In the Black-Scholes world, every derivative can be replicated by a dynamic hedge in the stock and bond — the market is complete, and the unique replicating strategy pins down a unique no-arbitrage price. Add jumps to the asset price process, and this completeness breaks immediately: the stock and bond cannot replicate a derivative whose payoff depends on a random jump, because the jump size is a second source of randomness that cannot be traded away with the stock alone.
In an incomplete market, the no-arbitrage principle constrains prices to a range (no single point), and any price within that range is consistent with no-arbitrage — but different risk preferences (different choices of risk premium for the unhedgeable jump risk) correspond to different prices. This topic develops the theory and practice of pricing and hedging when replication is impossible.
By the end of this topic you should be able to:
- explain why jump processes create market incompleteness and what "no-arbitrage range" means in this context;
- compute the Esscher transform as a canonical measure change for Lévy processes;
- state the mean-variance hedging problem and contrast it with perfect replication;
- set up and discretise the PIDE for European and barrier option prices under jump dynamics;
- describe the entropy-regularised calibration problem for the Lévy measure.
In this lesson
- Contents
- The big picture
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