Option Pricing Under the Markov-switching Framework Defined by Three States

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School of Education, Culture and Communication

Division of Applied Mathematics

BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Option Pricing under the Markov-switching Framework Defined

by Three States

by

Minna Castoe

Teo Raspudic

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

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School of Education, Culture and Communication

Division of Applied Mathematics

Bachelor thesis in mathematics / applied mathematics Date:

2020-06-12 Project name:

Option Pricing Under the Markov-switching Framework Defined by Three States Author(s): Minna Castoe Teo Raspudic Supervisor(s): Marko Dimitrov Reviewer: Milica Rancic Examiner: Linus Carlsson Comprising: 15 ECTS credits

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Abstract

An exact solution for the valuation of the options of the European style can be obtained using the Black-Scholes model. However, some of the limitations of the Black-Scholes model are said to be inconsistent such as the constant volatility of the stock price which is not the case in real life. In this thesis, the Black-Scholes model is extended to a model where the volatility is fully stochastic and changing over time, modelled by Markov chain with three states - high, medium and low. Under this model, we price options of both types, European and American, using Monte Carlo simulation.

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Acknowledgement

We would like to take the opportunity to express our greatest gratitude to all those who have helped and supported us during the completion of this thesis.

Firstly, we would like to express a deep thank to our supervisor Marko Dimitrov who led our progress, directed our focus and contributed with many useful advice. Secondly, we appreciate the feedback, guidance and suggestions from our reviewer Milica Rancic and her remarks on all aesthetic segments which significantly improved our mathematical writing.

Västerås, Sweden 2020-06-12

Minna Castoe Teo Raspudic

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List of Figures

2.1.1 Transition diagram and transition matrix for two-states MC . . . 8

2.1.1 Transition diagram and transition matrix for three-states MC . . . 8

2.1.1 Transition diagram and transition matrix for four-states MC . . . 9

4.2 Probability transition diagram for three-states Markov chain . . . 22

5.3 Volatility smile for options . . . 25

7.1 Volatility smile for initial high state . . . 34

7.1 Volatility smile for initial medium state . . . 34

7.1 Volatility smile for initial low state . . . 35

7.2.1 Volatility smile for initial high state . . . 36

7.2.1 Volatility smile for initial medium state . . . 37

7.2.1 Volatility smile for initial low state . . . 37

7.2.2 Volatility smile for high initial state (I TPM variation) . . . 38

7.2.2 Volatility smile for medium initial state (I TPM variation) . . . 39

7.2.2 Volatility smile for low initial state (I TPM variation) . . . 39

7.2.2 Volatility smile for high initial state (II TPM variation) . . . 40

7.2.2 Volatility smile for medium initial state (II TPM variation) . . . 41

7.2.2 Volatility smile for low initial state (II TPM variation) . . . 41

7.2.3 Volatility smile for high initial state (I state variation) . . . 43

7.2.3 Volatility smile for medium initial state (I state variation) . . . 43

7.2.3 Volatility smile for low initial state (I state variation)) . . . 44

7.2.3 Volatility smile for high state (II state variation) . . . 45

7.2.3 Volatility smile for medium initial state (II state variation) . . . 45

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List of Tables

1 European option pricing for 3 volatility states, P = [0.7 0.2 0.1; 0.3 0.6 0.1; 0.2 0.3 0.5], state = [0.7 0.5 0.3], S0= 90, r = 0.1, m = 25, I = 10 million . . 33

2 European option pricing for 3 volatility states, P = [0.7 0.2 0.1; 0.3 0.6 0.1; 0.2 0.3 0.5], state = [0.7 0.5 0.3], S0= 90, r = 0.1, m = 100, I = 10 million . 36

3 European option pricing for 3 volatility states, P = [0.5 0.25 0.25; 0.25 0.5 0.25; 0.25 0.25 0.5], state = [0.7 0.5 0.3], S0= 90, r = 0.1, m = 25, I = 10 million 38

4 European option pricing for 3 volatility states, P = [0.4 0.4 0.2; 0.2 0.4 0.4; 0.2 0.4 0.4], state = [0.7 0.5 0.3], S0= 90, r = 0.1, m = 25, I = 10 million . . . 40

7 American option pricing for 3 volatility states and 4 maturities, S0 = 90, r =

0.1, m = 25, I = 10 million . . . 47 8 American option pricing for 3 volatility states and 4 maturities, S0 = 90, r =

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Abbreviations and Acronyms

ARCH Autoregressive conditional heteroskedasticity ATMF At-the-money factor

AMR American option BSM Black-Scholes model CLT Central limit theorem EUR European option

GARCH Generalized autoregressive conditional heteroskedasticity GBM Geometric Brownian motion

LSM Least square Monte Carlo simulation method MC Markov Chain

MCMC Markov-switching model applied in Monte Carlo simulation PDE Partial differential equation

SABR Stochastic Alpha Beta Rho SDE Stochastic differential equation TPM Transition probability matrix

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1 Introduction

Options pricing models are powerful tools in financial mathematics, various models have been used widely to determine the theoretical value of an option that provides an estimation of what an option should be worth, i.e. the fair value of an option, which in turn helps traders to adjust their strategies and portfolios for better future trading.

Unlike the valuation process of the American options which tends to be much harder than the valuation of European options, some simple models that assume the volatility to be con-stant like the binomial model which is a discrete-time model and the Black-Scholes model (BSM) which is a continuous-time model can evaluate financial derivatives of the European-style options. The BSM requires five inputs variables: the current price of the stock, the strike price or the exercise price of an option, time to expiration, risk-free rate and the volatility. Some of the important assumptions made by this model are: the stock prices follow a log-normal distribution i.e. the asset prices cannot be negative, the model assumes also that there are no taxes and no transactions costs and also assumes that the volatility is constant which is unrealistic because it is well known that the volatility fluctuates throughout the life of the option.

Furthermore, the model used for option pricing depends on the choice of the options’ style, i.e. American, European or any other type. Most of the options traded on the exchange are of the American-style rather than European. For the American options, which can be exercised at any time up to the expiration date, the classical BSM cannot be used to value an option of this type and there is no closed-form solution of this model for the American options. However, the binomial option pricing model can be used to evaluate the value of an American option where the option can be priced at each point of a specified time frame under this model.

Since the evaluation of American option price tends to be more challenging and more complicated compared to European options, where closed-form solutions are available, some other methods are used such as the simulation methods. Simulation methods are used widely to deal with options of the American-style, these methods also give numerical result for the complicated options.

In addition to the simulation methods, there are also several methods used for that purpose, some other numerical and analytical methods that are used to evaluate options which tend to be more complicated like the American options, are: trinomial tree method which is an exten-sion of the simple binomial model, finite difference method, explicit and implicit, simulation random tree, Monte-Carlo methods and least square Monte Carlo simulation method (LSM) [Zhao J.,2018].

A very important concept which should be considered in the pricing of derivatives is the risk-neutral valuation.

Definition 1.1 (Risk-neutral valuation, Björk T., 2009). A probability measure Q is called a risk neutral measure or, alternatively, a risk adjusted measure or a martingale measure if the following condition holds

St=

1 1 + RE

Q_[S t+1],

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where R is the one-period interest rate, S_t represents the stock price at time "t", and S_t+1 is the value of the stock price at time "t+ 1".

This is also known as the risk-neutral valuation formula, and it states that today’s stock price is given by the discounted expected value of tomorrow’s stock price. Therefore, the risk-neutral valuation uses the Q-probabilities instead of objectives probabilities assuming the absence of arbitrage.

