . 101.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 A Review of Classical CRR Model 12

3 Model Implementation 17

3.1 Extended CRR Model . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Technical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Numerical Results and EEBs 26

4.1 Correction of Reference Paper . . . . . . . . . . . . . . . . . . . . . 264.2 Option Pricing under Extended CRR . . . . . . . . . . . . . . . . . 274.3 Approximations for Extended CRR Prices . . . . . . . . . . . . . . 304.4 Properties of EEB for American Put Option . . . . . . . . . . . . . 324.5 Properties of EEB for American Call Option with Dividend . . . . . 35

5 Conclusion 37

5.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A Code 42

A.1 MATLAB Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.2 Sub-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

B Proofs 48

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

C Bachelor Degree Objectives 50

2

List of Figures

3.1 three-steps tree of the extended CRR model with two volatilitystates i = H,L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 EEB of ATM American puts with underlying S(0) = 100, maturityT = 0.25, and risk-free rate r = 0.05. . . . . . . . . . . . . . . . . . 32

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



4.5 EEBs of ATM American calls with underlying S(0) = 100, maturityT = 0.25, risk-free rate r = 0.05 and dividend yield y = 0.07. . . . . 35

3

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. 294.4 Prices of European call options under the extended CRR, along with

three di�erent 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 strikeK and maturity T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

B.2 Two portfolios that include one option each, both options have strikeK and maturity T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4

Abbreviations and Acronyms

ATM At-The-Money

BSM Black-Scholes-Merton

CRR Cox-Ross-Rubinstein

EEB Early Exercise Boundary

ITM In-The-Money

OTM Out-of-The-Money

TPM Transition Probability Matrix

w.p. with probability

5

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

6

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 theworld. Whether it is for hedging, arbitrage or speculation, �nancial derivatives'transactions constitute a huge proportion of all the transactions in the �nancialmarkets. A �nancial derivative is a �nancial instrument whose value depends onanother underlying variable which is usually more basic (Hull, 2003). It is worthmentioning that some derivatives depend on another derivative as an underlyingvariable.

Options derivatives have grown more popular in the �nancial market since theChicago Board Options Exchange was established in 1973 (Hull, 2003). Todaymany 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 inthe following pair:

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

• A European put option gives the right but not the obligation to sell theunderlying 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

7

on maturity, American options gives its owner further control over the exerciseright, with the ability to exercise at any time up to the maturity date. Thisbroader 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 standardEuropean options1. It was not until 1979 that pricing American options becamepractically possible through the work of JC Cox, SA Ross and M Rubinstein onthe 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 constantsand 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, anddZ(t) is a standard Wiener increment, dZ(t) ∼ N(0, dt). Additionally, the CRRdiscretizes in time the stochastic process above so that the stock price follows theprocess,

∆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 numberof the equidistant sub-intervals of the �nite time interval [0, T ]. The time epochsin this discrete setting becomes 0,∆t, 2∆t, . . . , N∆t. For simplicity, we enumeratethe time epochs as time-steps 0, 1, 2, . . . , n− 1, n, n+ 1, . . . , N .

The solution of the stochastic di�erential equation (1.1) which is known fromBlack and Scholes (1973) and Merton (1973), can also be discretized as follows,

S(n+ 1) = S(n) exp{

∆t(µ− σ

2

2

)+ σ�√

∆t}, (1.2)

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

S(n+ 1) = S(n) exp{σ�√

∆t}. (1.3)

Equation (1.3) yields the binomial tree process, which can be easily evaluatednumerically. The CRR model captures the essence of the stock movement to ahighly accurate level and it is computationally possible for a reasonably large

1By a standard option we refer to an option with a stock as the underlying variable and astandard payo� functions, i.e. max (0, S(t)−K) for calls and max (0,K − S(t)) for puts.

8

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

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

In recent years the demand for stochastic volatility models increased and it isno 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 thestochastic volatility model. There are many other stochastic volatility modelssuch as the two factors volatility model that was introduced in Heston (1993)and later in Christo�ersen et al. (2009), which assumes that the volatility of therisk-neutral stock price process is determined by two factors, where the varianceof the stock return is the sum of the two variance factors. However, valuationof American option pricing is a very challenging problem under these types ofmodels. There are some results on both numerical and analytical approximationsunder these stochastic volatility models, but far too few results in properties ofearly exercising decisions of American options.

The early exercise boundary (EEB) is a term associated with American optionpricing. It can be de�ned 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 toexercise, at each time step up to the maturity date. In addition to being the besttool for optimizing early exercise decisions, the EEB is considered an importantelement of pricing of American options. This was demonstrated by Ramaswamyand Sundaresan (1985) and Brenner et al. (1985), where they used the implicit�nite di�erence method by Brennan and Schwartz (1978) to �nd the EEB andthe prices of American options on future contracts simultaneously. Kim (1990)and Jacka (1991) derived the EEB in implicit form and used the resulted integralequation to evaluate the price of American options. All methods mentioned aboveinvolving the EEB assumed constant volatility. On the other hand, Tzavalis andWang (2003) used Chebyshev polynomials to approximate the EEB under stochas-tic volatility model, and integral representation of the American option price interms of the boundary.

Regime-switching or switching-state models are considered a middle ground2And many other �nancial derivatives.

9

between deterministic models and fully stochastic ones. Such models have beenused in di�erent methods for option pricing. For example, in Naik (1993) and Guo(2001) the authors assumed a geometric Brownian motion with switching-statevolatility. In Elliott et al. (2005) and Fard et al. (2014) the authors argue that dueto the incompleteness that is introduced to the market by the uncertain economicstate, the risk-free measure is no longer unique and they use Esscher transformin order to �nd the minimal entropy martingale measure for pricing Americanoptions, which is beyond the scope of this thesis but certainly worth mentioning.

Finally, Aingworth et al. (2006) suggested that discrete-time Markov processwith a small number of states, is similar to a probabilistic model known as normalmixture di�usion (Alexander, 2004; Brigo and Mercurio, 2002), where the volatilityfollows a speci�c discrete distribution. Sato and Sawaki (2014) followed a similarmodel for pricing non-standard callable American options and even have a plot ofEEBs.

