Leseprobe

## Table of Contents

List of Abbreviations

List of Figures and Tables

1 Introduction

1.1 Background

1.2 Problem Definition and Objective

1.3 Course of the Investigation

2 Option Basics

2.1 Classification and Types of Options

2.2 Terminology & Payoff Characteristics

2.3 Put-Call Parity

3 Stochastic Processes

3.1 Markov Property

3.2 Wiener Process and Brownian Motion

3.3 Itô Process and Itô's Formula

3.4 Application of Itô's Lemma to Stock Price Movements

4 The Black-Scholes Model

4.1 Assumptions

4.2 Risk free Evaluation

4.3 Black-Scholes Partial Differential Equation

4.4 Black-Scholes Pricing Formula

4.5 Evaluation of Model Parameters and Assumptions

4.6 Volatility Smile

5 Stochastic Volatility Models

5.1 Overview of Stochastic Volatility Models

5.2 The Heston Model

5.2.1 The Mean-reverting Square Root Diffusion Process

5.2.2 Derivation of the Heston PDE

5.3 Heston Pricing Formula for European Calls

6 Empirical Analysis

6.1 Methodology

6.1.1 Closed Form approximations

6.1.2 Implementation

6.2 Calibration

6.2.1 Model parameters

6.2.2 Calibration methods

6.3 Data

6.4 Results

6.4.1 Calibration results

6.4.2 Pricing Results

7 Model evaluation

7.1 Interpretation of the Results

7.2 Drawbacks of the Model

7.3 Approaches for further Improvements

7.3.1 Time dependent Heston Model

7.3.2 Jump-Diffusion Model

8 Conclusion

Reference List

Appendix

## List of Abbreviations

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## List of Figures and Tables

Figure 1 - Payoff from a long European call. Strike price *K=40* 3

Figure 2 - Wiener process

Figure 3 - Generalized Wiener process

Figure 4 - Implied volatility smile for foreign currency options

Figure 5 - Volatility skew of SPX call options on 01/02/2008. TTM=1m; S=1447.16

Figure 6 - Implied volatility surface, Heston closed-form prices

Figure 7 - Values of the integrand as function of integration border b

Figure 8 - Discontinuities in for long maturities; From: (Kahl & Jäckel, 2005)

Figure 9 - VIX levels (1/1/2004 - 2/5/2010) and (Vol. of vol.) of VIX (2005-2008)

Figure 10 - IVS of SPX calls on 01/02/08 (left); IVS of Heston call prices (right)

Figure 11 - IVS of Black-Scholes call prices (left); SPX IVS vs. BS IVS (right)

Figure 12 - SPX IVS(multi-colored) vs. Heston IVS(single-colored), viewed from different angles

Figure 13 - Probability Densities with changing (left) and (right). From: (Heston, 1993)

Figure 14 - Ericsson stock price falls by 25%, 10/16/2007. From: (Haug & Taleb)

Figure 15 - Path of a Poisson process for , T=1

Figure 16 - Comparison market SPX IVS (upper, light grey) with Heston-SVJ Fit (9/15/2005) - Two perspectives. From: (Gatheral, 2006)

Table 1: Overview of Diffusion Models

Table 2 - Heston's default parameters for simulation of option prices

Table 3 - Parameters from calibration

Table 4 - Summary of the results

## 1 Introduction

### 1.1 Background

The Black-Scholes (or Black-Scholes-Merton) Model has become the standard model for the pricing of options and can surely be seen as one of the main reasons for the growth of the derivative market after the model's introduction in 1973. As a consequence, the inventors of the model, Robert Merton, Myron Scholes, and without doubt also Fischer Black, if he had not died in 1995, were awarded the Nobel prize for economics in 1997.