Proposition 1.1 (Björk T., 2009). The Market model is arbitrage free if and only if there exists a risk neutral measure Q.

The risk-neutral valuation method is also known as the martingale measure or the equilib-rium measure. The idea of this method came from some valuation models like the BSM where the value of the option under this model is independent of risk preferences. The method states that, when valuing a derivative or pricing options, it is valid to assume that the investors are risk-neutral and therefore the world is also risk-neutral.

Moreover, a risk-neutral world has two features:

• The expected return on the underlying asset is the risk-free rate.

• The discounted rate used for the expected payoff on an option is the risk-free rate.

1.1 Background and Literature Review

The exact and precise description of market fluctuations and economic cycles has always been an interesting but challenging task as well. The appearance of the Markov-switching framework explaining state transitions offered the new insight into the situation. Very famous BSM has been established by three economists, Fischer Black, Myron Scholes and Robert Merton and published in Boyle P. P. (1977). For this huge discovery, they were awarded Nobel prize for Economics in 1997. However, the main flaw of the BSM is the fact that it relies on constant instead of stochastic volatility and therefore, in reality, some adjustments needed to be added. Markov-switching model could be seen as the extension of the BSM in a way that it fully allows calculations with stochastic volatility, for both European and American option pricing. The main literature sources in which stochastic volatility models are more thoroughly analysed are Hull J.C. and White A. (1987) and Heston S.L. (1993). There exist several models intended for pricing stochastic volatility options as explained in Joubert P. and Vencatasawmy C. (2005), and Wang. P. (2008). Instead of working with constant volatilities, they assume that the volatility changes randomly. It is important to mention that for this thesis one of the main papers is Aingworth D., Das S. and Motwani R. (2006). In the paper, the authors describe their model which indeed has similarities with our topic, but instead with three states they work with two, high and low. In the article written by Fuh C., Wah K., Ho R., Hu I. and Wang R. (2012), the variation of the parameters is made concerning the state of the hidden Markov process.

After and in the years of World War II, the Monte Carlo method has been constructed by three scientists, Neumann J., Ulam S. and Metropolis N. (1949). They were working on the

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development of the Manhattan Project, a large and highly expensive project intended for pro-duction of nuclear weapons. At first, among them appeared just an interesting thought of the probability of getting the successful outcome in the game of solitaire. But this idea was only the basis for enormous discovery in the field of informatics, mathematics, physics, finance and many more. This name for the method is given thanks to Ulam’s uncle who had a bad habit of gambling in Monte Carlo casinos. The idea of making thousands and millions of simulations of the process and thus estimating probability for its happening brought a notable help for the progress of their project, but also for every science that used the same approach in their re-searches. Irish economist and professor Phelim Boyle is the well-known initiator of the usage of the Monte Carlo method in the financial field, with the main purpose of pricing options. His work and early research have been published in the Journal of Financial Economics 4 (1977). Among other significant authors who contributed with their researches, like Johnson H. and Shanno D. (1987), Canina L. and Figlewski S. (1993) and others, especially important for this thesis was Hull J.C. (2012) whose book helped a lot with the understanding of the problem and the principles it is based on. The contribution has been also done in Jia Q. (2009) who fo-cused on the application of the Monte Carlo method in American option pricing, emphasising the role of the Least Squares Monte Carlo method (LSM) for higher dimensions.

1.2 Aim of the Project

This paper aims to price both European and American options using Monte Carlo simula-tion, under a model called Markov-switching model where volatility is assumed to be fully stochastic and modelled by a so-called "Markov Chain" (MC). We consider three states of the stock volatility of this model (high, medium and low) and we include some numerical ex-amples. The numerical analysis is expected to provide some useful results and graphs, which would serve as a strong basis for making the conclusions about this model. Several vital para-meters are going to be varied inside the Monte Carlo simulation, thus offering a clearer picture of the way that those changes affect the outcome. The analysis is done using programming language MATLAB. For each of the variations, all-important option prices for every maturity and three different strike prices will be presented in the form of a table. Further, for European options, graphs of volatility smiles representing each volatility state will be shown and com-pared. The focus will be put on the behaviour of those smiles and the reasons why it is so.

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2 Markov Chains

In financial studies, Markov-switching models are used to analyze the dynamic behaviour of financial variables that are assumed to vary with time. The importance of the inconsistency of the volatility of the stock price has been an interesting area for researchers in the last few years, where the volatility parameter is assumed to be stochastic and fluctuates over time. These fluctuations of the volatility can be well described in regime-switching models letting volatility taking different values depending on the states of the model.

Recent renewed interest in Markov-switching models was largely studied by Hamilton J. D. (1989, 1994). These models have been developed to modify the assumption made about the volatility of the stock price under the BSM, that the volatility is constant. In addition to this model, two more models have received considerable attention to consider the fluctuations of the stock volatility: stochastic volatility models and ARCH models.

The volatility parameter under Markov-switching models is assumed to be modelled by the so-called Markov Chain.

2.1 Discrete-time Markov Chain

Definition 2.1 (Stochastic process, Kijima M., 2016). A stochastic process is a collection of random variables {Xt}t∈T, if T is countable then the process is said be be discrete-time

stochastic process, otherwise the stochastic process is said to be continuous-time stochastic process.

Definition 2.2 (Markov chain, Kijima M., 2016 and Milton J. et al., 1995). LetS be the finite state space. A stochastic process{Xt,t = 0, 1, 2, . . . } is called Markov chain (MC) if, for all

t≥ 0 and all states i0, i1. . . , it−1, it, j ∈S, it satisfies the Markovian property

P(Xt+1= j | X0= i0, X1= i1, . . . , Xt−1= it−1, Xt= it)

= P(Xt+1= j | Xt = it)

= pi j,

where pi jis the probability that the state i will transit to j, also called the transition probability

from i into j, X_t+1represents the future event, X_t is the current event, X₀, X1. . . Xt−1 represent

the past events.

MC is characterized by the Markov property, takes a finite number of values denoted by i, j, where the probability of any future value of Xt+1 equals j, that is, given the past states

X₀, X1, . . . , Xt−1 and the present state Xt, the conditional distribution of any future state Xt+1

depends only on the current state Xt= i, and independent of the past states.

Definition 2.3 (Time-homogeneous Markov Chain, Kijima M., 1997 and Milton J. et al., 1995). A Markov chain is said to be time-homogeneous if the transition probabilities are independent of time t, that is for all i, j ∈S,

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The transition matrix for a homogeneous Markov chain is also independent on time and is denoted byP = (pi j)i, j∈S. For a discrete-time homogeneous Markov chain, the one-step

trans-ition matrix is given by

P = P(t,t + 1), t= 0, 1, 2, . . . . The n-step transition matrix is defined by

P(t,t + n) = Pn, n= 0, 1, 2, . . . . 2.1.1 Components of Markov Chain

MC and its states are well described by three components: the transition matrix, the stationary distribution vector or steady-state vector and the transition diagram.

• The transition matrix which is also called the transition probability matrix (TPM) or the Markov matrix is a matrix consisting of transition probabilities. The transition probab-ility is the probabprobab-ility of transitioning from one state to another.

