8.2 Modern Pricing Models

The geometric Brownian motion (GBM) is the building block of modern finance. In particular, in the Black-Scholes model the underlying stock price is assumed to follow the GBM dynamics:

$\displaystyle dS_t = r S_t dt + \sigma S_t dW_t,$ (8.1)

which, applying Itô's lemma, can be written as:

$\displaystyle S_t = S_0\exp \left\{\left(r - \frac{\sigma^2}{2}\right)t + \sigma W_t \right\}.$ (8.2)

The empirical facts, however, do not confirm model assumptions. Financial returns exhibit much fatter tails than the Black-Scholes model postulates, see Chapter 1. The common big returns that are larger than six-standard deviations should appear less than once in a million years if the Black-Scholes framework were accurate. Squared returns, as a measure of volatility, display positive autocorrelation over several days, which contradicts the constant volatility assumption. Non-constant volatility can be observed as well in the option markets where ``smiles'' and ``skews'' in implied volatility occur. These properties of financial time series lead to more refined models. We introduce three such models in the following paragraphs.


8.2.1 Merton Model

If an important piece of information about the company becomes public it may cause a sudden change in the company's stock price. The information usually comes at a random time and the size of its impact on the stock price may be treated as a random variable. To cope with these observations Merton (1976) proposed a model that allows discontinuous trajectories of asset prices. The model extends (8.1) by adding jumps to the stock price dynamics:

$\displaystyle \frac{dS_t}{S_t} = r dt + \sigma dW_t + dZ_t,$ (8.3)

where $ Z_t$ is a compound Poisson process with a log-normal distribution of jump sizes. The jumps follow a (homogeneous) Poisson process $ N_t$ with intensity $ \lambda $ (see Chapter 14), which is independent of $ W_t$. The log-jump sizes $ Y_i\sim N( \mu, \delta^2 )$ are i.i.d random variables with mean $ \mu$ and variance $ \delta^2$, which are independent of both $ N_t$ and $ W_t$.

The model becomes incomplete which means that there are many possible ways to choose a risk-neutral measure such that the discounted price process is a martingale. Merton proposed to change the drift of the Wiener process and to leave the other ingredients unchanged. The asset price dynamics is then given by:

$\displaystyle S_t=S_0\exp\left(\mu^M t+ \sigma W_t + \sum_{i=1}^{N_t}Y_i \right),$    

where $ \mu^M=r-\sigma^2-\lambda\{\exp(\mu+\frac{\delta^2}{2})-1
\}$. Jump components add mass to the tails of the returns distribution. Increasing $ \delta$ adds mass to both tails, while a negative/positive $ \mu$ implies relatively more mass in the left/right tail.

For the purpose of Section 8.4 it is necessary to introduce the characteristic function (cf) of $ X_t=\ln \frac{S_t}{S_0}$:

$\displaystyle \phi_{X_t}(z)= \exp \left [t \left \{-\frac{\sigma^2z^2}{2} + i\mu^M z + \lambda \left(e^{-\delta^2 z^2/2+ i\mu z -1 } \right )\right \} \right ],$ (8.4)

where $ X_t=\mu^M t+ \sigma W_t + \sum_{i=1}^{N_t}Y_i$.


8.2.2 Heston Model

Another possible modification of (8.1) is to substitute the constant volatility parameter $ \sigma$ with a stochastic process. This leads to the so-called ``stochastic volatility'' models, where the price dynamics is driven by:

$\displaystyle \frac{dS_t}{S_t} = r dt + \sqrt{v_t} dW_t,$    

where $ v_t$ is another unobservable stochastic process. There are many possible ways of choosing the variance process $ v_t$. Hull and White (1987) proposed to use geometric Brownian motion:

$\displaystyle \frac{dv_t}{v_t} = c_1 dt + c_2 dW_t.$ (8.5)

However, geometric Brownian motion tends to increase exponentially which is an undesirable property for volatility. Volatility exhibits rather a mean reverting behavior. Therefore a model based on an Ornstein-Uhlenbeck-type process:

$\displaystyle dv_t = \kappa (\theta - v_t) dt + \beta dW_t,$ (8.6)

was suggested by Stein and Stein (1991). This process, however, admits negative values of the variance $ v_t$.

These deficiencies were eliminated in a stochastic volatility model introduced by Heston (1993):


$\displaystyle \frac{dS_t}{S_t}$ $\displaystyle =$ $\displaystyle r dt + \sqrt{v_t} dW_t^{(1)},$  
$\displaystyle dv_t$ $\displaystyle =$ $\displaystyle \kappa (\theta - v_t) dt + \sigma \sqrt{v_t} dW_t^{(2)},$ (8.7)

where the two Brownian components $ W_t^{(1)}$ and $ W_t^{(2)}$ are correlated with rate $ \rho$:

$\displaystyle \mathop{\hbox{Cov}}\left(dW_t^{(1)},dW_t^{(2)} \right )=\rho dt,$ (8.8)

for details see Chapter 7. The term $ \sqrt{v_t}$ in equation (8.7) simply ensures positive volatility. When the process touches the zero bound the stochastic part becomes zero and the non-stochastic part will push it up.