### 1.2 Problem Definition and Objective

The model, however, makes some strict assumptions that must hold true for accurate pricing of an option. The most important one is constant volatility, whereas empirical evidence shows that volatility is heteroscedastic. This leads to increased mispricing of options, especially in the case of out of the money options, as well as to a phenomenon known as *volatility smile*. As a consequence, researchers introduced various approaches to expand the model by allowing the volatility to be non-constant and to follow a stochastic process. It is the objective of this thesis to investigate if the pricing accuracy of the Black-Scholes model can be significantly improved by applying a stochastic volatility model.

### 1.3 Course of the Investigation

The outline of this thesis is as follows: In the first chapter, the very basics about options will be covered by introducing the terminology, payout profiles, and concepts such as the put-call parity, in order to obtain a basic intuition of what determines the price of an option. Secondly, the mathematical fundamentals, namely stochastic processes and differential equations, will be reviewed which leads directly to the introduction of the Black-Scholes(BS) Model, its assumptions, pricing formulas, and weaknesses. In the fifth chapter, which comprises the main part of this paper, the Heston model of stochastic volatility will be presented and interpreted. After that, the computational application of the model is shown by calculating call prices on the S&P500 index and comparing them to the BS prices. Finally, the obtained results will be interpreted in the seventh section, which is followed by a critical evaluation of the Heston models and suggestions for further improvement.

## 2 Option Basics

### 2.1 Classification and Types of Options

Within the taxonomy of derivatives, options belong to the conditional derivative instruments which means that, in contrast to a forward or future contract, the buyer of an option contract has the option but no obligation to settle the contract on the maturity date. Accordingly, the holder of a *call option* has the right to buy the underlying asset (i.e. stocks, bonds, swaps) for a certain price, which is known as *strike* or *exercise price*, by a certain date known as *expiration date* or *maturity (Hull, 2007, p. 6)*. Analogously, a *put option* gives the holder the right to sell the underlying asset with respect to the described conditions. The investor, however, who, by writing an option and earning a premium, takes the short position in the option contract, has the obligation to sell (in case of a *short call option*) or to buy (in case of a *short put option*) the underlying asset to or from the option holder.

The most commonly traded types of plain vanilla options are *American options* and *European options*. European options can be exercised only on maturity whereas American options can be exercised at any time before maturity. On top of that, there are also various other types of so-called *exotic option* contracts with very different payoff profiles. The scope of this thesis, however, will be on the valuation of plain vanilla European call options only.

### 2.2 Terminology & Payoff Characteristics

Due to their nature of being conditional derivatives, options have an asymmetric payoff in contrast to the symmetric payoff of, for instance, a forward contract: If the payoff is unfavorable for the holder of an option, he will, of course, decide not to exercise. That is, in case of a call option, if the strike price is greater than the current price of the underlying asset on maturity . The option is said to be *out of the money* (OTM) then.

Hence the payoff of a European long call is given by

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Figure 1 - Payoff from a long European call. Strike price *K=40*

Figure 1 shows that the payoff is zero as long as is greater or equal . The latter case is also known as *at the money* (ATM) whereas, when is smaller than , the option is *in the money* (ITM). The payoff then is indeed if the option is exercised and the underlying stock is sold immediately for the market price . The payoff of an ITM option can be also seen as an intuitive proxy for the value of that option, which is often referred to as *intrinsic value.*

### 2.3 Put-Call Parity

By constructing two portfolios A, consisting of a European call *c* and an amount of cash equal to the continuously discounted strike price , and B, consisting of a European put *p* and the underlying asset , it can be shown "that the value of a European call with a certain strike price and exercise date can be deduced from the value of a European put with the same strike price, exercise date, and vice versa" (Hull, 2007, p. 208). This relationship is also known as the *put-call parity* and is described by the following equation:

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The put-call parity implies the absence of any arbitrage opportunities which is, as will be shown later, also an assumption of the Black-Scholes model. Furthermore, it can be used to extend the closed form pricing formulas from Black-Scholes as well as from Heston to calculate the corresponding put prices, too.

