Extensions to the BS methodology

The above stylized facts gave rise through the years to a number of theoretical extensions to the Black-Scholes framework.

Figure 7.2: The normal distribution...

Figure 7.3: ...and the Oct-87 events

1. JUMP DIFFUSION The jump diffusion model of Merton (1976, Journal of Financial Economics) is motivated by the infrequent but substantial crashes that are observed in the asset price diffusions. For example, figures 7.2 and 7.3 illustrates the extreme nature of some market events. In fact, if the underlying distribution was log-normal, then and event such as the crash of 19-Oct-87 should happen on average once every $10^87$ years, a time period which is millions of times the age of our universe...

   According to Merton's model, the underlying price follows

   $$\frac{dS(t)}{S(t)}=\mu dt+\sigma dB(t)+J(t) dq(t)$$
   where $dq(t)$ is a Poisson random variable that takes value zero [no jump realized] with probability $1-\lambda dt$, and value one [jump realized] with probability $\lambda dt$.

   There are two sources of risk, the diffusion $dB(t)$ and the jumps $dq(t)$, but only one traded asset apart from the riskless one. The key assumption of Merton is that the jump component of the asset's return represents nonsystematic risk, that is to say the expected jumps do not affect the representative agents marginal utility of wealth. This risk is not priced in the economy, or equivalently its market price of risk is equal to zero. In this case, one can consider the diffusion under the risk neutral measure, with $\mu =r$, and take the appropriate expectation under this measure. Then, the pricing formula is

   $$C_{Merton}\left( S,t;K,\tau \right) =\sum_{n=0}^{\infty }\frac{e^... ...\left( \tau -t\right) \right] ^{n}}{n!}C_{n}\left( S,t;K,\tau \right) \text{,}$$
   where $C_{n}$ is the BS price when the instantaneous variance rate is
   $$\sigma ^{2}+\frac{n}{\tau -t}\sigma _{J}^{2}$$,
   and the risk free rate is
   $$r-\lambda \mu _{J}+\frac{n}{\tau -t}\log \left( 1+\mu _{J}\right)$$   .

   In the case where the price of risk is not equal to zero for the jump component, Bates (1998, Wharton School UPenn) shows that under risk neutrality the risk adjusted rate of the Poisson process driving the jumps is

   $$\bar{\lambda}=\lambda \mathbf{E}\left[ 1+d\mathcal{M}\right]$$   ,
   and the risk adjusted mean jump is
   $$\bar{\mu}_{J}=\bar{\mu}_{J}+\frac{d\left\langle q,\mathcal{M}\right\rangle }{\mathbf{E}\left[ 1+d\mathcal{M}\right] }\text{.}$$
   The quantity $d\mathcal{M}$ represents the percentage change of the marginal utility of nominal wealth, given that a jump has occurred in the interval of length $dt$.

2. STOCHASTIC VOLATILITY MODELS Hull and White (1987, Journal of Finance) propose a model where the volatility of the underlying asset evolves in a stochastic fashion under risk neutrality
   

   $$\frac{dS\left( t\right) }{S\left( t\right) } = rdt+\sqrt{V\left( t\right) }d\varepsilon \left( t\right) \text{,}$$ (7.1)

   $$dV\left( t\right) = \kappa _{V}\left[ \theta _{V}-V\left( t\right) \right] dt+\sigma _{V}\left[ V\left( t\right) \right] ^{\gamma }d\varepsilon _{V}\left( t\right)$$ (7.2)

   
   
   The errors $d\varepsilon \left( t\right)$ and $d\varepsilon _{V}\left( t\right)$ are considered uncorrelated in the above setup. The Hull and White model is by construction under risk neutrality and therefore the riskless asset need not be specified. Nevertheless, if the model was specified under the true measure, one should make assumptions in the spirit of Merton, since the market is once more incomplete. Under the above assumptions, Hull and White show that the European call option price is given by
   $$C_{HW}\left( S,t;K,\tau \right) =\int_{0}^{\infty }C_{\bar{V}}\left( S,t;K,\tau \right) g\left( \bar{V}\right) d\bar{V}\text{,}$$
   where $\bar{V}$ is the average value of the variance rate over the life of the option, $C_{\bar{V}}$ is the BS option price expressed as a function of the average variance $\bar{V}$, and $g$ represents the density of the mean variance under the diffusion above. Hull and White note that when the asset return and the volatility are correlated, a feature that would take into account the leverage effects, prices can only be obtained through Monte Carlo simulations. Hull and White give a series expansion and Heston (1993, Review of Financial Studies) gives a quasi-closed form solution for the particular case where $\gamma =0.5$.