Why the Adjusted Poisson Intensity Appears in Merton’s Jump-Diffusion Call Formula

Summary

This note asks how the adjusted Poisson parameter arises when a European call is priced under Merton’s jump-diffusion model. It starts from a valuation expressed as a sum over the possible number of jumps, with each term weighted by a Poisson probability and a conditional option payoff. It then presents a Black–Scholes mixture formula whose Poisson weights use an intensity adjusted by the average jump size.

The question focuses on deriving that adjusted intensity, while the supplied formula also shifts the rate and volatility for each jump count. The document contains no derivation or answer, so it does not establish the assumptions behind the parameterization or resolve notation such as the meaning of the average jump size. Its value is in identifying a key step in the mixture representation: conditioning on jump count changes the effective weights as well as the conditional Black–Scholes inputs. Readers should verify the convention for jump multipliers and risk-neutral drift before applying the displayed expression.

Key ideas

• The option value is represented as a sum of conditional values indexed by the number of jumps.
• The displayed mixture uses a Poisson intensity adjusted by the mean jump size.
• The conditional Black–Scholes inputs also vary with the number of jumps.
• The note poses the derivation question but supplies no solution, so parameter conventions require verification.

Tags

# Poisson parameter in Merton's Jump-Diffusion Model to price call option


# Poisson parameter in Merton's Jump-Diffusion Model to price call option












I've been taught the following European call valuation formula under jump-diffusion model: \begin{equation} price = E[e^{-rT}max(S_T-K,0)] =\sum_{j = 0}^\infty e^{-rT}P_j(\lambda)E[max(S_T-K,0)|J=j] \end{equation} where $J$ is the number of jumps, probability term $P$ is the poisson process: \begin{equation} e^{-\lambda T} \frac{(\lambda T)^j}{j!} \end{equation} And I've been given directly the final form of the valuation formula: \begin{equation} price = \sum_{j = 0}^\infty P_j(\lambda^{'})BS(S_0,r_j,\sigma_j^2) \end{equation} where: \begin{equation} \lambda^{'}=\lambda(1+k) \end{equation} $k$ is the average jump size, and: \begin{equation} r_j = r-\lambda k+j\ln(1+k) / T, \sigma_j^2=\sigma^2+j\sigma_s^2/T \end{equation} $\sigma^2 $ is original Black-Scholes volitility, $\sigma_s^2$ is the jump variance.

By expanding the Black-Scholes equation, I've managed to derive $r_j$ and $\sigma_j^2$, but I'm confused about where $\lambda^{'}$ comes from. Can someone help derive this term?

In case of anything unclear, the formula can also be found here and here (in the second link, $m=1+k$)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.