## 3 Stochastic Processes

The dynamics of interest rates, stock prices, and of many other figures in the financial markets cannot be calculated deterministically in a way it is possible in physics, for instance. Observed from an outside perspective, stock prices move rather randomly and are strongly fluctuating. Nevertheless, the behavior of the financial markets can be modeled using stochastic processes, that is a sequences of random variables having a well-defined joint distribution function. However, a difficulty that arises when describing stock price dynamics using stochastic processes is that ordinary, deterministic calculus can usually not be applied to random variables due to their different convergence behavior.

### 3.1 Markov Property

According to Neftci (1999, p. 108), a discrete time process, , with joint probability distribution, , is said to be a *Markov process* if the implied conditional probabilities satisfy

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where and is the probability conditional on the information set .

Assuming that stock prices follow a Markov process is thus also consistent with the hypothesis of weakly efficient markets since all information available at time *t* are already incorporated in the stock price at *t* and any additional knowledge about the past is irrelevant when trying to forecast future stock prices.

### 3.2 Wiener Process and Brownian Motion

In general, a Markov process is defined by its mean, the *drift parameter,* and its variance which is the *diffusion* parameter *.* The simplest Markov process, with zero mean and a constant variance of one, is called Wiener process or Brownian Motion.

It follows from the Markov property that all increments of the Wiener process, , are statistically independent and, furthermore, on account of its zero mean that is a martingale: No trend is observable and the best future forecast equals the last observed value of W. It follows from the martingale property that the increments follow a normal distribution with zero mean and variance one.

Figure 2 shows a plot of a Wiener process with the number of intervals , and T=1. The plot also illustrates geometrically the problem of applying deterministic calculus: The Wiener process is not at all a smooth function and thus nowhere differentiable.

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Figure 2 - Wiener process

However, a single Wiener Process is not appropriate to describe the dynamics of stock prices since they are assumed to have a trend, even though it is usually unobservable. Consequently, in order to model a more realistic behavior, the Wiener Process must be generalized by adding coefficients a and b to the drift and diffusion terms. This generalized Wiener process is then described by the following stochastic differential equation (SDE):

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Again, the a plot of the process along with its drift term and Wiener process is shown below:

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Figure 3 - Generalized Wiener process

A problem that remains even in this more general model are the still constant drift and variance rates. Evidence shows that variance is usually higher if the stock price is high and that a constant expected drift rate, which implies an absolute, infinitesimal, and stock price-independent change of , is not consistent with expectations of investors of approximately constant expected *return*.

Hull (2007) exemplified that "if investors require a 14% per annum expected return when the stock price is 10$, then, *ceteris paribus,* they will also require a 14% per annum expected return when it is $50" (p. 265). To adapt the drift and diffusion terms to the current stock price they can be multiplied by S which is referred to as *geometric Brownian motion* (GBM) *:*

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### 3.3 Itô Process and Itô's Formula

An generalized Wiener process can be even further generalized by allowing the parameters *a* and *b* to be functions of t and X . Such a process is called Itô process and is described by the following SDE:

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The GBM derived in the previous section follows an Itô process for and .

However, when trying to price derivates, one is usually not directly interested in the movement of the underlying asset itself but only in its influence on the price of the derivative, or, in other words, the total differential of the pricing equation with respect to, at least, the current price of the underlying asset, time, and optionally additional parameters. Since deterministic calculus, as already mentioned, is not applicable to random variables, one has to use stochastic concepts, such as Itô's formula.