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 ofstudy is the EEBs associated with the American option pricing and the fair pricesof both American and European options. By successfully constructing a recombin-ing tree, we obtain a convenient setting for in-depth analysis on EEB of Americanoptions under this model. We note that the study of EEBs was not carried out inAingworth et al. (2006) and that our numerical study on EEBs for the standardAmerican 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 afully detailed algorithm in the mentioned literature. We believe that our algorithmprovides a straightforward description of the model implementation problem. Incombination with our MATLAB code in Appendix A, our algorithm can be easilymodi�ed to examine other types of payo� functions or for further extending themodel.

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 extensivenumerical results for both American and European puts and calls, with di�erentvolatility states and di�erent transition probability matrices (TPM). In partic-ular, we obtain the EEBs for American put options and American call optionswith dividend-paying underlying stock. In conducting numerical studies on EEBswe vary the parameters and demonstrate the EEBs behavior under our model.Finally, we give three di�erent approximations for European call prices under our

10

studied model, using approximation formulas. As a by-product, we also make acorrection 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 CRRBinomial model. In Chapter 3 we describe our model, the extended CRR modelwith a Markov switching-state volatility. We include a proof of its complexitylevel under that setting. Chapter 3 also explains our algorithm under the two-state version of the model for pricing American options and obtaining the earlyboundary of exercise.

In Chapter 4 we present our experimental studies. The properties of the earlyexercise boundary under two-state model with di�erent volatility values and di�er-ent TPMs, are to be examined. We include pricing tables for both American andEuropean options. We also include approximations to the extended CRR usingapproximation formulas. Finally, in Chapter 5 we summarize our results and guidethe reader to some related further research topics.

11

Chapter 2

A Review of Classical CRR Model

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

Throughout this report we use the notation n to describe the discrete-time-stepnumber, 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 stochasticprocess given by Equation (1.3), we identify the following up- and down-factors,

u = exp{σ√

∆t}, d = exp

{− σ√

∆t}

=1

u.

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

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

V (n) =m∑i=0

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

where θi is the ith asset in the portfolio.

De�nition 2 (Kijima (2016)). A contingent claim X is said to be attainable ifthere exists a replicating portfolio {θ(n);n = 0, 1, . . . , N}, that is a self-�nancingportfolio such that V (N) = X.

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

12

De�nition 3 (Kijima (2016)). An arbitrage opportunity is the existence of someself-�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 pro�t with no risk with positive probability, i.e. V (N) ≥ 0with probability 1 (a.s.), and V (N) > 0 with probability p > 0.

Theorem 1 (No-Arbitrage Pricing, Kijima (2016)). For a given contingent claimX, 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 contingentclaim, i.e. V (0) = X.

De�nition 4 (Kijima (2016)). A stochastic process {X(n);n = 0, 1, . . . , N} thatis de�ned 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) |] 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] = S∗i (n) n =0, 1, . . . , N − 1 holds for all i and n, where E∗ is the expectation underthe risk-neutral probability measure, {d∗(n)} is the dividend process thatis adapted to the �ltration {Fn} and S∗(n) is the denominated stock pricewith 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 takein our portfolio a ∆ amount of the stock S, along with a correspondent B amountinvested in the risk-free account. Then we equate the value of the portfolio to thevalue of the option V , so that the portfolio is V (n) = S(n)∆ + BS0(0), whereS0(n) = e

nr∆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 = Vu, w.p. p,

∆dS(n) +Ber∆t = Vd, w.p. 1− p.(2.1)

13

where u and d denote the up- and down-factor, respectively. Solving the abovesystem 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 isself-�nancing. Substitute back in our portfolio, we get

V (n) =

(Vu − Vdu− d

)+

(uVd − dVu(u− d)er∆t

),

=e−r∆t

[er∆tVu − er∆tVd − uVu + uVd + uVu − dVu

u− d

],

=e−r∆t

[Vu

(er∆t − du− d

)+ Vd

(u− er∆t

u− d

)],

orV (n) = e−r∆t [Vup

∗ + Vd(1− p∗)] , (2.2)where

p∗ =er∆t − du− d

. (2.3)

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

Note that we can de�ne Q = {p∗, (1 − p∗)} as a probability measure, sinceit 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 takethe risk aversion of the investors in consideration. More importantly, it ful�lls theconditions in De�nition 5; �rstly Q(A) is zero if and only if r = σ = 0 which is areasonable assumption for the real-world measure P(A). Furthermore, it is readilyseen 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 ofasset pricing (Harrison and Kreps, 1979; Harrison and Pliska, 1981).

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

V (0) = E∗[

X

S0(T )

], (2.4)

14

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

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

De�nition 6 (Complete Market, Kijima (2016)). A security market is said to becomplete if and only if every contingent claim is attainable.

Theorem 3 (Kijima (2016)). Suppose that a securities market admits no arbitrageopportunities, 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 optionmust match the replicating portfolio value. Hence by Theorem 1 the value of theEuropean option is given by the following recursion,

V (n) = e−r∆tE∗[V (n+ 1)],

where n = 0, 1, 2, . . . , N − 1 denotes the time step number. At time T we have tocalculate 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 variablesXi ∼ B(p∗), that is Wn ≡ X1 + X2 + · · · + Xn, n = 1, 2, . . . and Wn takes thevalues 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.

De�nition 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 stochasticprocess {X(n)} de�ned on a probability space(Ω,F ,P) with �ltration {Fn}. If theprocess is a martingale, then for any stopping time τ ∈ τ0, where τn denotes theset of stopping times that take values in the set {n, n+ 1, . . . , N}, we have,

E[X(τ)] = E[X(n)], n = 0, 1, 2, . . . , N. (2.6)

15

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 toaccount for the freedom of early exercise and take the highest of the two values sothat the value of an American option Y with a payo� function h(x) is given by,

Y (n) = max{h(S(n)), e−r∆tE∗[Y (n+ 1)]

}, (2.7)

where S(n) is given by (2.5).

Theorem 5 (Kijima (2016)). Suppose that there exist a risk-neutral probabilitymeasure Q that the adapted process {Z(n);n = 0, 1, . . . , N} that is de�ned suchthat,

Z(n) = maxτ∈τn

E∗[Y ∗(τ) | Fn], n = 0, 1, . . . , T,

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

τ(n) ≡ min{s ≥ n : Z(s) = Y ∗(s)}. (2.8)

The algorithm for American options under the CRR starts by evaluating thestock and the option at maturity. Then we cover all the time steps excluding thematurity, with a loop. Inside the time level loop, we have another loop, that willcover all the nodes at that current level. At each node the algorithm will evaluateboth the intrinsic value of the option, if we are to exercise, and the payo�, if weare to continue. Then we compare the two values and assign the larger one to thevalue of the option. Once all nodes at the current time level are covered, we canrecover the price of the option, along with the stock prices that we can create theEEB from.

