Option Pricing Models

Research Proposal

Title: Option Pricing Models

Introduction

It is known that most of the option pricing models and techniques employeed by today's analysts are rooted in a model developed by Fischer Black and Myron Scholes in 1973. One basic assumption of BS model is that the stock price is log-normally distributed with constant volatility. However, Fama (1965) and Mandelbrot (1966) found that stock returns exhibit both fat-tailed marginal distributions and volatility clustering. These features are interpreted as evidence of stochastic volatility of financial asset prices. To overcome the shortcoming, many researchers have contributed to substantial new models that incorporate stochastic volatility in the last two decades. It is thus interesting to examine whether the stochastic-volatility option pricing models provide improvements to the BS model.

During the past decade, researchers have begun to study generalized autoregressive conditional heteroskedasticitic (GARCH) models for option pricing due to the superior ability of this class of models to describe asset return dynamics. Duan (1995) developed a theory with respect to which options can be priced when the evolution of the asset return follows the GARCH process. Empirically, Duan (1996), Heston and Nandi (2000), Hsieh and Ritchken (2000), Hardle and Hafner (2000), Duan and Zhang (2001) and Christoffersen and Jacobs (2002) have showed that the GARCH model can be used to capture the pricing behavior of exchange-traded European options. Analytically pricing European options requires the knowledge of the risk-neutral distribution of the cumulative return with respect to a given model. However, the analytical form of the distribution for the time aggregated return is unknown for all GARCH specifications, and thus computing option prices must rely on some time-consuming numerical procedures.

In recent years, researchers have tried to speed up the valuation of European options under GARCH by developing analytical solutions and analytical approximations for specific forms of the GARCH model. Heston and Nandi (2000) developed an analytical formula to price European options when the dynamic of the conditional variance is given by a specific GARCH process. In contrast, Duan, Gauthier and Simonato (1999) developed and analytical approximation for the European option price under GARCH. Their approach utilizes the idea of Jarrow and Rudd (1982) to find an approximate option price under general stochastic process.

Meanwhile, significant computational simplification is achieved when option pricing is approached through the change of numeraire technique. By pricing an asset in terms of another traded asset (the numeraire), this technique reduces the number of sources of risk that need to be accounted for.

Research Objectives

To examine the function of the Black-Scholes model and some new models developed from it in modern financial market.
To develop the models with less assumptions.
To apply the models to the China financial market.
To grope different numeraires to simplify the models in China financial market.
To compare different option pricing models, such as BS model, GARCH model and so on.
To make comment on the models within China financial market.

Literature Review

The Black and Scholes Model:

In order to understand the model itself, we divide it into ...

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Research Objectives

To examine the function of the Black-Scholes model and some new models developed from it in modern financial market.
To develop the models with less assumptions.
To apply the models to the China financial market.
To grope different numeraires to simplify the models in China financial market.
To compare different option pricing models, such as BS model, GARCH model and so on.
To make comment on the models within China financial market.

Literature Review

The Black and Scholes Model:

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, Ke(-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.

The GARCH models

There are two types of volatility models: continuous-time stochastic models and discrete-time stochastic generalized autoregressive conditional heteroskedasticity (GARCH) models. On one hand, the continuous-time model can serve as the limit of a certain GARCH model. For example, Nelson(1990a) showed that the GARCH (1,1) model converged to a certain diffusion model. Duan (1996) argued that most of the existing bivariate diffusion models that had been used to model asset returns and volatility could be represented as limits of a family of GARCH models. As a special case, the particular GARCH option model proposed by Heston and Nandi (2000) was proved to contain Heston’s (1993) stochastic volatility models as a continuous-time limit. On the other hand, the GARCH model has an advantage over the continuous-time model in that the volatility is readily observable in the history of asset prices. As a result, it is possible to price an option only using the information from the observations of asset prices. In contrast, the continuous-time stochastic model has an inherent disadvantage that it assumes that volatility is observable, but it is impossible to exactly filter volatility from discrete observations of spot asset prices in a continuous-time stochastic volatility model. Consequently, it is impossible to price an option solely on the basis of the history of asset prices. Since volatility is unobservable, one has to use the volatility implied from one option to value other options. Unfortunately, this method is not always feasible especially when the related options are thinly traded. Thus, the GARCH model is chosen over the continuous-time model when comparing the empirical performance of the stochastic option model and the discrete-time model.

