# The Black–Scholes approach

### Introduction

Generally, asset returns are assumed to follow normal distribution. However, a number of authors proved that stock returns commonly present kurtosis higher than the normal distribution and that they regularly are negative skewed. Thus, the Black - Scholes model is not suitable for option pricing when returns do not follow normal distribution. Consequently, the prices obtained under the Black and Scholes model are not consistent with market option prices as the input to the Black - Scholes formula, such as moneyness and time to maturity, vary. Jarrow and Rudd and Heston notice, that the mispricing of the classic model is attributable to the skewness and the kurtosis in the distribution of the underlying asset's returns. Rubinstein indicates that the put options are usually underpriced under the Black and Scholes model as the result of the risk neutral distribution of returns being probably negatively skewed with a fatter left tail. Furthermore, Nandi finds evidence that because the kurtosis is higher than the normal distribution makes the model to underprice out of the money options. The effect of the incorrect assumption of the underlying process causes the need of using different levels of volatilities to price options on the same asset under the Black - Scholes approach. As a consequence of the limits of the Black - Scholes model, several authors proposed alternative option models.

The necessity for both stochastic volatility and jumps has been proven by empirical studies over these models. In order to capture the variation in strike at long term periods stochastic volatility appears to be needed to explain the variation in strike at longer terms, while jumps are needed to explain the variation in strike at shorter terms. Furthermore, jumps and the inability to trade continuously are usually the favored explanations for the existence of substantial hedging errors, whose persistence has been documented repeatedly.

An important feature of stochastic volatility models is the so-called "leverage effect" in which returns and volatility are negatively correlated. This correlation has been incorporated in two basic ways in the current literature: by allowing contemporaneous correlation and allowing returns to be correlated with future volatility. An important contribution to this literature is by Jacquier, Polson, and Rossi (2004) (JPR hereafter) who develop a stochastic volatility model that simultaneously allows for the leverage effect, fat tails and as a byproduct generates negative skewness in stock returns. The leverage effect is modeled by allowing the shocks to returns and volatility to be contemporaneously correlated. They capture excess kurtosis by modeling unexpected returns as an inverse-chi squared mixture of normals (the normals are correlated with the innovations to log-volatility, which generates the leverage effect), so stock returns are effectively modeled as a Student t random variable that is correlated contemporaneously with volatility. Because returns and volatility are negatively contemporaneously correlated this model generates negative skewness in returns, which sets this model apart from virtually all other stochastic volatility models. Because returns and volatility are negatively correlated a negative unexpected return is generally related to an increase in volatility that pronounces the left tail of the distribution while positive unexpected returns are dampened by a contemporaneous decrease in volatility. This is the only current stochastic volatility model that we are aware of that can capture all these three important characteristics.

One approach has been to use a stochastic volatility model: Hull and White16, Johnson and Shanno17, Scott18, Wiggins19, Bailey and Stulz20, Melino and Turnbull21, Stein and Stein22, Amin and Ng23, Heston24, Nandi25, Bates26 and Singleton28, for example, followed this approach.

An important class of extensions of the classical option pricing is given by the presence of jumps in the underlying process. Discontinuities in the underlying price in the form of jumps have a long history in the financial literature. Merton considered in 197635 the addition of a jump component to the classical geometric Brownian motion model for option pricing. Even, before that time, the possibility of jumps in asset prices has been considered by Mandelbrot36 and Press37. The importance of introducing a jump component in the underlying process has been noted for example by Bakshi, Cao and Chen38 who argue that models based on pure diffusion processes have difficulties in explaining volatility smiles in general and in particular for short dated option prices. Moreover Broadie, Chernov and Johannes39 find strong evidence in favor of both jumps in the underlying returns and in the volatility of the returns using all S&P 500 future option transactions from 1987 to 2003. Inside the class of processes presenting jumps in the underlying process, it is possible to further distinguish between jump diffusion processes and pure jump processes. Models presenting a Poisson jump on top of the diffusion process, like the one presented by Merton in 1976, are examples of jump diffusion models. The assumption that the stock returns follow a jump diffusion process has been used to improve option pricing as well as other theories like the capital asset pricing model. Authors who studied jump diffusion processes include Cox and Ross40, Jarrow and Rosenfeld41, Ahn and Thompson42, Naik and Lee43, Aase44, Amin45, Bates46, Bakshi and Chen47, Scott48, Kou49 and Duffie, Pan and Singleton50. Pure jump processes, on the other side, lack of the di®usion elements and hence of the continuous component. The presence of an infinite number of discontinuities in the pure jump processes makes them a quite different class of processes compared with the traditional Black, Scholes and Merton approach.