Our model is the extended CRR model. Since our volatility is a stochasticprocess, as proven by Romano and Touzi (1997), we have an incomplete market.When the market is incomplete, there is more than one equivalent martingalemeasure. Hence we face a problem of selecting a martingale measure for optionpricing. In practice, this is not a problem as one can calibrate the model to optiondata to obtain the parameters under the market martingale measure. However, asour 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.

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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 substantialamount of important details.

3.1 Extended CRR Model

The CRR can be easily extended to regime-switching volatility, so that under thesame 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{σi√

∆t},

di = exp{− σi√

∆t},

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

17

switching-state is given by,

P =

1 . . . κ 1 p11 . . . p1κ... ... . . . ...κ pκ1 . . . pκκ

.

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

V i(n+ 1) =

{∆u1S(n) +Be

r∆t = V 1u , w.p. p1,

∆d1S(n) +Ber∆t = V 1d , w.p. 1− p1.

w.p. pi1,{...{...{...{∆uκS(n) +Be

r∆t = V κu , w.p. pκ,

∆dκS(n) +Ber∆t = V κd , w.p. 1− pκ.

w.p. piκ,

where V i is the value of the option (or the replicating portfolio). Notice thatthe 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 volatilitystates and the stock movement probabilities are independent, solving the abovesubsystem separately yields κ di�erent amounts of ∆ and B. By substituting theresulted amounts weighted by their transition probabilities and simplifying in thesame manner that we did earlier in Chapter 2 we get,

V i(n) =e−r∆t{

[V 1u p1 + V 1d (1− p1)]pi1 + [V 2u p2 + V 2d (1− p2)]pi2 + . . .

+ [Vuκpκ + Vdκ(1− pκ)]piκ

},

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

V i(n) =e−r∆t{V 1u p

1pi1 + V1d (1− p1)pi1 + V 2u p2pi2 + V 2d (1− p2)pi2 + . . .

+ Vuκpκpiκ + Vdκ(1− pκ)piκ

}.

(3.1)

Assuming that the TPM that we are using, has risk-neutral probabilities, weget that the risk-neutral probability measure vector that corresponds to the initialstates vector is as follows,

18

Q =

p1pi1,

q1pi1,... i = 1, 2, . . . , κ,pκpiκ,

qκpiκ,

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

pi =er∆t − diui − di

,

qi = 1− pi.

We can see thatQ ful�lls the martingale condition since it is a weighted averageof 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 e�ect that we explained earlier. The TPM in thiscase is of the form,

P =

H L[ ]H pHH pHLL pLH pLL

where, i =H,L are the volatility states. Since there are only two states, we referto 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 theoriginal CRR, nevertheless, it does not have exponential complexity but ratherpolynomial one. To analyze the complexity of this model, we �nd the count systemof 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. thenumber of steps taken in each movement factor.

1Here we are not accounting for the further combination that is allowed due to the fact thatud = 1, see Putri et al. (2018) for example.

19

u3H

u2H u2HuL

u2HdH

uHuL u2HdL

uH uHu2L

uHdH uHuLdH

uHuLdL

uHdL uHd2H

uL uHdHdL

u2L uHd2L

H,L

uLdH u3L

dH u2LdH

uLdL u2LdL

uLdHdL

d2H uLd2H

dL uLd2L

dHdL d3H

d2HdL

d2L dHd2L

d3L

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

20

It is given that the sum of all the powers is equal to n, keeping in considerationthe powers can take values 0, 1, . . . , n. We also know that there are 2κ di�erentpossible 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 to2κ− 1 since in this setting we can repeat zeros 2κ− 1 since, it is required that atleast one of the powers is greater than zero. Hence, we can conclude that we havein 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∑j=0

(j + 2κ− 1

2κ− 1

)=

(2κ− 12κ− 1

)+

(2κ

2κ− 1

)+

n∑j=2

(j + 2κ− 1

2κ− 1

),

expanding the �rst two terms using the de�nition of combination, we get,n∑j=0

(j + 2κ− 1

2κ− 1

)=

(2κ− 1)!(2κ− 1)!

+(2κ)!

(2κ− 1)!+

n∑j=2

(j + 2κ− 1

2κ− 1

)

=(2κ)!

(2κ)!+

2κ(2κ)!

(2κ)!+

n∑j=2

(j + 2κ− 1

2κ− 1

)

=(2κ)!(1 + 2κ)

(2κ)!+

n∑j=2

(j + 2κ− 1

2κ− 1

)

=(2κ+ 1)!

(2κ)!+

n∑j=2

(j + 2κ− 1

2κ− 1

)

=(2κ+ 1)!

(2κ)!+

2κ(2κ+ 1)!

(2κ)!(2!)+

n∑j=3

(j + 2κ− 1

2κ− 1

)

=2(2κ+ 1)! + 2κ(2κ+ 1)!

(2κ)!(2!)+

n∑j=3

(j + 2κ− 1

2κ− 1

)

=(2κ+ 2)!

(2κ)!(2!)+

n∑j=3

(j + 2κ− 1

2κ− 1

)...

=(2κ+ n)!

(2κ)!n!

21

Proposition 2 tells us that the tree under the extended CRR has a polynomialgrowth, 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)!

=

∏2κ−1i=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 insigni�cant e�ect onthe probability distribution of the existing securities. The �nancial market isfrictionless, that is, there are neither transaction fees nor taxation. The securitiesthat we are using are assumed to be in�nitesimally divisible, short selling is allowed,with no restrictions. We also assume that the rates of the risk-free securities arethe 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 looping backward toward the present value, takingadvantage of the combining e�ect we mentioned earlier. We start by de�ning zeromatrices for the stock values S and option values fκ, and zero vectors for the earlyexercise values.

The algorithm starts using Equation (3.2) and the payo� function to evaluatethe stock and the option values at the last time level, i.e. at the maturity. Thenwe 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 atthat current level. At each node the algorithm evaluates both the payo� valueof the option through Equation (3.1) if we continue and its intrinsic value if we

22

exercise, then compare the two values and hold the larger as the value of theoption. Furthermore, if the intrinsic value is larger than the payo� then we take acopy of the value at current node and put it in a vector Eκ. After that all nodesat the current time level are covered, the function takes the minimum value thatwe 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 pathfor the same stock.