LetS be the finite state space S = {s0, s1, . . . , sN}. The transition probability matrix P

for N-states is defined by

P =      p₁₁ p₁₂ · · · p_1N p₂₁ p₂₂ · · · p_2N .. . ... . .. ... p_N1 p_N2 · · · p_NN     

where the transition probabilities (pi j)i, j=1,2,...,N for stationary or time-homogeneous

Markov chain is given by

pi j = P{X1= j|X0= i}. = P{X₂= j|X₁= i}. = P{X3= j|X2= i}. .. . = P{Xt+1= j|Xt= i}.

Also, pi j is non-negative, i.e.

p_{i j}≥ 0 and sum up to 1 in each row, that is

N

∑

j=1

pi j= 1, for all i = 1, 2, . . . , N.

These are a one-step transition probabilities, which can be extended to n-step transitions. The n-step transition probability is defined as

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whereas, for computing multi-step transition probabilities, Chapman-Kolmogorov equa-tion is used p(m+n)_{i j} = N

∑

k=1 p(n)_ik p(m)_{k j} , for all i, j = 1, 2, . . . , N.

Definition 2.4 (Distribution Vector, Kijima M., 2016 and Milton J. et al., 1995). The probability distribution of a random variable being(X ), taking values in a finite space S = {s1, s2, . . . , sN}, can be represented by the row distribution vector πππ

π

ππ = (π1, π2, . . . , πN)

where,

πi= P(X = si), for all i= 1, 2, . . . , N.

Let the initial probability distribution of the MC on a finite state spaceS = {s0, s1, . . . sN}

given by the row vector πππ(0) and defined by π ππ(0)= [P(X0= s1) P(X0= s2) . . . P(X0= sN)] where, P(X0= si) ≥ 0 for every i= 1, 2, . . . , N. or in vector notation, πππ(0)≥ 0 and πππ(0)× 1|= 1.

where 0 denotes the zero vector, the row vector whose components are all zero and 1| denotes the one vector, the column vector whose components are all one.

Now, in order to obtain the probability distribution of X1, X2, . . . , XN, we use the law of

total probability. Particularly, for any j ∈S = {s0, s1, . . . sN} we have

P(X1= j) = N

∑

i=1 P(X1= j | X0= i)P(X0= i) = N

∑

i=1 p_{i j}P(X₀= i).

In general, as the n-step state vector for MC is defined as follows π

ππ(n)= [P(Xn= s1) P(Xn= s2) . . . P(Xn= sN)]

which can be written in a form of matrix multiplication. That is, the one-step state vector can be written in a matrix form as

πππ(1)= πππ(0)P. The two-step state vector

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More generally, the n-step state vector π π π(n)= πππ(n−1)P = πππ(n−2)P2= . . . = πππ(0)Pn. That is, πππ(n+1)= πππ(n)P πππ(n)= πππ Pn.

• The steady-state vector or the stationary distribution vector for MC represents the long-term probabilities being in each state, i.e. the fraction of times of MC being in each state as n increases.

Definition 2.5 (Stationary Distribution, Ibe O., 2014). Let πππ(n)= (π₁(n), π₂(n), . . . , π_N(n)) be the n-step probability distribution of the Markov chain, then πππ = (π1, π2, . . . , πN) is

called the steady-state vector or the stationary distribution of Markov chain if πj= lim

n→∞P(Xn= j | X0= i), for all i, j ∈S

where,

∑

j

πj= 1.

Theorem 1 (Ibe O.,2014). Let Xn be a Markov chain on the finite state space S =

{s0, s1, . . . sN} and let πππ(n)= (π (n) 1 , π (n) 2 , . . . , π (n)

N ) be the distribution vector of the Markov

chain. Then the stationary distribution vector or the steady-state vector is given by π

ππ = πππ P.

Proof. If πππ(n)= (π₁(n), π₂(n), . . . , π_N(n)) is limiting distribution for a Markov chain, then πππ = lim n→∞πππ (n) = lim n→∞(πππ (0)_Pn_). Similarly, π ππ = lim n→∞πππ (n+1) = lim n→∞(πππ (0)_Pn+1₎ = lim n→∞(πππ (0) PnP) = ( lim n→∞(πππ (0) Pn)P = πππ P.

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• Transition diagram is another way of describing the switches of the states of the Markov chain. Each node represents each state of Markov chain with arrows that explain the connection between the states. For example, assume that the initial state is i = 1, then the probability that the future state would be j = 2 is described by an arrow going from the first state to the second state. Otherwise, if the state does not switch to another state then the probability that it stays at the same state is described with an arrow which goes around this state.

Three different types of states are illustrated in figures below, where two-, three- and four- states transition diagrams are included with the corresponding TPM.

P = p11 p12 p₂₁ p₂₂ State₁ State₂ P22 P11 P21 P12

Figure 1: Transition diagram and transition matrix for two-states MC

P =   p11 p12 p13 p₂₁ p₂₂ p₂₃ p₃₁ p₃₂ p₃₃   State₁ State₂ State₃ P31 P12 P22 P23 P32 P21 P13 P33 P11

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P =     p₁₁ p₁₂ p₁₃ p₁₄ p₂₁ p₂₂ p₂₃ p₂₄ p31 p32 p33 p34 p₄₁ p₄₂ p₄₃ p₄₄     State₁ State₂ State 3 State 4 P43 P34 P13 P42 P24 P41 P14 P44 P33 P22 P11 P21 P32 P23 P31 P12

Figure 3: Transition diagram and transition matrix for four-states MC

2.2 Continuous-time Markov Chain

So far, the discrete-time Markov chain has been discussed. Another important type of MC is the continuous-time Markov chain or time-varying Markov chain.

Definition 2.6 (Continuous-time Markov chain, Kijima M., 1997 and Milton J. et al., 1995). A stochastic process{Xt,t ≥ 0} is a continuous-time Markov chain if for all s,t ≥ 0, and all

states i, j, x(u) ∈_{S, 0 6 u < s}

P_{{X(t + s) = j | X(s) = i, X(u) = x(u), 0 6 u < s}} = P{X (t + s) = j | X (s) = i}.

In other words, a continuous-time Markov chain is a stochastic process satisfying the Markovian property which states that given the present state at time s and all the past states, the conditional distribution of the future state at time t + s is dependent only on the present state and is independent of the past states. In addition, if P{X (t + s) = j | X (s) = i} is inde-pendent of time s, then the continuous-time Markov chain is said to have either homogeneous or stationary transition probabilities, and non-homogeneous otherwise (Kijima M.,1997).

The transition probability pi j for the continuous-time Markov chain in the homogeneous

case is defined by

p_{i j}(t) = P{Xt+s= j|Xs= i}

= P{Xt= j|X0= i}, for all s,t ∈ [0, ∞).

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by P(t) =      p₁₁(t) p₁₂(t) · · · p_1N(t) p21(t) p22(t) · · · p2N(t) .. . ... . .. ... pN1(t) pN2(t) · · · pNN(t)      .

One useful concept to be considered when analyzing the continuous-time Markov chain is the generator matrix.

Definition 2.7 (The Generator Matrix, Kijima M., 1997 and Ibe O., 2014). The generator matrixG for a continuous-time Markov chain for (i, j)thentry of the transition matrix is given by

g_{i j}= (

λipi j, if i 6= j

−λi, if i = j.

Here λi is called the transition rate out of state i. The quantity gi j = λipi j represents the

switch from state i to state j and therefore it is called the transition rate from state i to state j and

∑

j6=i

p_{i j}= 1.

