Abstract

The dynamics of the asset process and variance process are driven by continuous time processes in the Information Based Asset Pricing Framework as proposed by Brody, Hughson and Macrina, also known as the BHM Model. To make use of numerical simulation, the continuous time processes can be discretized to discrete time processes. Here, two discretization schemes will be looked at: Euler scheme and Milstein scheme. The main objective of this study is to apply the two discretization schemes to the Information Based Asset Picing Framework. The two schemes will first be applied to the Black-Scholes and the Heston models and then extended to the BHM model. Studies have shown that the Euler scheme approach to discretization can be inefficient which makes the use of the Milstein scheme approach to discretization more accurate due to the expansion of the coefficients involved in the stochastic differential equation.

1. Introduction

The classical model of asset prices which was published in 1973 is known as the Black-Scholes model. One of the key assumptions made under the model is that the volatility of asset returns is constant. This implies that the dynamics of the prices of risky assets can be modelled by geometric Brownian motions (GBMs). This results in a distribution which is lognormal.

Empirical studies demonstrate that the volatility of asset returns is not constant with Rosenberg ¹ coming to the same conclusion through studies of implied volatility. Much work has followed with Scott ² also providing empirical evidence showing that volatility changes with time and that the changes are unpredictable. The studies also show that volatility has a tendency to revert to a long-run average.

The Heston model is an extension of the Black-Scholes model which makes the assumption of stochastic volatility and that the volatility and the underlying asset price are correlated. In so doing, the Heston model is able to capture various properties of the financial information which the Black-Scholes model doesn’t. However, the reliability of the model is questionable because the assumptions on the volatility and the underlying asset price dynamics display an ad-hoc nature.

Another approach suggested later by Brody, Hughson and Macrina called the BHM approach or model obtains the asset price dynamics using a more realistic approach towards the structure of the market unlike the Heston model where the dynamics of the volatility and price are pre-specified. The approach specifies a model for the structure of the information available in the market since asset prices are determined by expectations on the future cash flows given the market information available.

The later approach doesn’t assume any dynamic model for the asset prices, it’s observed that the asset price dynamics derived with the assumed information structure naturally have stochastic volatility giving a different view of the volatility nature. Accordingly, the model illustrates that the volatility of volatility is stochastic. In the BHM approach, the stochastic process governing the dynamics of an asset are deduced as compared to being imposed at the beginning of the process in an arbitrary way.

The aim of this study is to make the BHM Model suitable for numerical simulation through discretization of its stochastic differential equations (SDEs). Discretization refers to this process of approximating a continuous time SDE to a discrete time SDE. Both in literature and in practice, a lot of focus has been directed to discrete time approximations of SDEs which has led to the development of schemes that accomplish this.

The Euler scheme approach to discretization was first studied by Maruyama ³. In order to simulate a realization of the Euler approximation, the independent random variables involved need to be generated. Gatheral ⁴ shows that a Euler discretization of the variance process may result in a negative variance.

There are several approaches that can be taken to fix the problem of the variance becoming negative with a non zero probability which makes the computation of the square root of the variance impossible. One approach is by the reflection scheme which involves taking the absolute value of the variance, that is if denotes the variance, then, its absolute value, will be used.

Another approach that can be used to counter the problem of the variance becoming negative is the full truncation scheme which involves taking the positive value of the variance which is denoted by that is This study adopts the second approach, that is the full truncation scheme.

According to Platen ⁵, the Euler scheme approach can be inefficient and often shows poor stability properties. This means that other stochastic numerical methods need to be considered in discretization of the BHM model. The Milstein scheme increases the accuracy of the discretization by considering expansion of the coefficients involved in the SDE. In addition, the scheme significantly alleviates the problem of the variance becoming negative which makes it preferred to the Euler scheme.

The study starts by looking at the existing work in discretization as applied to the Black-Scholes and Heston Model as shown in Gatheral ⁴. A similar approach is then extended to the BHM Model. Both the Euler scheme and Milstein scheme will be looked at in the discretization process.

An assumption is made that the asset price, is driven by the SDE:

(1.1)

where is a Brownian motion.

is simulated over the interval [0, T], which is assumed to be discretized as The time intervals are equally spaced with width .

Integrating equation 1.1 over the interval leads to:

(1.2)

(1.3)

The value at time t for in equation 1.3 is known, the problem is to find the value for .

2. Euler Scheme Discretization

Euler discretization is equivalent to getting an approximate integral value using the left-point rule since the value at time t is known. From integral 1.3:

Similarly:

where and have the same distribution.

Thus,

(2.1)

Under this model, the asset price dynamics under the risk neutral measure, are:

where is a Brownian motion, denotes the underlying asset price, is the volatility of the underlying asset and is the risk free rate of interest.

Integrating equation 2.2 over the interval gives:

Thus, the Euler discretization for the Black-Scholes model is given as:

(2.3)

Under a risk-neutral measure, , the Heston ⁶ model assumes that an underlying asset price, has a stochastic variance, , that follows a Cox, Ingersoll and Ross ⁷ process with long-run mean and rate or reversion while is the volatility of the volatility. This process is represented by the following dynamical system:

(2.4)

(2.5)

where is a constant risk-free interest rate and is a constant dividend. All the parameters and are positive constant. The terms and are Wiener processes that must be correlated with each other, that is

(2.6)

In equation 2.6, the term $\rho$ is the correlation coefficient between the return of the underlying asset and the changes in the variance. The correlation, which is often negative, will ensure that the volatility for example will rise if the underlying asset value falls dramatically. In addition the variance is also mean-reverting, which is also evident in the market. The mean-reverting process is the term .