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

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 andapplied mathematics in general. For an inexperienced programmer, MATLABrequires 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 computationspeed. 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. Thelargest number of steps we experimented with is 120 which took about 20 hours.Given the complexity of the tree, our function's e�ciency is satisfactory.

The most challenging part of the algorithm is in line 18, where it is requiredto identify the parent nodes in the n + 1 time-step. In order to achieve this inan e�cient manner and without the need of tracing complex indices, we used asimple fact about the change in the power vector by which we mean the set of theexponents that corresponds to the set of movement factor. For example, in thebasic case of two states, as in Figure 3.1, we have that the price of the stock atany 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 timecorresponds to increase in one of the values in this vector, subtracting the currentnode's power vector from all the power vectors in the next time step identi�es allthe parent nodes.

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

23

2 0 0 01 1 0 01 0 1 01 0 0 10 2 0 00 1 1 00 1 0 10 0 2 00 0 1 10 0 0 2

.

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

1 0 0 00 1 0 00 0 1 00 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 varietyof methods one can use to obtain the required indices from the matrix above.With this method we do not need to have a speci�c order to the nodes. We canorder the obtained nodes with the help of the resulting non-negative vectors. Theposition of the digit 1 in the non-negative vector corresponds to the type of theparent node, i.e. the type of the movement towards this parent node.

The MATLAB function provided by Isaac (2019) is an e�cient method thatour MATLAB function utilizes for obtaining power matrices, i.e. sets of powervectors.

24

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 =

(Nsteps+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 do9 for j ← 1 to Nsteps do10 Evaluate S;11 end

12 end

13 V1(all, Nsteps) = · · · = Vκ = h(S(all, Nsteps));14 for n← (UNsteps− 1) to 1 do15 Define NLeaves =

(n+2κ−1

2κ−1

);

16 Define nLeaves =(n−1+2κ−1

2κ−1

);

17 for h← 1 to NLeaves do18 Identify all 2κ parent nodes;19 Evaluate V1, . . . , Vκ;20 V1, . . . , Vκ = max((V1, . . . , Vκ), h(S));21 if V1, . . . , Vκ < h(S) then22 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);

25

Chapter 4

Numerical Results and EEBs

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

Here we mention that in the case of no dividend and non-negative risk-freerate, American calls are never optimally exercised before maturity. Since giventhe 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 andshort 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 thischapter we assume that there is no dividend on our underlying stock, with theexception 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 violatethe put-call parity for American options (Hull, 2003).

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

S(0)−K ≤ C − P ≤ S(0)−Ke−rT ,

26

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

Given the proposition above and the fact that American and European callshave 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 someof the prices in Aingworth et al. (2006). For example, we take the call price inthe last entry of their Table 1, Call(H)= 10.642 and the American put price in thelast entry of their Table 2, Put(H)= 37.019. The prices correspond to the sameparameters K = 115.64, T = 1, r = 0.05 and S(0) = 100. Substitute in the leftpart 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 weshow here, is not a unique case. In Table 4.11 we redo the last three rows fromthe mentioned tables in Aingworth et al. (2006) and we get signi�cantly di�erentprices.

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)European Americans

Put(H) Put(L) Put(H) Put(L)1 94.61 19.126 12.062 9.122 2.058 9.607 2.1861 105.13 14.229 5.693 14.231 5.696 15.070 6.4781 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 saythat an option can exist in three states at any time during its life (Hull, 2003):

• at-the-money (ATM), when K = S,

• in-the-money (ITM), when S > K for calls and S < K for puts,

• out-of-the-money (OTM), when K > S for calls and S > K for puts.1the staying probabilities for this table are obtained using the generator matrix given in

Aingworth et al. (2006) paper.

27

We say that the option is deeper in-the-money or deeper out-of-the-money as thevalues of S and K gets further apart. This relation between K and S, is usuallyreferred to as the moneyness (Hull, 2003). We also need to denote CallH and PutHas call and put option prices given that the initial state is the High state, and CallLand 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 re�ecton the results. In Table 4.2, we vary the moneyness and the maturity and �xpHH = 0.9 the probability of staying at the High state, pLL = 0.95 the probabilityof staying at the Low state. We evaluate ATM, ITM and OTM puts and calls,with di�erent 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 CallLEuropean Americans

PutH PutL PutH PutL0.25 80 21.34 21.23 0.34 0.24 0.35 0.240.25 90 12.83 12.51 1.71 1.39 1.74 1.410.25 100 6.49 6.02 5.25 4.78 5.36 4.870.25 110 2.78 2.37 11.42 11.00 11.74 11.290.25 120 1.04 0.79 19.55 19.30 20.27 20.040.5 80 23.09 22.82 1.11 0.84 1.14 0.860.5 90 15.42 14.91 3.20 2.69 3.29 2.760.5 100 9.51 8.84 7.04 6.38 7.29 6.580.5 110 5.48 4.82 12.76 12.10 13.33 12.590.5 120 2.97 2.45 20.01 19.48 21.11 20.521 80 26.42 25.91 2.52 2.01 2.63 2.081 90 19.59 18.84 5.20 4.45 5.47 4.651 100 14.06 13.14 9.18 8.27 9.76 8.721 110 9.85 8.89 14.48 13.52 15.55 14.421 120 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 theoryof option pricing, e.g. the deeper in the money the option is the higher its value andthe longer the maturity of the option, the higher its value. Most importantly, forthe European options, our prices are not a direct source of arbitrage opportunitiesand 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 thecorresponding TPM and �x the rest of the parameters for ATM options.

28

Table 4.3: Prices of options with, N = 50, S(0) = K = 100, r = 0.05, T = 0.25.