The intuition behind these stochastic volatility models is related to the scaling property of Brownian motion, which says that changes in time are related to changes in scale; so it is possible to randomly change the volatility by randomly change the time. The rate of time change has to be mean reverting to allow the random time changes to persist, moreover the instantaneous rate of time change should be positive otherwise the time would be decreasing. The author in particular generate these new processes which have the desired volatility properties by subordinating the variance gamma and the CGMY to the time integral of a Cox, Ingersoll and Ross20 square root process. Define the instantaneous rate of time change, y(t), as the solution of the differential equation where W(t) is a standard Brownian motion independent of all the other processes encountered so far; ´ is the long run rate of time change; k is the rate of mean reversion and ¸ control the volatility of the time change. The randomness of the process induces stochastic volatility, while the mean reversion in this process creates the volatility clustering desired. We can see that here changes in volatility are independent of asset returns. This is not consistent with empirical evidence showing a correlation, which is often negative in the case of stocks, between returns and their volatilities. Carr and Wu21 extend this approach by allowing that changes in volatility are independent of asset returns. In particular Carr and Wu work with Levy processes in general, and employ a measure change in the complex plane to obtain the characteristic function of the time changed Levy process and then they use fast Fourier transform to price contingent claims.

On the theoretical side, arguments have been proposed by Geman, Madan, and Yor [20] which suggest that price processes for financial assets must have a jump component, while they need not have a diffusion component. Their argument rests on recognizing that all price processes of interest may be regarded as Brownian motion subordinated to a random clock. This clock may be regarded as a cumulative measure of economic activity, as conjectured by Clark [12], and as estimated by Ane and Geman [1]. As time must be increasing, the random clock can be modelled as a pure jump increasing process, or alternatively as a time integral of a positive diffusion process, and thus devoid of a continuous martingale component. If jumps are suppressed, then the clock is locally deterministic which they rule out a priori. Thus, the required jumps in the clock induce jumps in the price process, while no argument similarly requires that prices have a diffusion component.

The explanation usually given for the use of jump diffusion models is that jumps are needed to capture the large moves that occasionally occur, while diffusions are needed to capture the small moves which occur much more frequently. However, since at least the pioneering work of Mandelbrot [26] on stable processes, it has been recognized that many pure jump models are able to capture both rare large moves and frequent small moves. Motivated by the possibility that price processes could be pure jump, several authors have focused attention on pure jump models in the Levy class. Technically, these processes can capture frequent small moves through the use of a Levy density whose spatial integral is infinite

There are at least three examples of such pure jump infinite activity Levy processes. First, we have the normal inverse Gaussian (NIG) model of Barndorff-Nielsen [4], and its generalization to the generalized hyperbolic class by Eberlein, Keller, and Prause [15]. Second, we have the symmetric variance gamma (VG) model studied by Madan and Seneta [29] and its asymmetric extension studied by Madan and Milne [28], Madan, Carr, and Chang [27].

### The two models

- Variance Gamma
- Normal Inverse Gaussian Process

Carr, Madan, Geman and Yor19 present extensions of the variance gamma model and of the CGMY models which incorporate stochastic and mean reverting volatilities.