In other words, if i = j, the (i, j)th elements of the generator matrix are diagonal elements, so that, these elements of the generator matrix are chosen such that the sum of each row of this matrix (G) is equal to zero. Otherwise, if i 6= j, the (i, j)th elements of the generator matrix are given by gi j.

Furthermore, the generator matrix can be useful to obtain the steady-state vector because this vector is an eigenvector for the eigenvalues λ = 1, since the transition matrix P(t) has the property that every row sums to 1, it follows that λ = 1 is an eigenvalue for the transition matrix. The generator matrix is then given by

G = VVVlog(D)V−1

where V is the eignvector matrix and D is the eigenvalues matrix of the transition probability matrix P.

Theorem 2 (Ibe O., 2014 and Milton J. et al., 1995). Let Xt be a continuous-time Markov

chain on the finite state spaceS = {s0, s1, . . . sN} with generator matrix G. The vector

π

ππ = (π1, π2, . . . , πN), πi> 0, i = 1, 2, . . . , N is the steady-state vector, i.e. stationary distribution

if and only if it satisfies the following system of equations (

πππ G = 0, ∑N₁πi= 1.

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Proof. Assume that πππ is stationary distribution, then πππ = πππ P(t). Differentiating both sides, gives

0 = d

dt[πππ P(t)] = πππ P0(t)

= πππ GP(t). (By backward equation)1 Now, for t = 0, it holds that

0 = πππ GP(0) = πππ G, where P(0) = I (Identity matrix).

Now assume that πππ satisfies the condition πππ G = 0, then, by backward equations, stands P0(t) = GP(t).

Multiplying both sides by πππ , gives π π

π P0(t) = πππ GP(t) = 0, where πππ P0(t) is the derivative of πππ P(t).

This implies that πππ P(t) is independent of t, that is, for any t ≥ 0 π

ππ P(t) = πππ P(0) = πππ . Thus, πππ is a stationary distribution.

1_{Backward equation states that the derivative of the transition matrix equals the matrix multiplication of}

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3 Option Pricing

Options are that type of financial contracts which give the holder the right, but not the ob-ligations to buy or sell the underlying assets (opposite to forward and futures contracts) at a mutually agreeable exercise price called the strike price, on or before a specific future date, which is known as maturity. By exercising an option means that the holder of an option can buy or sell the stock at the strike price. There are many types of options, three of which are the most common ones, American, European and Asian options. Even though they are called by geographical regions, they do not have any such connections. There are also different types of each of these options, the two most popular types are the call and the put options. The main distinction is that buyer of a call option is hoping that the stock price will increase, while the buyer of a put option wants it to decrease. In this thesis, the focus will be put on American and European options.

3.1 American and European Options

The main characteristic of the American options is that they are considered as a continuous-time instrument since they could be exercised at any continuous-time from the starting date up to the maturity. The decision of exercising the option before reaching an expiration date is called early exercising. Thanks to its exercising time flexibility, the most common options that are traded on exchanges are American. Therefore, American options are always more valuable, or at least worth as much as European.

Denote the American call and put options by CAMR and PAMR and European call and put

options by CEU Rand PEU Rrespectively. It holds that

C_AMR= C_{EU R}.

However, since it is not possible to know when exactly the American option will be exercised, or will it be at all, the predictions and evaluations of those options are getting more complex than the European ones.

Because the option provides the holder with the right, but not the obligation to buy or sell the asset, he will decide to exercise the option only in case when it is profitable to do so. In other words, the holder of a call option will exercise it only if the market price of the asset is higher than the strike price. By following the same logic, he will exercise a put option exclusively if the market price of the asset is smaller than the market price. Any option has some certain value because it offers its holder the chance to gain unlimited profit at the risk of a limited loss.

Since the American options are the continuous-time instrument and the value of the option changes over time in an uncertain way so it is said to follow a stochastic process classified as the continuous-time where the changes of the value of the option can take place at any time between specified time interval. For that type of options, stock prices are usually assumed to follow a Markov process where the predicted future value of an option is only dependent on the present value of the stock, it is not dependent on the particular path followed by the price in the past and all other past values of the stock are irrelevant.

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Unlike American options, European options contracts do not allow the flexibility in timing, so they restrict the execution exclusively to the date of expiration. This characteristic makes its evaluation slightly simpler because the date of potential execution is known. The BSM could be very helpful for pricing of European options since one of the assumptions on which the model is based on, is that the options will not be exercised early.

Now turn the attention to the bounds of the American and European call and put options. Since the call options give the holder the right to buy one share of stock for a certain price S, that leads to that the option can never be worth more than the stock, i.e.

C_AMR≤ S0,

C_{EU R}≤ S0.

On the other hand, the put option, either American or European, give the holder the right to sell one share of stock for so-called the strike or exercise price K, and thus the option can never be worth more than K for the American options and more than the present value of K for the European options, i.e.

P_AMR≤ K, PEU R≤ Ke−rT,

where r represents the interest rate.

Moreover, the value of an option can either have certain value or expire worthlessly, there-fore CEU R≥ 0 and PEU R≥ 0, so that the values of the options are

C_{EU R}≥ max(S0− Ke−rT, 0),

P_{EU R}≥ max(Ke−rT − S₀, 0).

The European option is often characterized in terms of its payoff, that what is assumed to be the lower bound for European call and put options with no-dividend-paying stocks, assuming that K is the exercise price, ST is the final price of the underlying asset, then the payoff of a

European call option is

C_{EU R}= max(S_T− K, 0), and the payoff of a European put option is

P_{EU R}= max(K − S_T, 0).

These two payoff functions can be applied for the American call and put options also if and only if the value of the option is exercised at a specific time before or at the maturity. How-ever, the valuation of the European style options tends to be easier and less dramatic than the American style options. There is a relationship between the European call and put options, which is described in the put-call parity

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Using the fact that

C_{EU R}≥ S₀− Ke−rT,

it follows that the value of an American option must be larger than the corresponding European call option,

CAMR≥ S0− Ke−rT.

Given that r > 0 and T > 0, it follows that

CAMR> S0− K.

The left-hand side determines the value of the American option at any time before the maturity time, whereas the right-hand side explains the value of exercising the option immediately before the expiration date since the value of the option is strictly greater than the value of exercising the option at time t < T , so it can never be optimal to exercise the option before the maturity date T . Thus the price of the American option is equal to the corresponding European option. This argument is valid for American option that pays no dividend. However, for American option with dividend yield, it can be optimal to exercise the option early.

That means that the value of exercising the American call option is S0− K, which is less

than the value of CAMR, i.e. the value of selling or keeping the American call option. Since

we know that we can buy an American call option for the price K and sell it immediately at the market for S0, therefore it can never be optimal to exercise the American call option

which pays no dividend early. In that case, the American call options are equivalent to the corresponding European options, i.e.

C_AMR= CEU R.

Furthermore, it can be optimal to exercise an American put option on a non-dividend-paying stock early, in that case, an American put option must be worth more than the corresponding European put option, i.e.

P_AMR> PEU R.

3.2 Options with Dividends

Dividend yields are usually denoted by q or δ and they represent the shareholder’s (or stock-holder’s) reward for investing their money into a certain company. Firms which tend to con-stantly pay dividends without decreasing the dividend yield are showing their power and strength in that way. Sometimes, for the same reason, even though they temporarily gain a lower profit than before, they continue maintaining or even raising the level of dividends. Since the company obliges to make this payout, the value of the dividend is subtracted from the stock’s growth rate on an ex-dividend date. For European options that have the maturity at time T and dividend yield q, there are two steps during the pricing of the option. Firstly, the initial stock price S0 offers a dividend yield q. Secondly, the initial stock price decreases

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that the call option cannot have a negative value (the lowest reachable value can be 0, which means it expired worthless). Mathematically it could be written as

C≥ max(S0− Ke−rt, 0).

