Monte Carlo and PDE Approaches to Pricing Under State-Dependent Volatility

Summary

The document poses an interview problem for estimating the expectation of a payoff at a future time when the underlying follows a diffusion whose volatility depends on its current value. It asks for two approximate algorithms: a stochastic simulation and a partial differential equation method, with an implementation in C++ or Java.

The proposed response sketches a simulation by generating underlying paths and evaluating volatility along them, then suggests assuming a distribution for the payoff and using a moment-generating function. It does not provide the requested PDE approach, code, or a complete Monte Carlo estimator. The distributional assumption is not justified for arbitrary volatility and payoff functions, so the response is an incomplete starting point rather than a reliable pricing recipe.

Key ideas

• The target quantity is the expected payoff at the terminal time under a diffusion with state-dependent volatility.
• A stochastic algorithm can approximate the expectation by simulating paths and averaging terminal payoffs.
• A PDE approach is requested, but the document does not describe one.
• Assuming a normal payoff distribution requires justification and may not apply to arbitrary payoff functions.

Tags

# Stochastic Vol simulation - Quant job interview question


# Stochastic Vol simulation - Quant job interview question












this is a question from a quant interview (FO quant for IR Exotics for a big 4). First it might be useful when preparing your interviews, second, any brainstorming will be appreciated. Note that no more information other than the below was given.

Let us consider the following SDE:

$dS_t = \sigma(S_t) dW_t$

$S_0 = S(0)$

We are after $P = E(phi(S_1))$, the expectation of $phi(S_1)$.

Sigma and phi are some arbitrary functions, W is a Brownian motion.

Describe two algorithms that would (approximatively) compute P. One algorithm is a stochastic approach, the other one is a partial differential equation approach.

Code it (in C++ or Java).

- My answer:

1.Simulate the underlying $S_i$

2.Get the $\sigma_i$ using some assumption on the functional form of $\sigma(S_t)$

3.Assume some distribution for the function PHI (for example that is normal) and get the price using the Moment Generating Function of the Normal -i.e. $E(e^X) = e^{\mu+\frac{1}{2}\sigma}$

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