σH σL pHH pLL CallH CallLEuropean Americans

PutH PutL PutH PutL0.3 0.25 0.99 0.90 6.49 6.34 5.17 4.68 5.36 5.220.3 0.25 0.90 0.85 6.23 6.18 4.99 4.93 5.10 5.040.3 0.25 0.90 0.80 6.29 6.25 5.05 5.00 5.15 5.110.3 0.25 0.90 0.70 6.36 6.33 5.12 5.09 5.22 5.200.3 0.25 0.80 0.70 6.22 6.20 4.98 4.96 5.08 5.070.3 0.25 0.70 0.80 6.03 6.01 4.79 4.77 4.89 4.870.3 0.25 0.70 0.90 5.89 5.86 4.64 4.61 4.75 4.720.3 0.25 0.60 0.95 5.74 5.71 4.50 0.47 4.61 4.580.3 0.25 0.40 0.95 5.69 5.67 4.45 4.43 4.55 4.540.3 0.25 0.30 0.90 5.74 5.73 4.49 4.49 4.60 4.600.3 0.25 0.10 0.15 6.10 6.11 4.86 4.87 4.96 4.970.3 0.25 0.80 0.15 6.41 6.41 5.17 5.17 5.27 5.270.5 0.15 0.99 0.90 9.59 6.30 8.35 5.06 8.99 8.160.5 0.15 0.95 0.80 9.62 9.32 8.38 8.07 8.48 8.170.5 0.15 0.65 0.60 8.16 8.12 6.92 6.88 7.02 6.980.5 0.15 0.52 0.65 7.47 7.44 6.22 6.20 6.32 6.290.5 0.15 0.52 0.55 7.85 7.84 6.61 6.60 6.71 6.700.5 0.15 0.50 0.98 7.79 7.78 6.55 6.54 3.22 2.970.5 0.15 0.15 0.95 4.38 4.36 3.14 3.11 3.25 3.220.5 0.15 0.15 0.90 4.92 4.91 3.68 3.67 3.78 3.770.5 0.15 0.15 0.85 5.35 5.35 4.12 4.12 4.22 4.220.5 0.15 0.15 0.80 5.72 5.73 4.48 4.49 4.58 4.590.5 0.15 0.15 0.70 6.30 6.32 5.06 5.08 5.16 5.180.5 0.15 0.15 0.60 6.74 6.77 5.50 5.53 5.60 5.630.5 0.15 0.15 0.20 7.84 7.89 6.60 6.65 6.71 6.760.5 0.15 0.15 0.10 8.02 8.07 6.78 6.83 6.89 6.940.5 0.15 0.05 0.10 7.85 7.91 6.61 6.66 6.71 6.770.5 0.15 0.01 0.02 7.92 7.97 6.68 6.73 6.79 6.840.75 0.1 0.9 0.05 14.72 14.73 13.48 13.49 13.59 13.590.75 0.1 0.9 0.3 14.50 14.46 13.26 13.22 13.36 13.320.75 0.1 0.9 0.5 14.18 14.08 12.94 12.84 13.04 12.940.75 0.1 0.9 0.7 13.53 13.28 12.29 12.04 12.39 12.130.75 0.1 0.9 0.9 11.43 10.56 10.19 9.32 10.28 9.400.75 0.1 0.7 0.9 8.42 7.98 7.18 6.74 7.26 6.810.75 0.1 0.5 0.9 7.06 6.83 5.82 5.58 5.90 5.660.75 0.1 0.3 0.9 6.24 6.14 5.00 4.90 5.09 4.990.75 0.1 0.05 0.9 5.57 5.59 4.33 4.35 4.42 4.44

29

The prices in Table 4.3 holds for the put-call parity and re�ect mostly intu-itive results. The gap between the prices of the same option with di�erent initialvolatility state, is larger with higher probabilities of staying and smaller with lowerprobabilities of staying. The gap becomes smaller as the TPM approaches a sta-tionary distribution and it disappears completely when the TPM is at a stationarydistribution. 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. Wegive 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 probability of transiting toHigh, the prices at High are lower than the prices at Low. It is mentioned in Satoand Sawaki (2014) in Assumption 2.1, that the TPM is assumed to be stochasticincreasing matrix, that is, for each i ≤ κ,

∑κλ=i phλ is non-decreasing in h. This

is to conciliate with the following su�cient condition for option prices: With ahigh volatility state the option prices should be greater than with a low volatilitystate. Our results show the importance of the stochastic increasing TPM assump-tion since validation of this assumption results in an over-weighted Low state thatleads to prices that violate the overly mentioned su�cient condition for optionpricing.

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 stepsthe initial state becomes less important as the chain settles into its stationarydistribution. The stationary distribution (limiting distribution) π is calculated bysolving the system of equations π> = π>P under the condition

∑for all i πi = 1.

The stationary distribution always exists in the case of two states, as long theprobabilities of staying are not equal to 1 or 02. Also, in Fuh et al. (2012) theauthors had a similar switching-state model, where they mention that in the caseof the unknown current state, a weighted average of the resulted prices can beused 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 denotethe weighted average for a European call price by Callavg and our approximationbecomes,

2Classi�cation of Markov chains and TPMs are beyond the scope of this thesis, thereforewe chose to omit the theoretical ground for the limiting distribution and its existence, see Ross(2014) for details.

30

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 inBall and Roma (1994), that the average variance can be used as an approximationof a stochastic variance in the case of European options if the variance σ2 is notcorrelated with the underlying asset. In our case, we can take the square rootof the average of the volatility states weighted by the limiting distribution. Wedenote σ̄ as our approximation for two volatility states and we get,

σ̄ =√πH(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 bethe approximations of a European call, resulted from using σ̄ 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 itas a benchmark for the last two. We denote RECRR and REBSM to be the relativeerrors 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.

31

Table 4.4: Prices of European call options under the extended CRR, along withthree di�erent 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 REBSM0.3 0.25 0.99 0.90 6.494 6.359 6.482 1.7× 10−3 2.8× 10−30.3 0.25 0.90 0.85 6.235 6.177 6.212 4.8× 10−3 3.2× 10−40.3 0.25 0.70 0.90 5.887 5.856 5.864 4.8× 10−3 3.4× 10−40.3 0.25 0.60 0.95 5.738 5.711 5.714 3.8× 10−3 5.2× 10−40.3 0.25 0.40 0.95 5.688 5.675 5.676 3.5× 10−3 8.8× 10−40.3 0.25 0.30 0.90 5.736 5.731 5.731 4.4× 10−3 1.7× 10−40.5 0.15 0.99 0.90 10.121 9.321 10.048 8.9× 10−4 5.6× 10−30.5 0.15 0.95 0.80 9.619 9.316 9.559 2.5× 10−3 2.2× 10−30.5 0.15 0.15 0.85 5.358 5.358 5.358 3.7× 10−3 5.6× 10−40.5 0.15 0.15 0.80 5.721 5.730 5.728 4.4× 10−3 5.2× 10−50.5 0.15 0.15 0.70 6.299 6.320 6.315 5.4× 10−3 9.5× 10−40.5 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 di�erent parameters. Wehave chosen a handful of EEB plots to include in our report.