Similarly, put option looks like:

P≥ max(Ke−rt− S0, 0).

Combining above mentioned two steps for pricing options with dividend yield and these two formulas, it is not hard to find lower bounds for both call and put options. What is needed is to replace S0by S0e−qT to get the following formula for the lower bound for the call option

C≥ max(S0e−qT− Ke−rt, 0)

and for the put option:

P≥ max(Ke−rt− S0e−qT, 0).

Furthermore, pricing formulas for European options with dividend yield can be obtained by slightly modifying the formulas for call and put options in the Black-Scholes model. Again substitute S0by S0e−qT and get

C= S0e−qTN(d1) − Ke−rTN(d2)

and

P= Ke−rTN(−d2) − S0e−qTN(−d1),

where the function N(x) stands for the cumulative probability distribution function for a stand-ardized normal distribution. From this, d1and d2could be defined as

d₁= ln(S0/K) + (r − q + σ 2_/2)T σ √ T , d₂= ln(S0/K) + (r − q − σ 2_/2)T σ √ T = d1− σ √ T, where σ represents stock price volatility.

Knowing that American options could be exercised at any time before maturity, the ques-tion is what would happen if the dividend yield is taken into consideraques-tion and as well, when is the optimal time for that exercise?

Consider now a short analysis by Hull J. C.,(2012) to find the most profitable time for the early exercise of an American call option when discrete dividends are included instead of dividend yield.

Suppose that there exist n ex-dividend times named as t1,t2,t3, . . .tn, and t1 < t2 < t3<

. . . < tn, for every corresponding discrete dividends D1, D2, D3, . . . Dn, respectively. Denote

stock price at time t as St and strike price as K.

Assume exercising the American option at time tn. The gain for the investor would be

equal to

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However, in case that the option is not exercised, price of the stock would be S_t_n− Dn.

Knowing the property of non-negativity of American call option, the lower bound for the non-exercised option is S_t_n− D_n− Ke−r(T −tn)_. Furthermore in addition S_t_n− Dn− Ke−r(T −tn)≥ Stn− K, Dn≤ K h 1 − e−r(T −tn)i_.

From this inequality, it is obvious that exercising at time tn is not the best way to go. But in

case that St is sufficiently high and

D_n> K h

1 − e−r(T −tn) i

,

it would be optimal to exercise this American option at time tn. This inequality holds when

dividends are significantly large and, as well, the exercising time (ex-dividend date) is quite near to the maturity date, which means T − tnis sufficiently small.

Similarly, let us check the same proof for the time tn−1 instead of ex-dividend date tn. If

exercise happens, the gain for the investor would be equal to Stn−1− K.

However, in case that the option is not exercised, price of the stock would be S_t_n−1− Dn−1.

It is important to keep in mind that, after tn−1, the next possible time for option exercising is tn

Following the same steps as for time tn, the lower bound for the non-exercised option at time

tn−1is

S_t_n−1− Dn−1− Ke−r(tn−tn−1).

and thus

S_t_n−1− Dn−1− Ke−r(tn−tn−1)≥ Stn−1− K, D_n−1≤ Kh1 − e−r(tn−tn−1)i_.

Again, like in previous example, exercising immediately prior to time tn−1is not optimal. So,

there is a general pattern for that will apply in all possible cases. For any i < n, if

D_i≤ Kh1 − e−r(ti+1−ti)i ₍₁₎ optimal is to not exercise the option just prior to time ti.

When the strike price (K) is close enough to the value of the current stock price St, the

inequality (1) is satisfied when the risk-free rate of interest (r) is higher than dividend yield on the stock (Di).

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4 Model Formulation

In this section, firstly we consider the most popular model used for valuing the options of the European-style, the standard Black-Scholes option pricing model, thereafter we extend this model to a more useful one under which both American and European options can be valued where the volatility varies randomly, called the Markov-switching model.

4.1 Classical Black-Scholes Option Pricing Model

Consider the dynamics of the stock price under the standard Black-Scholes model dSt= µStdt+ σ StdWt,

where, µ and σ are both constant, representing the drift and the volatility respectively and {dWt,t ≥ 0} is the standard Wiener process or Brownian motion.

Definition 4.1 (Brownian Motion, Kijima M., 2016). Let {S(t),t ≥ 0} be a stochastic process defined on the probability space(Ω,F ,P). The process {S(t)} is called a standard Brownian motion if

• the increment S(t + s) − S(t) is normally distributed with mean 0 and variance equals to s, i.e.{S(t),t ≥ 0} has stationary increments, independently of time t,

• it has independent increments,

• it has continuous sample path, and S(0) = 0.

Remark1. A geometric Brownian motion Xt is the solution of an stochastic differential

equa-tion with linear drift and diffusion coefficients, i.e. dXt= µXtdt+ σ XtdWt,

It is well known that the process of the stock price under the BSM is described in a stochastic differential equation (SDE) and governed under risk-natural probability measure. The stock price {S(t),t ≥ 0} is modelled by Geometric Brownian motion (GBM) with drift µ and volatility σ . Furthermore, the drift and the volatility under the classical model of the BSM are assumed to be deterministic constant. However, this is not the case in reality since the real-life volatility is never constant, and it varies and changes concerning different strike prices and maturity dates of the option. The volatility appears to be smile shaped and BSM failed to reflect this shape since volatility is assumed to be constant.

Since the dynamics of the stock price is described in the SDE above, so applying Itô lemma to the SDE gives the stock price at time T .

Definition 4.2 (Itô Process, Kijima M., 2016). Let {X = (Xt)t≥0} be a stochastic process that

solves the following SDE

dX_t = µ(Xt,t)dt + σ (Xt,t)dWt, Xt= X0+ Z t 0 µ (Xs, s)ds + Z t 0 σ (Xs, s)dWs.

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The Itô process Xt can also be written in a shorter differential form

dX_t= µdt + σ dWt.

Theorem 3 (Itô Formula, Oksendal B., 2013). Let {X (t)} be an Itô process given by dX_t= µdt + σ dWt.

Let f (t, x) ∈ C2([0, ∞)×R) (i.e. function f is twice continuously differentiable on ([0, ∞)×R), then Yt= f (t, Xt) is also an Itô process, where

dY_t= ∂ f ∂ t (t, Xt) dt + ∂ f ∂ x(t, Xt) dXt+ 1 2 ∂2f ∂ x2(t, Xt) (dXt) 2 . where (dXt)2= (dXt) · (dXt) is computed according to,

dt· dt = dt · dWt = dWt· dt = 0, dWt· dWt= dt.

4.1.1 A Stock-Price Model

This section derives the price of the underlying asset S at time T by applying Itô formula to the SDE which describes the dynamics of the stock price under the standard BSM.

Let F be a function of the stock price St and time t,

F = f (t, St). By Itô lemma dF= ∂ f ∂ t (t, St) dt + ∂ f ∂ St (t, St) dSt+ 1 2 ∂2f ∂ S2_t (t, St) dS 2 t.

Now assume that

X = f (t, St) = ln(St).

Applying Itô lemma to the function above gives: dX dt = 0, dX dS_t = 1 S_t, d2X dS2_t = − 1 2 1 S_t2. That is, dX= 1 St dS_t−1 2 1 S2_t dS 2 t.