Integrating equation 2.4 over the interval :

Euler scheme approximates the integrals as follows:

Thus:

(2.7)

To counter the occurrence of a negative variance, the full truncation scheme is applied which involves substituting with which results in:

(2.8)

Integrating equation 2.5 over the interval gives:

Thus:

(2.9)

where

To counter the occurrence of a negative variance, the full truncation scheme is applied which involves substituting with which results in:

(2.10)

With initial values for the asset price and for the variance, equations 2.13 and 2.14 can be used to obtain and .

and have a correlation two independent standard normal variables and are generated such that:

According to Brody, Hughson and Macrina, ⁸, the dynamics of the price process are given as:

In this study, an assumption will be made that is a constant which implies that . Thus;

(2.11)

where denotes the absolute volatility process:

denotes the discount factor and denotes the short rate.

The approach by Macrina ⁹ is used to obtain the dynamics for the volatility in the BHM model which are given as:

(2.12)

where and

Integrating equation 2.11 over the interval , results in:

Thus, the Euler discretization is given as:

(2.13)

Integrating equation 2.12 over the interval , gives:

Thus, the Euler discretization is given as:

(2.14)

With initial values for the asset price and for the variance, equations 2.13 and 2.14 can be used to obtain and .

3. Milstein Scheme Discretization

Glasserman ¹⁰ shows that the Milstein scheme refines the Euler scheme by the introduction of an additional term.

This scheme is applicable where the coefficients and depend only on This implies that an assumption will be made such that:

(3.1)

Integrating equation 3.1 over the interval , results in:

(3.2)

Through the application of Ito's lemma, the accuracy of the Milstein scheme in discretization is enhanced by considering the expansions of the coefficients and as follows:

Using Ito's lemma:

(3.3)

(3.4)

Getting the square of equation 3.1 and substituting the result in equation 3.4, leads to:

Thus:

(3.5)

Using a similar approach, from Ito's lemma:

Thus:

(3.6)

Integrating equation 3.5 over the interval [t, s] where , results in:

(3.7)

Integrating equation 3.6 over the interval [t, s] where , leads to:

(3.8)

Substituting for equation 3.7 and 3.8 in equation 3.2 gives the following integral:

Ignoring the terms drds, and since they are higher than order one results in:

(3.9)

Applying Euler scheme discretization to the last term in equation 3.9 leads to:

Thus

(3.10)

Let , using Ito's lemma,

(3.11)

This implies that:

(3.12)

Substituting the result of equation 3.12 to equation 3.10 results in:

(3.13)

From equation 3.10 and 3.13, the Milstein discretization is given by:

Thus:

(3.14)

Using the assumption made by the Milstein scheme in equation 3.1, under the Black-Scholes model, and , such that

(3.15)

Substituting for and in equation 3.14leads to:

Thus

(3.16)

This scheme adds a correction term of to the Euler Scheme result in equation 2.3.

For the Heston model asset price process, a substitution is done in equation 3.14, such that and resulting in:

Thus

(3.17)

A correction term of is added to the Euler discretization given in equation 2.7.

Applying the full truncation scheme, equation 3.17 becomes:

(3.18)

For the Heston model variance process, a substitution is done in equation 3.14 such that and which leads to:

Thus:

(3.19)

This approach produces less negative values for the variance as compared to Euler discretization. To counter the occurrence of a negative variance, the full truncation scheme is applied which involves substituting with in equation 3.19 resulting in:

(3.20)

The result in Equation 3.19 is the same result obtained in Gatheral ⁴ for the Milstein scheme discretization of the Heston Model variance process.

Comparing the result obtained in equation 3.20 and that obtained under the Euler scheme in equation 2.10, there's an additional term of .

With initial values for the asset price and for the variance, equation 3.18 and equation 3.20 can be used to obtain and .

and have a correlation two independent standard normal variables and are generated such that:

Using the assumption made by the Milstein scheme in equation 3.1, under the BHM model, and , such that

(3.21)

Substituting for and in equation 3.14 leads to:

Thus:

(3.22)

The result obtained in equation 3.22 is the same as the result obtained under the Euler scheme in equation 2.13. This means that for the BHM model, the asset process discretization remains the same under both the Euler scheme and Milstein scheme.

For the BHM model variance process, a substitution in equation 3.14 is made such that, and leading to:

Thus:

(3.23)

The result obtained in equation 3.23 is the same as the result obtained under the Euler scheme in equation 2.14. This means that for the BHM model, the variance process discretization remains the same under both the Euler scheme and Milstein scheme.

The dynamics under the BHM model Milstein discretization scheme will take the same form as the BHM model under the Euler discretization scheme.

4. Conclusion

A brief discussion of the asset pricing models is given in the introduction starting from the Black-Scholes model which is the basic model from which the stochastic volatility models are derived from. The Heston model which addresses the biggest weakness of the Black-Scholes model which is the asumption of constant volatility is also looked at. The BHM model is then presented which addresses the weakness of the other two models of imposing the asset dynamics.

The Euler discretization is applied to the Black-Scholes model and the Heston Model. This approach is then extended to the BHM Model which is the main objective of this study. One of the disadvantages of the Euler Scheme is that the variance can become negative. To counter this problem, the full truncation scheme approach is applied.

Another discretization scheme, Milstein scheme is shown to be more accurate since it considers the expansion of the coefficients involved in the SDE. In addition, the Milstein scheme significantly alleviates the problem of the variance becoming negative which makes it preferred to the Euler approach to discretization.

From this study, it is observed that the asset process and variance process in the information based asset pricing framework take a similar form under both the Euler Scheme and Milstein Scheme.

Further works will be to extend this approach to the multi-asset framework for the information based asset pricing framework and to consider the case where wich denotes the interest rate is not assumed to be constant.

References