Itô's formula, also widely known as Itô's lemma, gives the dynamics of a function of an Itô process *X* and time t:

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### 3.4 Application of Itô's Lemma to Stock Price Movements

Following Hull's (2007) approach, it can be shown that the price of a stock *S* is lognormally distributed, which is one of the key assumptions of the Black-Scholes model. Then, following the notation used above, is a function of a geometric Brownian Motion *S* and time *t* which is given by

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In a first step the necessary partial derivatives must be calculated

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which are then inserted into equation (8) resulting in

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Equation (11) now has the typical structure of a generalized Wiener process with constant drift coefficient a= and a constant diffusion term b= . As a consequence, increments in G, that can be equally written as (i.e. logreturns), are normally distributed with a mean of and a variance of :

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Equation (12) implies that the natural logarithm of a variable , denoted as , is normally distributed, which means that variable itself must be lognormally distributed. This assumption is also quite intuitive in so far as the stock price cannot become negative, which is also an important property of the lognormal distribution.

## 4 The Black-Scholes Model

### 4.1 Assumptions

The assumptions under which the model holds true can be summarized as follows:

- The stock price follows a geometric Brownian motion described by equation (6) with constant and . As shown in the last section, stock prices are assumed to be lognormally distributed. It also follows from the GBM property that the stock prize is not subject to jumps.

- No arbitrage opportunities, consequently the put-call parity must hold

- Borrowing and lending at a constant risk-free rate of interest *r*

- Securities can be traded continuously and are perfectly divisible, that means any fraction of a security can be bought or sold.

- There are no dividend payments occurring during the life of the option

- No transaction costs or taxes, especially no penalty to short selling

### 4.2 Risk free Evaluation

As it was shown in the previous sections, the Wiener process is part of every stochastic process describing the dynamics of stock prices and their derivatives. Owing to its martingale property, no assumptions about its future value can be made, except for the zero mean normal distribution of its increments. It is a source of uncertainty that makes it very difficult to calculate the fair price of a derivative.

Nevertheless, by applying Itô's formula one can see that the price of the derivative and the price of the underlying stock are subject to the same source of uncertainty. The Wiener processes in the derivative pricing formula, as well as the one in the stock price formula are perfectly correlated. Therefore, a portfolio can be constructed consisting of a long position in the stock and a short position in a call option on that stock. In their effects on the total payoff of such a portfolio, the sources of uncertainty are offsetting each other and the derivative pricing function can be determined by applying the no-arbitrage argument, which states that a risk-free portfolio must yield the risk-free interest rate *r.*

### 4.3 Black-Scholes Partial Differential Equation

As explained in the previous sections, the stock price follows a GBM:

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while denotes the price of a derivative on *S* at time *t*. After applying Itô's formula, the infinitesimal change, , with respect to *S* and *t* is given by the following SDE:

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In order to create a risk-free portfolio, the diffusion terms of (in the stock price) and (in the option price) must offset each other which is the case if there is a long position of shares. The value of the resulting portfolio is then

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whereas the change in value, , must equal the value of compounded at the risk free rate *r*:

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Inserting (13), (14), and (15) into (16) gives the Black-Scholes-Merton partial differential equation:

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Since the dynamic of this hedge is expressed by an differential equation, it must be noted that the portfolio is risk-free only for an infinitesimal period of time and must be rebalanced continuously to maintain risk-free.

### 4.4 Black-Scholes Pricing Formula

The partial differential equation (17) that was derived above can be used to price any derivative with an underlying asset S. For a specific derivative certain boundary conditions are necessary to find a pricing formula that satisfies the PDE. The key boundary condition of a plain vanilla European call option is given by its payout at maturity T:

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Solving the BSM-PDE for a European call option C with respect to the boundary conditions finally yields the Black-Scholes pricing formula as

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where *S* and *K* denote the stock and strike price, respectively, *r* is the risk-free interest rate, is the time to maturity, the volatility of *S*, and represents the probabilities given by the cumulative normal distribution function adapted to a risk-neutral environment.

The first term, , of the formula can be interpreted as the expected value of S under the condition that *S>K* while the second term, ,is the continuously discounted strike price times the probability of S>K.