Substituting the dynamics of the stock price dStwhich follows GBM described in the equation

above, we get dX= 1 S_t(µSt+ σ StdWt) − 1 2 1 S2_t σ 2_S2 tdt.

Simplifying the expression, taking into account that

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we get dX= µ −1 2σ 2 dt+ σ dWt.

Now, since we have X = ln(St), that gives

dln(St) = µ −1 2σ 2 dt+ σ dWt.

Integrating both sides from 0 to T , i.e. Z T 0 dln(St) = Z T 0 µ −1 2σ 2 dt+ Z T 0 σ dWt gives ln(ST) − ln(S0) = µ −1 2σ 2 T+ σWT.

Rearranging this equation, we get

S_T = S0e(µ −

1

2σ2)T+σWT_,

where WT is a normally distributed random variable, with mean 0 and variance equals to T

WT ∼ N(0, T ).

Then the distribution of√T Zif Z is a standard normal random distribution, i.e. Z∼ N(0, 1).

The terminal stock price can, therefore, be written as follows, ST = S0e(r−

1 2σ

2₎_T_+σ√_{T Z} ,

where µ can be replaced by the risk-free rate r since the assumption under BSM is made so that the market is arbitrage free, and hence, the drift of the risk-free asset is assumed to be r that is µ is equal to r (Björk T., 2009).

4.1.2 The Black-Scholes Equation

Considering the assumptions behind the BSM, that the stock price process follows a GBM, we can derive the Black-Scholes equation which is a partial differential equation (PDE) which governs the price of the European call and put options.

Assume that the market consists of two assets, riskless asset Bt, and risky asset St with

dynamics given by

dB_t= rBtdt,

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Denote the price of a call option or any other derivative at the time "t" given the current stock price St= S by a smooth function V (S,t), then from Itô lemma, it follows that the process

followed by a function V of S and t is

dV = ∂V (S, t) ∂ t + ∂V (S, t) ∂ S µ S + 1 2 ∂2V(S,t) ∂ S2 σ 2_S2 dt+ ∂V (S, t) ∂ S σ S dW.

After constructing a portfolio based on the two assets, the underlying stock and the deriv-ative asset, the Black.Scholes boundary value problem or the Black-Scholes equation can be obtained and it is given by, [Björk T., 2009]

∂ ∂ tV(S,t) + rS ∂ ∂ SV(S,t) + 1 2σ 2_S2 ∂2 ∂ S2V(S,t) = rV (S,t).

A closed-form solution to the above PDE, i.e. the price of European put and call options is obtained in the Black-Scholes formula together with the payoff functions being the boundary conditions, i.e. c(St,t) = max(St− K, 0) for a European call and p(St,t) = max(K − St, 0) for

a European put option, where c(S,t) and p(S,t) representing the value of the call and the put European option respectively.

The value of a call option at time “t” for a non-dividend paying stock in terms of Black-Scholes parameters is expressed as

c(S,t) = SN(d1) − Ke−r(T −t)N(d2),

and the price of a corresponding put options is expressed as

p(S,t) = Ke−r(T −t)N(−d2) − SN(−d1),

where N(·) is the cumulative standard normal distribution function, d1and d2are given as

follows d₁= ln(S0/K) + (r + σ 2_/2)T σ √ T , d₂= ln(S0/K) + (r − σ 2_/2)T σ √ T = d1− σ √ T.

This model was developed by Black F. et al.,(1973) can be applied only for options of European style which can be exercised only at the maturity date. Moreover, Cox J. et al.,(1979) de-veloped a new approach to value options of different styles assuming the volatility is constant.

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4.2 The Three-States Markov-switching Framework

In this section, the standard Black-Scholes option pricing model is extended to a useful model of changing volatility, a model that can be applied to both styles of options, American and European. The main idea behind the Markov-switching model is to obtain a model that de-scribes the fluctuations of the volatility of the stock price by allowing the volatility being fully stochastic, changes randomly in time and takes different values depending on the states of Markov chain.

The stock price under the standard BSM is extended to be as follows, dSt = µStdt+ σXtStdWt,

where Xt is a stochastic process representing the states of the volatility process, dWt is the

standard Wiener process and it is independent of Xt, the drift term µ is constant, where the

volatility term σXt takes different values when Xt is in different states, i.e. Xt follows a so called “Markov Chain” which contains finite number of states.

Let the number of states be k, and denote them by the state spaceS, so that S = {σ1, σ2, σ3, . . . , σk}.

In this paper, the dynamics of the volatility is described by three states, high, medium and low, i.e. k = 3. It follows that,

Xt=     

σXt = σ1, the high volatility state σXt = σ2, the medium volatility state σXt = σ3, the low volatility state

As Xt is stochastic and jumps randomly between different states, the model of stock price

dynamics will also change randomly according to the different states of the volatility. The transitions from one state to another for volatility that follows Markov chain are driven by so-called the Markov matrix or probability transition matrix which governs the random behaviour of the state variable with k = 3. The three-states volatility Markov matrix is defined as follows,

which follows that,

P =     p₁₁ p₁₂ p₁₃ p₂₁ p₂₂ p₂₃ p₃₁ p₃₂ p₃₃    

where pi j, (i, j = 1, 2, 3) denote the probability of transition of Xt= σj given that Xt−1= σi,

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This model with three-states can easily be extended to k-states model as well as the probab-ility transition matrix which can be transformed for different time intervals using the generator matrix (G) for Markov chain.

The transition between any pair of states can be also described in a so-called transition diagram. The three-states volatility transition diagram is as follows,

p11 𝜎2 𝜎3 𝜎1 p12 p23 p32 p21 p31 p13 p22 p33

Figure 4: Probability transition diagram for three-states Markov chain

The figure above can also be extended to k-states volatility so that sequence models can be obtained which gets closer to the models of the stochastic volatility. However, two or three states volatility process are often enough to capture the issues behind pricing options in the financial market.

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5 Stochastic Volatility

The volatility of returns of the underlying asset is one of the most important factors that affect the option prices. It is the most central notation in derivatives and options analytics. How-ever, the assumption under the BSM is made that the volatility of the stock price is constant, which has not been considered in the stochastic volatility models where these models take into account the fact that volatility of the stock price is a random variable.

The model developed by Hull J. et al.,(1987) is one of the models which have suggested and examined that volatility in the asset prices is varying with time, for that reason, the atten-tion was turned to the stochastic volatility rather than constant volatility. The term stochastic volatility refers to volatility that cannot be foreseen precisely and therefore is determined ran-domly.

Different models for pricing options with stochastic volatility were developed to take into account the fluctuations of the volatility of the stock price and to modify the BSM for option pricing, so that the price volatility is varying randomly, instead of letting it being constant.

Stochastic volatility models which are also called time-varying volatility models have been popular in finance, the fluctuations of the price volatility are considered and characterized by a stochastic process. Examples of these models include the Heston model, SABR and GARCH models. However, choosing the right estimate of volatility is also important when calculating the option pricing using different models.

To be able to use different option pricing models with different input parameters, numerical estimation of all the inputs parameters is required. Some of the input parameters are easy to estimate while others tend to be complex such as the volatility parameter. Obtaining an estimate of the volatility can be over-complicated. Therefore, different volatility approaches are used to estimate the volatility, two of which are used concerning stochastic volatility, historical volatility and implied volatility.