The standard Black-Scholes (BS) formula prices a European option on an asset that follows geometric Brownian motion. The asset’s uncertainty is the only risk factor in the model. A more general approach developed by Black Merton-Scholes leads to a partial differential equation. The most general method developed so far for the pricing of contingent clains is the martingale approach to arbitrage theory developed by Harrison and Kreps (1981) and others.

Whether one use the PDE or the standard risk-neutral valuation formulas of the martingale method, it is in most cases very hard to obtain analytic pricing formulas. Thus, for many important cases, special formulas (typically modifications of the original BS formula) have been developed.

One of the most typical cases with multiple risk factors occurs when an option involves a choice between two assets with stochastic prices. In this case, it is often of considerable advantage to use a change of numeraire in the pricing of the option.

The basic idea of the numeraire approach can be described as follows. Suppose that an option’s price depends on several (say, n) sources of risk. We may then compute the price of the option according to this scheme:

Pick a security that embodies one of the sources of risk, and choose this security as the numeraire.
Express all prices in the market, including that of the option, in terms of the chosen numeraire. In other words, perform all the computations in a relative price system.
Since the numeraire asset in the new price system is riskless (by definition), we have reduced the number of risk factors by one, from n to n-1.
We thus derive the option price in terms of numeraire. A simple translation from the numeraire back to the local currency will then give the price of the option in monetary terms.

Assumption 1. Given a priori are:

An empirically observable (k+1)-dimensional stochastic process:

X = (X1, …, Xk+1)

with the notational convention that the process k+1 is the riskless rate:

Xk+1(t) = r(t)

We assume that under a fixed risk-neutral martingale measure Q the factor dynamics have the form:

dXi(t) = μi(t, X(t))dt + δi(t, X(t))dW(t)

i = 1, …, k+1

where W = (W1, …, Wd)́ is a standard d-dimensional Q-Wiener process and δi = (δi1, δi2, …, δid) is a row vector. The superscript ́ denotes transpose.

A risk-free asset (money account) with the dynamics:

dB(t) = r(t)B(t)dt

The interpretation is that the components of the vector process X are the underlying factors in the economy. We make no a priori market assumptions, so whether a particular component is the price process of a traded asset in the market will depend on the particular application.

Assumption 2 introduces asset prices, driven by the underlying factors in the economy.

Assumption 2

We consider a fixed set of price process S0(t), …, Sn(t), each assumed to be the arbitrage-free price process for some traded asset without dividends.
Under the risk-neutral measure Q, the S dynamics have the form

dSi(t) = r(t)Si(t)dt + Si(t)Σσij(t, X(t))dWj(t) (1)

for i = 0, …, n-1, j = 1, …, d.

The n-th asset price is always given by

Sn(t) = B(t)

And thus (1) also holds for i = n with σij = 0 for j = 1, …, d.

We now fix and arbitrary asset as the numeraire, and for notational convenience we assume that it is S0. We may then express all other asset prices in terms of the numeraire S0, thus obtaining the normalized price vector Z = (Z0, Z1, …, Zn), defined by Zi(t) = Si(t)/ S0(t)

We now have two formal economies: the S economy where prices are measured in the local currency, and the Z economy, where prices are measured in terms of the numeraire S0.

The main result is a theorem that shows how to price an arbitrary contingent claim in terms of the chosen numeraire. For brevity, we henceforth refer to a contigent claim with exercise data T as a T-claim.

