Jumps: Poisson and Merton

When prices jump

Brownian motion moves in small steps, but real prices sometimes jump: an earnings surprise, a central bank decision, a crash. A Poisson process counts such events. It has a rate $\lambda$, the average number of jumps per year, and the chance of exactly $k$ jumps in time $t$ is $e^{-\lambda t}(\lambda t)^{k}/k!$. In the Merton model the stock follows the usual diffusion and, at each jump time, is multiplied by a random factor $J$. Here $k=E[J-1]$ is the average jump size, and the term $-\lambda k$ takes back the drift that jumps add on average, so the expected return stays $\mu$.

Jumps change what a model can do. Returns get fat tails: most days look like the diffusion, but a few are far bigger. The variance has two sources, the diffusion part $\sigma^{2}T$ and the jump part $\lambda T(m^{2}+\delta^{2})$. And hedging becomes harder: you cannot follow a sudden jump with small trades in the stock, so an option's price depends on your view of the jump risk and not only on $\sigma$.

• Poisson process: counts jumps at rate $\lambda$. The expected count in time $t$ is $\lambda t$.
• Merton model: diffusion plus jumps of random size, with a drift correction $-\lambda k$.
• Fat tails: jumps add rare big moves, and a jump cannot be fully hedged.

Lab · what's graded

• No jump at all15 pts
• The average jump15 pts
• Variance with jumps15 pts
• What jumps add15 pts
• Match the symbols15 pts
• Simulating a jump path10 pts
• Why jumps are hard to hedge15 pts