Exotic Options and Complex Derivatives › Greeks for Exotic Options

Risk management for exotic options requires computing the Greeks, as with vanilla options. However, path dependence and discontinuities in exotics create additional complexity.

The simplest approach for computing Greeks uses finite differences. We bump inputs and reprice to estimate sensitivities:

$$\begin{aligned} \Delta &= \frac{V(S_0 + \epsilon) - V(S_0 - \epsilon)}{2\epsilon} \\ \Gamma &= \frac{V(S_0 + \epsilon) - 2V(S_0) + V(S_0 - \epsilon)}{\epsilon^2} \end{aligned}$$

where:

• $V(S)$: option price for spot price $S$
• $S_0$: current spot price
• $\epsilon$: small change in price (bump size)

The central difference formula for delta provides second-order accuracy, meaning the error decreases quadratically as the bump size shrinks. The choice of bump size involves a trade-off: too large a bump introduces truncation error from ignoring higher-order terms, while too small a bump amplifies numerical noise in the price calculation. For Monte Carlo prices with inherent simulation noise, bump sizes of around 1% of the spot price typically work well.

The gamma formula uses a symmetric second difference, which also achieves second-order accuracy. However, gamma is inherently more difficult to estimate than delta because it involves the second derivative of price with respect to spot. Small errors in the individual price calculations become magnified when taking differences of differences, making gamma estimates noisier than delta estimates for the same computational effort.

For Monte Carlo pricing, this approach requires new sets of random paths for each bump, which can be computationally expensive and introduces noise.

A more efficient approach reuses the same random numbers across bumped prices. For path-dependent options with smooth payoffs, we can compute delta as:

$$\Delta = e^{-rT} \mathbb{E}\left[\frac{\partial \text{Payoff}}{\partial S_0}\right]$$

where:

• $r$: risk-free rate
• $T$: time to expiration
• $\mathbb{E}[\cdot]$: expectation under the risk-neutral measure
• $\frac{\partial \text{Payoff}}{\partial S_0}$: sensitivity of the specific path's payoff to the initial price

This pathwise approach, sometimes called infinitesimal perturbation analysis, exploits the fact that we can often differentiate through the payoff function. For an Asian call with arithmetic average, the payoff is $\max(A - K, 0)$ where $A = (1/n)\sum S_{t_i}$. Each price $S_{t_i}$ on the path scales proportionally with the initial price $S_0$, so $\partial S_{t_i}/\partial S_0 = S_{t_i}/S_0$. When the option finishes in the money, the derivative of the payoff with respect to $S_0$ is simply $A/S_0$, the average price divided by the initial price.

The pathwise method fails when the payoff function is not differentiable, such as at the exercise boundary for options or at the barrier for barrier options. In these cases, the likelihood ratio method provides an alternative approach that differentiates the probability distribution rather than the payoff.