(a) σH = 0.4, σL = 0.2 andpHH = 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

Und

erly

ing

Ass

et p

rice

Continuation Region

Exercise Region

EEB when H

is the active state

EEB when L is the active state

(b) σH = 0.3, σL = 0.25 andpHH = 0.9, pLL = 0.95.

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

32

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

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

0.05 0.1 0.15 0.2 0.25

t

65

70

75

80

85

90

95

100

Und

erly

ing

Ass

et p

rice Continuation Region

Exercise Region

EEB when H

is the active state


(a) σH = 0.5, σL = 0.15 andpHH = 0.9, pLL = 0.4.

(b) σH = 0.5, σL = 0.15 andpHH = 0.99, pLL = 0.98.

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

We notice in Figure 4.2 that the spread of the boundaries is also being a�ectedby the values of the TPM. The higher the staying probabilities, the more spreadthe boundaries are.

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

33

0 0.05 0.1 0.15 0.2 0.25

t

88

90

92

94

96

98

100

Und

erly

ing

Ass

et p


Exercise Region

EEB when H

is the active state


(a) σH = 0.5, σL = 0.15 andpHH = 0.01, pLL = 0.99.

0 0.05 0.1 0.15 0.2 0.25

t

75

80

85

90

95

100

Und

erly

ing

Ass

et p

rice

Continuation Region

Exercise Region

EEB when H

is the active state


(b) σH = 0.5, σL = 0.15 andpHH = 0.3, pLL = 0.7.


(a) σH = 0.5, σL = 0.15 andpHH = 0.01, pLL = 0.02.

0 0.05 0.1 0.15 0.2 0.25

t

75

80

85

90

95

100

Und

erly

ing

Ass

et p


Exercise Region

EEB when H

is the active state


(b) σH = 0.5, σL = 0.15 andpHH = 0.2, pLL = 0.6.


In Figure 4.4 we notice again the counter e�ect that occurs when we have aprobability of staying at the High state is less than the probability of transitingto High. That is, when we violate the assumption of a stochastic increasing TPM

34

mentioned in Section 4.2. Therefore, we see that the boundary in the High stateis 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 ofdividend, 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 equalto 100 and 3−months maturity. As expected, the obtained boundary has anasymptotic behavior where it approaches the maturity from above, as opposedto the asymptotic behavior in the put case where it approaches the maturity frombelow, which coincides with results obtained in Jönsson (2001) and in Sato andSawaki (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 thecontinuous dividend yield rate (Hull, 2003; Kijima, 2016).

We are free to choose any positive value for dividend yield and the risk-freerate, 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 = 0.15 andpHH = 0.7, pLL = 0.8.

(b) σH = 0.5, σL = 0.15 andpHH = 0.98, pLL = 0.99.

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

35

In the �gure above we see that the EEBs for calls have a similar behavior withrespect 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 todemonstrate other important variations. Nevertheless, it is worth mentioning thatvarying the moneyness with the other parameters �xed results in shifts in the stockprice axis. As expected, the deeper in-the-money the option is initially the largerthe exercise region is.

36

Chapter 5

Conclusion

In this report, we conclude that the switching-regime volatility extended CRRmodel is a simple and e�ective way of obtaining the early exercise boundariesalong with the fair prices of options. We demonstrate that the model is suitablefor pricing both American and European options. We also show how the boundarychanges as the parameter of the underlying and we illustrated how the changescoincide with the theoretical ground of American options. We analyze numericallythe 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 CRRmodel are consistent and they have a positive relationship with the moneyness,the maturity duration, and the volatility values. The gap between two pricesthat correspond to two initial volatility states, has a positive correlation with thestaying probabilities in the TPM (Table 4.3). In Section 4.3 we demonstrate howcan we use a simple approximation in the case of the unknown initial state. Wealso demonstrate approximations that use �xed volatility value, by approximatingthe volatility states into a �xed value. In Table 4.4 we give a brief comparisonbetween these approximations.

The EEBs plots we obtained show an asymptotic behavior as it approachesmaturity. Our results demonstrate how the spread of the two EEBs is e�ected bythe volatility values and the staying probabilities. We even show how validatingthe assumption of a stochastic increasing TPM, could result in an inverse positionof the EEBs.

37

5.2 Further Research

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

One can also do numerical analysis and parameterizations to match the TPMof the volatility states with the market. Furthermore, numerical comparisons withother 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 payo�s can be step-wise, logarithmic orsquare of a standard payo�, as studied in Jönsson (2001). The algorithm and thecorresponding MATLAB function that we provide can be modi�ed to accompanymost payo�s and it is of interest to demonstrate the limitation of this model, interms of non-standard payo�s.

38

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41

Appendix A

Code

A.1 MATLAB Function

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

1 f unc t i on [ pr i ce1 , p r i c e 2 ] = . . .2 Bintree_M(S0 , K, r , y , T, sigma , P, Nsteps , opttype )3

4 % A Function that re turn the p r i c e o f Both5 % American and European Option6 % With Markov Switching two State v a r i a b l e s .7 %8 %9 %Input

10 % S0 − cur rent s tock p r i c e11 % K − s t r i k e12 % T − exp i ry time13 % r − i n t e r e s t r a t e14 % y − div idend y i e l d15 % sigma − v o l a t i l i t y va lue s [H,L ]16 % P − The t r a n s i t i o n matrix corre spond ing to the v o l a t i l i t y s t a t e s17 % opttype − 0 f o r a c a l l , o the rw i se a put18 % Nsteps − number o f t imes teps19 %20 %Output21 % pr i c e 1 : opt ion 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 e22 % pr i c e 2 : opt ion p r i c e in the second v o l a t i l i t y s t a t e23 % K1 : the ea r l y e x e r c i s e boundary vec to r f o r the f i r s t s t a t e24 % K2 : the ea r l y e x e r c i s e boundary vec to r f o r the second s t a t e25 %

42

26 %27 % Account f o r the div idend ra t e28 R=r−y ;29 %time−s tep s i z e ( Delta t ) and t r e e parameters30 de l t = T/Nsteps ;31

32 % compute the up and down movements33 u = exp ( sigma ∗ s q r t ( d e l t ) ) ;34 d = 1 ./u ;35 % the d i scount f a c t o r36 a = exp ( (R) ∗ de l t ) ;37 % the p r obab i l i t y o f the up and down movements ac co rd ing ly38 p = ( a − d) . / ( u − d) ;39 q = 1−p ;40 % number o f v o l a t i l i t y va lue s41 Nsigma = length ( sigma ) ;42 % the complexity o f the t r e e as g iven in the model43 NLeaves= nchoosek ( Nsteps−1+(2∗Nsigma ) , (2∗Nsigma )−1) ;44