5.1 Historical Volatility

The historical volatility is estimated from the historical data over a period of the stock price, using the standard Black-Scholes Geometric Brownian motion model with the assumption that the stock price process is assumed to follow a log-normal distribution, i.e.

r_i= log S(t_i) − log S(t_i−1) = log S(ti) S(ti−1)

,

where r1, r2, . . . , rnare the log return, independent and normally distributed random variables.

The volatility then may be estimated as

ˆ σ = s 1 n− 1 n

∑

i=1 (ri− ˆµ )2,

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ˆ µ =1 n n

∑

i=1 r_i.

However, it can be complicated estimating the volatility using the historical volatility when the volatility term is not constant and is assumed to vary randomly. Thus, in that case, implied volatility is used as an estimate for the volatility parameter rather than the historical volatility.

5.2 Implied Volatility

Implied volatility refers to the fluctuation in the market price of the underlying assets and it is estimated by using the market expectation of the volatility, which is in its turn obtained by considering the market price data, i.e. price of the benchmark option.

Assume that the price of the benchmark option is denoted by Pr. Assume also that the strike price is denoted by K, today’s value of the underlying asset is denoted by S, the interest rate is denoted by r and the time to maturity is denoted by T . Then, by considering the Black-Scholes pricing formula for a European call option, which is given by

c(S, K, T, r, σimp)

the implied volatility can be obtained by setting the price of the benchmark option to be equal to the Black-Scholes pricing formula for a European call option, i.e. solving the following equation for σimp,

Pr= c(S, K, T, r, σimp).

The value of σimp which solves this implicit equation is called the implied volatility, where

we try to find a value of σimp which the market has implicitly used for valuing the benchmark

option. c(S, K, T, r, σimp) = SN(d1) − Ke−r(T −t)N(d2) where, d₁=ln(S0/K) + (r + σ 2 imp/2)T σimp √ T , d₂=ln(S0/K) + (r − σ 2 imp/2)T σimp √ T = d1− σimp √ T.

The implied volatility tends to be more important for traders than the historical volatility since it is forward-looking. As a contrast, the historical volatility is estimated from the histor-ical data, so it is backwards-looking. For that reason, the implied volatility is used by traders to estimate the volatility of the underlying assets which leads that most of option pricing models estimate the volatility in terms of the implied volatility.

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5.3 Volatility Smile

When focusing on the stochastic volatility rather than constant volatility, an important concept related to the stochastic implied volatility should be considered, and that is volatility smile.

Volatility smile illustrates the relationship between the implied volatility of the option and the strike price. It is a graph showing the implied volatility of an option as a function of the strike price during the life of the option.

It is well known that the three factors that compose the value of an option are the strike price, time to maturity and the implied volatility. Therefore all these factors are summed up in a graph and the pricing of options which assumes to be complicated in some cases is well reflected in this graph, i.e. the volatility smile.

The volatility smile for both call and put options is the same. However, the term volatility smile came from that out-of-the-money and in-the-money options are traded at a higher level of implied volatility than the at-the-money options. Thus the graph of the implied volatility often looks like a smile which explains the term volatility smile. The volatility smile for options is illustrated in the figure below.

Remark2. An option is said to be in-the-money at time t if S(t) > K, and out-of-the-money if S(t) < K. for put option the inequality is reversed. Also, if S(t) = K the option is said to be at-the-money. At-The-Money

Strike Price

Im

pli

ed

Vo

lat

ili

ty

ITM Call ITM Put

OTM Call OTM Put

Figure 5: Volatility smile for options

Moreover, by plotting the implied volatility against the exercise price under the BSM, it appears a flat curve since the volatility is assumed to be constant under this model and it is not

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varying with time. That explains why the plot of the implied volatility can be used to test the BSM.

The volatility smile in this paper is considered to show the connection between the BSM and the model used for pricing options, so instead of inputting a parameter for the volatility in the BSM to compute the option’s fair value, we reverse the calculations. There we try to obtain the fair value using any other model letting the volatility being stochastic and thereafter substitute the value obtained from this model in the BSM to calculate the implied volatility with each stock price and to obtain the volatility smile, that is, the volatility is treated as an output and the option value as an input in the BSM in this case.

Furthermore, the corresponding implied volatility needed to obtain the volatility smile is calculated using the Black-Scholes formula where the European option price is computed with the three states Markov-switching model using Monte Carlo simulation, i.e.

S_BS= S0N(d1) − Ke−rTN(d2) = SMCMC.

where, SBS represents the stock price calculated with the Black-Scholes formula, and SMCMC

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6 Monte Carlo Method for Option Pricing

This section highlights the method used to price both European and American options with stochastic volatility modelled by the three states Markov chain, i.e. the Monte Carlo method. Description of the method, as well as the algorithm, is included in this section.

6.1 Description of the Method

Monte Carlo simulation has enough power to deal with complex stochastic processes and payoffs. Of course, many factors affect the choice which method to select. When working with three or more stochastic variables, Monte Carlo method offers high numerical efficiency. The reason is the following: the time needed to complete the whole process of Monte Carlo simulation follows linear instead of exponential increment. Furthermore, Monte Carlo Method is capable of offering a standard error for the estimates gained during the process. Its efficiency and correctness of approximation raise as the number of iterations increases. However, some of the main flaws and weaknesses are that computations and the whole process is quite a time consuming and complexity raises rapidly in cases when early exercise opportunities are present.

Unlike binomial trees and finite different approaches, which start from the end and follows the life of a derivative backwards, Monte Carlo simulation follows natural path starting at the beginning and finishing at the end of the life of a derivative. Basically, this method establishes its calculations sampling the possible paths that can happen in a risk-neutral world. After cal-culating the payoff of each path, it gets discounted at the risk-free interest rate. For European options, the goal is to get the values of mean (µ) of those discounted payoffs together with standard deviation (σ ) and discounting the expected payoff at the risk-free interest rate to find an estimation of a value of a derivative.

To make the process of this simulation clear and visible, let us represent it in five main steps: [Hull J. C.,2012]

• Step 1: Sampling the possible random path for market variable in a risk-neutral world according to the given model.

• Step 2: Calculating a payoff for the random path.

• Step 3: Doing previous two steps many times to form a sample with enough amount of values of the payoff.

• Step 4: Approximating expected payoff by finding the average of the estimated payoffs and standard deviation by finding the variance.

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Monte Carlo simulation has some very close connections the central limit theorem, one of the highly important parts of probability theory and statistics. To gain desired convergence to the expected value, or mean, it is needed to have a significantly big amount of estimates of option value and to take their average. A lot of simulations are made and each obtained value is one small piece of the distribution of possible outcomes of the derivative. All those results are based on the output of random number generator, and thus there does not exist correlation of any kind between them. The estimate of the average can be written as follows

ˆ S= 1 n n

∑

i=1 S_i.

As previously said, the bigger amount of estimates n (n → ∞), the closer convergence to the real expected value is present. For the mean there could be some error in the estimate of the value of the derivative. This is called standard error and is calculated in the following way

σ √

n, where σ is standard deviation.

From this formula it is obvious that higher number of simulations (n) will cause higher accuracy of the outcome.

6.2 Monte Carlo for European Options

Considering the assumption under BSM, the stock price follows GBM with constant drift and volatility, that is,

dS_t= µSt+ σ StdWt.

For the European options, the focus is exclusively put on the stock price at the maturity time (S_T). The reason is that the European options can only be exercised at the maturity, whereas for the American options it is needed to simulate the paths and express the stock price at each path since American option is a path-dependent option.

