Jump Diffusion Option Pricing

This question focuses on extending the classic continuous-time option pricing setup to models where the underlying asset can exhibit sudden jumps. The candidate is asked to distinguish between pure jump processes and jump diffusion processes, and to reason about what these extra discontinuities imply for pricing and hedging derivative claims. It probes whether the Black-Scholes/Merton machinery can still be applied when the price path is no longer purely continuous, and what structural features of the model determine whether familiar risk-neutral valuation arguments remain valid.

To answer well, a candidate must connect stochastic process characteristics to market completeness, replication, and the existence and uniqueness of an equivalent martingale measure. It leans on concepts such as Brownian motion versus Poisson-driven jumps, additional sources of randomness, state-space dimensionality, and the role of tradable assets in spanning risks. Interviewers watch for clear articulation of when and why hedging breaks down, how many independent risk factors are present relative to the number of traded instruments, and whether the candidate can frame the conditions for no-arbitrage pricing in terms of replicating strategies rather than memorized formulas.

What it tests

The core structure underlying these problems is the relationship between market completeness and the ability to construct riskless hedges. In a complete market, every contingent claim can be perfectly replicated by trading in available assets, which is the foundation of the Black-Scholes/Merton no-arbitrage pricing. This completeness relies on the dynamics of the underlying asset being continuous and driven by Brownian motion, so that small changes in the asset price can be offset by adjusting the hedge. When jumps are introduced, especially with random sizes or additional sources of randomness, the market often becomes incomplete because not all risks can be hedged away using the available instruments. The principle is that the applicability of no-arbitrage pricing depends fundamentally on whether replication is possible, which is determined by the structure of the underlying stochastic processes.