45 % i n i t i a t i n g payo f f s and Stock p r i c e s in the t r e e in add i t i on to46 % ear l y e x e r c i s e v e c t o r s .47 f 1 = ze ro s (NLeaves , Nsteps+1) ;48 f 2 = ze ro s (NLeaves , Nsteps+1) ;49 S = ze ro s (NLeaves , Nsteps+1) ;50 B = ones (NLeaves , 1 ) ;51 E1 = ze ro s ( Nsteps+1 ,1) ;52 E2 = ze ro s ( Nsteps+1 ,1) ;53

54 % here we match the order o f the p r obab i l i t y vec to r and55 % the movement vec to r56 % The Probab i l i t y ve 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 vec to r61 ud = [ u d ] ;62

63 % Obtaining the powers o f the movements va lue f o r the l e av e s64 [ Nmatrix ] = mult inomial_powers_recurs ive ( Nsteps , Nsigma ∗2) ;65

66 % Compute the s tock and the opt ions va lue s in the l eave nodes at the67 % maturity68 f o r h=1:NLeaves69 f o r j =1: 470 B(h) = ud( j ) ^(Nmatrix (h,5− j ) ) ∗B(h) ;71 end72 S(h , 1 )= S0 ∗ B(h) ;

43

73 end74 f 1 ( : , 1 ) = opttype ∗max(K−S ( : , 1 ) , 0 ) + (1− opttype ) ∗max(S ( : , 1 )−K, 0 ) ;75

76 f 2 ( : , 1 ) = f1 ( : , 1 ) ;77

78 % the l a s t va lue o f the EEB i s always equal to the s t r i k e79 E1( Nsteps+1)=K;80 E2( Nsteps+1)=K;81

82 % va r i a b l e s f o r ho ld ing the ea r l y e x e r c i s e va lue s at each l e v e l83 e1=ze ro s (NLeaves+1 ,1) ;84 e2=ze ro s (NLeaves+1 ,1) ;85

86 % Star t the backward r e cu r s i on87

88 f o r n=1:Nsteps89 % the number o f nodes at the cur rent time step90 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 cur rent one92 Nleaves= nchoosek ( Nsteps−n+(2∗Nsigma ) , (2∗Nsigma )−1) ;93 % power matrix that corresponds to the time s t ep s mentioned above94 [ Nmatrix ] = mult inomial_powers_recurs ive ( Nsteps+1−n , Nsigma ∗2) ;95 [ Nmatri ] = mult inomial_powers_recurs ive ( Nsteps−n , Nsigma ∗2) ;96

97 B = ones (Nnodes , 1 ) ;98 f o r h=1:Nnodes99

100 % B i s the movement f a c t o r s accumulation at the cur rent node .101 f o r j =1:4102 B(h) = ud ( ( j ) ) ^(Nmatri (h,5− j ) ) ∗B(h) ;103 end104 % S i s the expected value o f the a s s e t at the cur rent node105 S(h , n+1)= S0 ∗ B(h) ;106 % phi i s the i n t r i n s i c va lue o f the opt ion at the cur rent

node107 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 prev ious step ,110 % subt rac t the powers o f the cur rent node111 % 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 valued113 % rows .114 Trans = bsxfun (@ (A,B) A−B ,Nmatrix , Nmatri (h , : ) ) ;115 [Row,~ ] = f i nd (Trans

121 % fo r c=1:Nleaves122 % i f Nmatri (h , : ) ps i 1147 e1 (h)= f1 (h , n+1) ;148 end149 i f phi1 > ps i 2150 e2 (h)= f2 (h , n+1) ;151 end152 end153

154 % f i l t e r out the minimum of the ea r l y e x e r c i s e va lue s155 e1 ( e1==0)=NaN;156 e2 ( e2==0)=NaN;157 [ ee1 , I1 ]=min ( e1 ) ;158 [ ee2 , I2 ]=min ( e2 ) ;159 % take the s tock p r i c e s that corresponds to160 % the minimum ea r l y e x e r c i s e va lue s161 E1( Nsteps+1−n)=S( I1 , n+1) ;162 E2( Nsteps+1−n)=S( I2 , n+1) ;163

164 % f i l l the empty va lue s o f the ve c t o r s E with z e ro s165 i f isempty ( ee1 ) | | i snan ( ee1 )166 E1( Nsteps+1−n)=0;167 end168 i f isempty ( ee2 ) | | i snan ( ee2 )

45

169 E2( Nsteps+1−n)=0;170 end171 % re s e t the ve c to r s e f o r the next round172 e1=ze ro s ( Nleaves +1 ,1) ;173 e2=ze ro s ( Nleaves +1 ,1) ;174 end175

176 % the value o f the opt ion at time 0 ( the l a s t l e v e l )177 pr i c e 1 = f1 (1 , Nsteps+1) ;178 pr i c e 2 = f2 (1 , Nsteps+1) ;179

180 % plo t o f the e a r l y e x e r c i s e boundar ies181 k1 = f i nd (E1) ;182 k2 = f i nd (E2) ;183 q1 = length ( k1 ) ;184 q2 = length ( k2 ) ;185 X1 = l i n s p a c e (0 ,T, Nsteps+1) ;186 X2 = l i n s p a c e (0 ,T, Nsteps+1) ;187 Q1 = X1( Nsteps−q1+2:Nsteps+1) ;188 Q2 = X2( Nsteps−q2+2:Nsteps+1) ;189 K1 = E1( Nsteps−q1+2:Nsteps+1) ;190 K2 = E2( Nsteps−q2+2:Nsteps+1) ;191

192 f i g u r e (1 )193 p lo t (Q1,K1, ' c o l o r ' , [ 0 . 8 5 0 0 , 0 .3250 , 0 . 0 9 8 0 ] ) ;194 hold on195 p lo t (Q2,K2, ' c o l o r ' , [ 0 , 0 .4470 , 0 . 7 4 1 0 ] ) ;196 l egend ( 'EEB when \sigma_H i s the a c t i v e s t a t e ' , . . .197 'EEB when \sigma_L i s the a c t i v e s t a t e ' , ' Locat ion ' , ' Best ' )198 x l ab e l ( ' t ' )199 y l ab e l ( ' Stock Pr i ce ' )200 hold o f f201 end

46

A.2 Sub-Function

Our function uses the following function actively to �nd the matrices of the powersof the movement factors at each time level. This function is quoted from Isaac(2019).

