6.1 Option Pricing

The option-pricing model by Black and Scholes (1973) is based on the assumptions that the underlying asset follows a geometric Brownian motion with a constant volatility $ \sigma$:

$\displaystyle \frac{dS_{t}}{S_{t}}$ $\displaystyle =$ $\displaystyle \mu dt + \sigma dW_{t},$ (6.1)

where $ S_t$ denotes the underlying-price process, $ \mu$ is the expected return, and $ W_t$ stands for the standard Wiener process. As a consequence, the distribution of $ S_t$ is lognormal. More importantly, the volatility $ \sigma$ is the only parameter of the Black-Scholes formula which is not explicitly observable on the market. Thus, we infer on $ \sigma$ by matching the observed option prices. A solution $ \sigma_{I}$, ``implied'' by options prices, is called the implied volatility (or Black-Scholes equivalent). In general, implied volatilities vary both with respect to the exercise price (the skew structure) and expiration time (the term structure). Both dependencies are illustrated in Figure 6.1, with the first one representing the volatility smile. Let us add that the implied volatility of an option is the market's estimate of the average future underlying volatility during the life of that option. We refer to the market's estimate of an underlying volatility at a particular time and price point as the local volatility.

Figure 6.2: Two levels of a CRR binomial tree.

\includegraphics[width=1.0\defpicwidth]{ITT_fig04-crr.ps}

Binomial trees, as a discretization of the Black-Scholes model, can be constructed in several alternative ways. Here we recall the classic Cox, Ross, and Rubinstein's (1979) scheme (CRR), which has a constant logarithmic spacing between nodes on the same level (this spacing represents the future price volatility). A standard CRR tree is depicted in Figure 6.2. Starting at a node $ S$, the price of an underlying asset can either increase to $ S_u$ with probability $ p$ or decrease to $ S_d$ with probability $ 1-p$:

$\displaystyle S_u$ $\displaystyle =$ $\displaystyle S e^{\sigma\sqrt{\Delta t}},$ (6.2)
$\displaystyle S_d$ $\displaystyle =$ $\displaystyle S e^{-\sigma\sqrt{\Delta t}},$ (6.3)
$\displaystyle p$ $\displaystyle =$ $\displaystyle \frac{F-S_d}{S_u-S_d},$ (6.4)

where $ \Delta t$ refers to the time step and $ \sigma$ is the (constant) volatility. The forward price $ F=e^{r\Delta t}S$ in the node $ S$ is determined by the the continuous interest rate $ r$ (for the sake of simplicity, we assume that the dividend yield equals zero; see Cox, Ross, and Rubinstein, 1979, for treatment of dividends).

A binomial tree corresponding to the risk-neutral underlying evaluation process is the same for all options on this asset, no matter what the strike price or time to expiration is. There are many extensions of the original Black-Scholes approach that try to capture the volatility variation and to price options consistently with the market prices (that is, to account for the volatility smile). Some extensions incorporate a stochastic volatility factor or discontinuous jumps in the underlying price; see for instance Franke, Härdle, and Hafner (2004) and Chapters 5 and 7. In the next section, we discuss an extension of the Black-Scholes model developed by Derman and Kani (1994) - the implied trees.