Parameter $ \kappa$ measures the speed of mean reversion, $ \theta$ is the average level of volatility, and $ \sigma$ is the volatility of volatility. In (8.8) the correlation $ \rho$ is typically negative, which is consistent with empirical observations (Cont; 2001). This negative dependence between returns and volatility is known in the market as the ``leverage effect.''

The risk neutral dynamics is given in a similar way as in the Black-Scholes model. For the logarithm of the asset price process $ X_t=\ln \frac{S_t}{S_0}$ one obtains the equation:

$\displaystyle dX_t = \left(r - \frac{1}{2}v_t\right ) dt + \sqrt{v_t} dW_t^{(1)}.$    

The cf is given by:


$\displaystyle \phi_{X_t}(z)$ $\displaystyle =$ $\displaystyle \frac{\exp\{\frac{\kappa\theta t(\kappa -i\rho \sigma z)} {\sigma...
...\rho\sigma z}{\gamma}\sinh \frac{\gamma t}{2} )^\frac{2\kappa\theta}{\sigma^2}}$  
  $\displaystyle \cdot$ $\displaystyle \exp \left \{-\frac{(z^2+ iz)v_0}{\gamma \coth \frac{\gamma t}{2} + \kappa - i\rho \sigma z}
\right \},$ (8.9)

where $ \gamma= \sqrt{\sigma^2 (z^2+ iz) + (\kappa- i\rho\sigma z)^2}$, and $ x_0$ and $ v_0$ are the initial values for the log-price process and the volatility process, respectively.


8.2.3 Bates Model

The Merton and Heston approaches were combined by Bates (1996), who proposed a model with stochastic volatility and jumps:

$\displaystyle \frac{dS_t}{S_t}$ $\displaystyle =$ $\displaystyle r dt + \sqrt{v_t} dW_t^{(1)} + dZ_t,$ (8.10)
$\displaystyle dv_t$ $\displaystyle =$ $\displaystyle \kappa (\theta - v_t) dt + \sigma \sqrt{v_t} dW_t^{(2)},$  
$\displaystyle \mathop{\hbox{Cov}}(dW_t^{(1)},dW_t^{(2)})$ $\displaystyle =$ $\displaystyle \rho\,dt.$  

As in (8.3) $ Z_t$ is a compound Poisson process with intensity $ \lambda $ and log-normal distribution of jump sizes independent of $ W_t^{(1)}$ (and $ W_t^{(2)}$). If $ J$ denotes the jump size then $ \ln(1+J)\sim
N(\ln(1+\overline{k})-\frac{1}{2}\delta^2, \delta^2 )$ for some $ \bar{k}$. Under the risk neutral probability one obtains the equation for the logarithm of the asset price:

$\displaystyle dX_t = (r - \lambda \overline{k}- \frac{1}{2}v_t )dt + \sqrt{v_t} dW_t^{(1)} + \tilde{Z}_t,$    

where $ \tilde{Z}_t$ is a compound Poisson process with normal distribution of jump magnitudes.

Since the jumps are independent of the diffusion part in (8.10), the characteristic function for the log-price process can be obtained as:

$\displaystyle \phi_{X_t}(z)=\phi_{X_t}^D(z)\phi_{X_t}^J(z),$    

where:


$\displaystyle \phi_{X_t}^D(z)$ $\displaystyle =$ $\displaystyle \frac{\exp \left \{\frac{\kappa\theta t(\kappa - i\rho \sigma z)}...
...gma z}{\gamma}\sinh \frac{\gamma t}{2} \right )^\frac{2\kappa\theta}{\sigma^2}}$  
  $\displaystyle \cdot$ $\displaystyle \exp \left \{-\frac{(z^2+ iz)v_0}{\gamma \coth \frac{\gamma t}{2} + \kappa - i\rho \sigma z} \right \}$ (8.11)

is the diffusion part cf and

$\displaystyle \phi_{X_t}^J(z)= \exp \{t\lambda (e^{-\delta^2 z^2/2 + i(\ln(1+\overline{k})-\frac{1}{2}\delta^2) z}-1) \},$ (8.12)

is the jump part cf. Note that (8.9) and (8.11) are very similar. The difference lies in the shift $ \lambda \overline{k}$ (risk neutral correction). Formula (8.12) has a similar structure as the jump part in (8.4), however, $ \mu$ is substituted with $ \ln(1+\overline{k})-\frac{1}{2}\delta^2$.