1 f unc t i on [ Nmatrix ] = mult inomial_powers_recurs ive (pow , ndim)2

3 % computes the mult inomial expansion o f4 % (x_0 + x_1 + x_2 + . . . + x_ndim)^pow5 % Nmatrix i s a matrix o f powers6 % This i s equ iva l en t to f i nd i n g a l l multi−i n d i c e s with norm=17

8 % Need another th ing to c a l c u l a t e the c o e f f i c i e n t s , but that i s easy9

10 % re cu r s i v e on dimension !11

12 i f ndim==113 Nmatrix = pow ;14 e l s e15 % recu r s e16 Nmatrix = [ ] ;17 f o r pow_on_x1 = 0 :pow18 % say we f i x the power in the f i r s t dimension to be19 % "pow_on_x1" ( 0 , 1 , 2 , . . . )20 % then the p o s s i b l e terms are a l l terms f o r [ ( pow−pow_on_x1) ,21 % ndim−1]22 [ newsubterms ] = mult inomial_powers_recurs ive . . .23 (pow−pow_on_x1 , ndim−1) ;24 % s t i c k on the power f o r the x1 part and add to Nmatrix25 Nmatrix = [ Nmatrix ; [ pow_on_x1∗ ones ( s i z e ( newsubterms , 1 ) ,1 ) . . .26 , newsubterms ] ] ;27 end28 end

47

Appendix B

Proofs

B.1 Proof of Proposition 3

Proposition 3. Given no dividend the prices of American options must obey thefollowing relation,

S(0)−K ≤ C − P ≤ S(0)−Ke−rT ,

where P and C are the price of American call and put, respectively, Both with thesame underlying S(0) and same strike K. Otherwise, arbitrage opportunities canbe produced from those options (Hull, 2003).

Proof. Starting with the right side of the relation, we consider two portfolios thathave the following evaluation at time zero and at time T , where we denote theprice of European call and put to be p and c, respectively,

Table B.1: Two portfolios that include one option each, both options have strikeK and maturity T .

Time 0 At time TS > K S ≤ K

Portfolio A Long European call c S(T )−K 0K unit in risk-free account Ke−rT K K

Total c+Ke−rT S(T ) KPortfolio B Long European put p 0 K − S(T )

Long stock S(0) S(T ) S(T )Total p+ S(0) S(T ) K

Since the values of the two portfolios are equal at time T , then they must beequal at time zero, i.e.

p+ S(0) = c+Ke−rT ,

48

otherwise we can combine the two portfolios and create an arbitrage opportunity.Furthermore, we have C = c since there is no dividend, and P ≥ p, because of

the added value of the early exercise, we get

P + S(0) ≥ C +Ke−rT or S(0)−Ke−rT ≥ C − P.

Secondly, to prove the left side of the relation, in similar manner we take thefollowing two portfolios

Table B.2: Two portfolios that include one option each, both options have strikeK and maturity T .

Time 0 At time TS > K S ≤ K

Portfolio CLong European call c S(T )−K 0

K unit in risk-free account K KerT KerT

Total c+K S(T ) +K(erT − 1) KerTPortfolio D if no early exercise occur

Long American put P 0 K − S(T )Long stock S(0) S(T ) S(T )

Total P + S(0) S(T ) KPortfolio D if an early exercise occur

Long American put P er(T−τ)(K − S(τ))Long stock S(0) er(T−τ)S(τ)

Total P + S(0) er(T−τ)K

Clearly at time T portfolio C is worth at least as much as portfolio D and itmust do so at time zero as well, hence,

S(0)−K ≤ C − P.

49

Appendix C

Bachelor Degree Objectives

Here I list, objectives from the requirements for obtaining a Bachelor degree inSweden, that I ful�lled through working on my thesis. I also include a brief moti-vation after each objective.

• Objective 1: The student should demonstrate knowledge and understandingin the major �eld of study, including knowledge of the �eld's scienti�c basis,knowledge of applicable methods in the �eld, specialization in some part ofthe �eld and orientation in current research questions.

This thesis report indeed demonstrates this objective. I utilize and re�ectupon some of the most important theorems and de�nitions in �nancial math-ematics in Chapter 2. I also reviewed thoroughly one of the most importantapplicable models in the �eld. My report specializes in the problem of earlyexercise boundaries for American options and pricing of both American andEuropean options, which I demonstrated knowledge and understanding of. Ialso provide a brief orientation in current research questions in my literaturereview in Section 1.1.

• Objective 2: The student should demonstrate the ability to search, collect,evaluate and critically interpret relevant information in a problem formula-tion and to critically discuss phenomena, problem formulations and situa-tions.

Through my work on this thesis we evaluate and interpret a lot of numer-ical and theoretical results. This was a specially di�cult task since a fulldescription and similar numerical research for our model, to the best of myknowledge, were not available in the literature. In Chapter 4 I carry outcritical discussions on my results. The problem formulation is presented at

50

the end of Section 1.1. I also give a description of the model in Chapter 3.

• Objective 3: The student should demonstrate the ability to independentlyidentify, formulate and solve problems and to perform tasks within speci�edtime frames.

This objective is demonstrated in deriving the model and the algorithm inChapter 3. It was challenging to produce the results in Chapter 4, since itinvolved a lot of high level problem solving formulating and understanding.I managed to solve all the involved problems in a relativity short time andthere was no delay in terms of deadlines.

• Objective 4: The student should demonstrate the ability to present orally andin writing and discuss information, problems and solutions in dialogue withdi�erent groups.

I ful�ll this objective on writing abilities throughout the report, especially inChapter 5, where I summarize adequately my results, their implications andpossible continuation of the research. As for oral abilities I have a numberof constructive discussions with my supervisor Ying Ni. Furthermore, in thethesis presentation I intend to communicate verbally a clear vision of myreport.

• Objective 5: The student should demonstrate ability in the major �eld ofstudy make judgments with respect to scienti�c, societal and ethical aspects.

This objective is ful�lled through the presentation of project and its re-sults in Section 1.1 and through our commitment to correctly use citationsthroughout the report. My report respects the area of research and presentsopportunities for further research, with an advantage of detailed descriptionof the model on a basic level.

51

option pricing and early exercise boundary of american...

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