### 4.5 Evaluation of Model Parameters and Assumptions

Concerning its assumptions, the Black-Scholes model is very restrictive and also Black and Scholes (1973) talk about "'ideal conditions' in the market for the stock and for the option" (p. 640). As a consequence, empirical evidence shows that in reality many of the model's assumption only hold to a certain degree.

First of all, securities are not perfectly divisible: It is not possible to buy a fraction of a share. Nonetheless, this problem can be solved be increasing the number of options in the risk-free portfolio in such a way that becomes an integer.

Secondly, in reality, there are of course transaction costs, as well as taxes and margin requirements for short-selling. Hence a single arbitrage-free price can usually not be calculated, but only an arbitrage-free price span, which in practice is reflected in the bid-ask spread of an option. Additionally, there is also often a spread between risk-free interest rates for borrowing and lending which, moreover, cannot be assumed to be constant until maturity of the option.

The major drawback of the model is its assumption about the stock price movement and the corresponding normal distribution of stock returns which implies a constant volatility. In combination with the continuous trading assumption, which neither holds true in reality, sudden stock price jumps are denied, as well. Empirical studies, however, show that large fluctuations in the stock price occur significantly more often than expected. This leads to the dissenting assumption that the distribution is subject to heavy tails, whereas jumps should be modeled using a Poisson diffusion term.

Even though the volatility is the only parameter in the model that is not directly observable, there is sufficient reason to believe that the volatility of the underlying stock is changing during the life time of the option. Financial time series analysis of the returns of the S&P 500 index, for instance, also supports the assumption of a time-varying volatility process.

### 4.6 Volatility Smile

The previously described shortcomings of the model, in particular, the assumption of constant volatility leads to a phenomenon which is known as *volatility smile.* This term originates from the shape of a graph, where the implied volatility is a function of the strike price of an option. The smile is created by the effect that implied volatility is higher for ITM and OTM options, whereas it is low for ATM options. This is illustrated by the following figure:

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Figure 4 - Implied volatility smile for foreign currency options.

The implied volatility is the volatility that is implied by the Black-Scholes pricing formula if the price of an traded option, and all parameters other than the volatility are given. This can be also be expressed by the inverse of the BS pricing formula:

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Since the inverse function cannot be found analytically, the equation must be solved numerically in such a way that the market price of the option equals the BS price for a given volatility. By choosing options with different strike prices and maturities, the implied volatilities can then be calculated in order to construct the volatility smile or surface, respectively.

The volatility smile's characteristic shape is most pronounced in the case of foreign currency options. In the case of equity options, the implied volatilities for OTM options are smaller than for ITM and ATM options. Thus, with increasing strike price, the function tends downwards, resulting in a shape that is often referred to *volatility skew.* Figure 5 shows an illustration of the implied volatility skew of SPX call options. The blue curve connects the actual implied volatilities whereas the red line is a fitted curve for better illustration of the skew.

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Figure 5 - Volatility skew of SPX call options on 01/02/2008. TTM=1m; S=1447.16

## 5 Stochastic Volatility Models

### 5.1 Overview of Stochastic Volatility Models

The existence and persistence of the volatility smile for varying types of underlying assets suggests that there is a systematic bias in the Black-Scholes-Merton model that is mainly caused by assuming a constant, deterministic volatility. Taking the rather leptokurtic distribution of stock returns into account, one can assume that the volatility itself follows a stochastic process. The stock price dynamics can then be generally described by the following two coupled SDEs, (21) and (22):

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As the parameters , and already imply, there are various possibilities for a specific model of volatility process. A first classification of the different approaches can be done by the diffusion parameter .

The most common approaches for are for the lognormal diffusion process of the earlier introduced GBM, and for a square-root or Cox-Ingersoll-Ross (CIR) diffusion, where in both cases is the volatility of the volatility.

**[...]**

- Arbeit zitieren
- Pascal Debus (Autor), 2010, Application of Stochastic Volatility Models in Option Pricing, München, GRIN Verlag, https://www.grin.com/document/232541

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