Main Theorem. Let the numeraire S0 be the price process for a traded asset with S0(t) > 0 for all t. Then there exists a probability measure, denoted by Q0, with properties as follows:

For every T-claim Y, the corresponding arbitrage free price process Π(t; Y) in the S economy is given by

Π(t; Y) = S0(t)ΠZ(t; Y/S0(T)) (2)

where ΠZ denotes the arbitrage-free price in the Z economy.

For any T-claim Y (Y = Y/ S0(T), for example) its arbitrage-free price process ΠZ in the Z economy is given by

ΠZ(t; Y) = E0t, X(t)[Y] (3)

where E0 denotes expectations with regard to Q0. The pricing formula (2) can be written

Π(t; Y) = S0(t) E0t, X(t)[ Y/S0(T)] (4)

The Q0 dynamics of the Z processes are given by

dZi = Zi[σi-σ0]dW0, i = 0, …, n (5)

where σi = (σi1, σi2, …, σid), and σ0 is defined similarly.

The Q0 dynamics of the price processes are given by

dSi = Si(r +σiσ΄0)dt + Siσi dW0 (6)

where W0 is a Q-Wiener process.

The Q0 dynamics of the X processes are given by

dXi = (μi + δiσ΄0)dt + δidW0 (7)

In passing, note that if we use the money account B as the numeraire, the pricing formula above reduces to the well-known standard risk-neutral valuation formula

Π(t; Y) = B(t) E0t, X(t)[ Y/B(T)] = E0t, X(t)[e-∫ r(s)dsY] [t, T] (8)

In more pedestrian terms, the main points of the Theorem above are as follow:

Equation (2) shows that the measure Q0 takes care of the stochasticity related to the numeraire S0(t)—we simply use the observed market price.

We also see that if the claim Y is of the form Y = Y0S0(T) (where Y0 is some T-claim) then the change of numeraire is a huge simplification of the standard risk-neutral formula (8). Instead of computing the joint distribution of ∫Ttr(s)ds and Y (under Q), we have only to compute the distribution of Y0 (under Q0).

Equation (3) shows that in the Z economy, prices are computed as expected values of the claim. Observe that there is no discounting fact or in (3). The reason is that in the Z economy, the price process Z0 has the property that Z0(t) = 1 for all t. Thus, in the Z economy the short rate equals zero.
Equation (5) says that the normalized price processes are martingales under Q0, and identifies the relevant volatility.
Equations (6)-(7) show how the dynamics of the asset prices and the underlying factors change when we move from Q to Q0. Note that the crucial object is the volatility σ0 of the numeraire asset.

Methodology

There are three research philosophies, which all have an important part to play in business and management research. Which one to choose depends on the way thinking about the development of knowledge. Positivism, “works with an observable social reality and that the end product of such research can be law-like generalizations similar to those produced by the physical and natural scientists” (Remenyi, 1998). There will be an emphasis on a highly structured methodology to facilitate replication (Gill and Johnson, 1997) and on quantifiable observations that lend themselves to statistical analysis. Interpretivism, it claims that rich insights into this complex world are lost if such complexity is reduced entirely to a series of law-like generalizations. And for realism, it is based on the belief that a reality exists that is independent of human thoughts and beliefs. According to the different characters of these three philosophies, I choose the positivism to do may research work. In this research, I assume the role of an objective analyst, coolly making detached interpretations about those data that have been collected in an apparently value-free manner.

Most of the data collected are the quantitative data. To be useful these data need to be analysed and interpreted. And the quantitative data can be incorporated into relatively inexpensive personal-computer-based analysis software. These range from spreadsheets such as Excel and Lotus 1-2-3 to more advanced data management and statistical analysis software packages such as Minitab, SAS, SPSS for Windows and Statiew, and the software from the website as well, such as Excel add-in, options strategy evaluation model, and on-line pricing calculators.

Data Resources