To obtain the stock price at the maturity time T , we need first to simulate Wt and multiply

it by√T, which is a standard normal random variable with mean zero and variance equals to 1. Thereafter we plug it into the stock price equation to get the stock price at time T . That is, for every unique simulation of standard normal random variable a unique stock price at time T is obtained.

The next step of Monte Carlo simulation is to apply the payoff function and find the cash flow of each stock price obtained from the simulation. For European options, however, we need to find the payoff only at the maturity date, i.e. at time T , whereas for the American option we need to look at each path and find the payoff at each path taking into account the optimal early exercise.

After applying the payoff function and finding the cash flow of each stock price obtained from the simulation, we need now to take the average over all these payoffs, which in that case

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gives the average expected payoff at time T , i.e. in the future, therefore we need to discount it back to today to obtain the stock price at time ”0”.

The value of the call option today is then,

C₀= e−rT " 1 I I

∑

i=1 (max(ST − K, 0)) # ⇒ Ee−rT_(S T− K)+ .

and the value of the put option today is,

P₀= e−rT " 1 I I

∑

i=1 (max(K − S_T, 0)) # ⇒ Ee−rT_{(K − S} T)+ .

Algorithm 1: Monte Carlo simulation for European options for i = 1 to I do

for j = 1 to m do generating zi, j

setting S(i, j + 1) = S(i, j) × exp r−1₂σ_{i, j}2 ∆ + σi, j √ ∆zi, j

setting Ci= e−rT(S(i, end) − K)+

setting ˆC_n= (C1+C2+ · · · +Cn)/n

In the algorithm, I is number of iterations (rows), Zi, jthe sequence of independent standard

normal random variables, m is number of time-steps (columns), ∆ is the step-size (the fraction between maturity T and m), Stasset price, Cidiscounted payoff, and ˆCnis the final estimation.

6.3 Monte Carlo for American Options

For the American options, the optimal exercise strategy should be considered, i.e. whether it is optimal or not to exercise the option at any time before the maturity date. For that reason, some additional steps are required compared to these in the European option case.

In order to price American options using Monte Carlo simulation, other methods are used to capture the early exercise strategy of the stock options and Least Square approach is one of them.

LSM is a type of simulation methods that is used to value options. This method was introduced by Longstaff F. A. et al.,(2001), is a method which has been popular recently that deals with options of the American-style. It is an appropriate method used for valuing path

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dependent options, e.g. American options and other types of options which can be exercised at any time before the expiration date.

For better understanding of this approach, LSM is described in the steps below [Longstaff F. A. et al.,2001], followed by the algorithm for an American put option:

• Step 1: Determining the payoff of each path at maturity time T , this is the same payoff that would be received in the European option case.

• Step 2: Working backwards, calculating the holding value and the exercise value at each time point, i.e. at each t.

• Step 3: Taking the present values of the option values at the next time point, i.e. the holding values for time point before the maturity time is calculated as the present values of this time point’s payoff.

• Step 4: Performing a regression analysis for in-the-money options to estimate the ex-pected holding value.

• Step 5: Obtaining the exercise value at this time point which is nothing but the payoff at this point.

• Step 6: Comparing the obtained expected holding value with the exercise value.

• Step 7: Deciding whether or not it is optimal to early exercise for these paths. If the exercise value at maturity time is greater than the holding value in some paths, then the option values are the corresponding exercise value at the time point before the maturity time. Otherwise, the option value is holding value.

• Step 8: Iterating the previous steps for each and every time-step up to time t.

• Step 9: Computing the option value today, i.e., at time t = 0 by taking the mean of the payoff at time t.

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Algorithm 2: LSM for American put option for i = 1 to I do

for j = 1 to m do generating zi, j

setting St(i, j + 1) = S(i, j) × exp

r−1₂σ_{i, j}2 ∆ + σi, j √ ∆Zi, j) setting V = zeros(I, m + 1) setting Y = (K − S, 0)+ setting f = Y (:, m + 1) itm = Y > 0 work backward for l = m : −1to 2 do S_t= S(:, l) EV = (K − St, 0)+ itm1 = itm(:, I + 1)

Fit and compute the continuation value (run regression) fit=polyfit (St(itm1), f (itm1) × e−rT), 2)

cont=polyval (fit, St)

for a = 1 to I do

if EV(a)>cont(a) then

Exercise, and set V (a, l) = EV (a) else

Continue, and set V (a, l) = f (a) × e−rT f = V (:, l)

setting ˆP_n= (e−rT(V1+V2+ · · · +Vn))/n

In the algorithm, besides already explained notation in the previous algorithm, Y stands for the holding value, f is the cashflow at time T , EV is the exercise value, itm stands for in-the-money, f it is the regression, cont is the continuous value, V is the payoff, and ˆPnis the final

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7 Numerical Study and Model Implementation

In this section, the focus will be pointed into the implementation of the Markov-switching model with three states and numerical analysis from which will be possible to construct con-clusions. Firstly we will implement the model with some initial parameters and obtain the res-ults. After the first step is completed, we will cover some possible variations of the parameters and compare the new results with all previous ones and in the end, emphasize everything im-portant that has been noticed throughout the process. Several initial inputs were taken into calculations and they are as follows:

• Four different maturities measured in years - 0.25, 0.5, 0.75 and 1, respectively.

• Three selected values for different volatility states - high volatility state (H) is equal to 0.7, medium volatility state (M) is equal to 0.5 and low volatility state (L) is equal to 0.3.

• Probability transition matrix of dimensions 3 × 3:

P =   0.7 0.2 0.1 0.3 0.6 0.1 0.2 0.3 0.5  

From this matrix it is visible that the probability for high state to remain on the same level is 70%, remaining of medium state is 60% and low state remains with probability of 50%.

• Number of simulations is I = 10 million. • Number of time-steps is m = 25.

• Interest rate r = 0.1. • Initial stock price S0= 90.

• Dividend yield q = 0.06 that satisfies non-arbitrage in later analysis.

• Seven different strike prices presented as a percentage of S0, starting from 70% until

130% with 10% difference, and then discounted -K1= 0.7S0erT,

K₂= 0.8S0erT,

.. .

Option Pricing Under the Markov-switching Framework Defined by Three States

School of Education, Culture and Communication

Division of Applied Mathematics

BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Option Pricing under the Markov-switching Framework Defined

by Three States

by

Minna Castoe

Teo Raspudic

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICS

School of Education, Culture and Communication

Division of Applied Mathematics

Acknowledgement

Contents

List of Figures

List of Tables

Abbreviations and Acronyms

1

Introduction

1.1

Background and Literature Review

1.2

Aim of the Project

2

Markov Chains

2.1

Discrete-time Markov Chain

∑

∑

∑

∑

∑

2.2

Continuous-time Markov Chain

∑

3

Option Pricing

3.1

American and European Options

3.2

Options with Dividends

4

Model Formulation

4.1

Classical Black-Scholes Option Pricing Model

4.2

The Three-States Markov-switching Framework

5

Stochastic Volatility

5.1

Historical Volatility

∑

∑

5.2

Implied Volatility

5.3

Volatility Smile

Strike Price

Im

pli

ed

Vo

lat

ili

ty

6

Monte Carlo Method for Option Pricing

6.1

Description of the Method

∑

6.2

Monte Carlo for European Options

∑

∑

6.3

Monte Carlo for American Options

7

Numerical Study and Model Implementation