Question

Consider the case of Merton's jump-diffusion model where jumps always reduce the asset price to zero. Assume that the average number of jumps per year is $\lambda$. Show that the price of a European call option is the same as in a world with no jumps except that the risk-free rate is $r+\lambda$ rather than $r$. Does the possibility of jumps increase or reduce the value of the call option in this case? (Hint: Value the option assuming no jumps and assuming one or more jumps. The probability of no jumps in time $T$ is $e^{-\lambda T}$ ).

   Consider the case of Merton's jump-diffusion model where jumps always reduce the asset price to zero. Assume that the average number of jumps per year is $\lambda$. Show that the price of a European call option is the same as in a world with no jumps except that the risk-free rate is $r+\lambda$ rather than $r$. Does the possibility of jumps increase or reduce the value of the call option in this case? (Hint: Value the option assuming no jumps and assuming one or more jumps. The probability of no jumps in time $T$ is $e^{-\lambda T}$ ).

Options, Futures, and Other Derivatives

John C. Hull 10th Edition

Chapter 27, Problem 5

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Step 1

In this case, the price of a European call option can be calculated using the Black-Scholes formula: $C = S_0e^{-qT}N(d_1) - Xe^{-rT}N(d_2)$ where: - $C$ is the price of the call option - $S_0$ is the current price of the underlying asset - $q$ is the continuous Show more…

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Consider the case of Merton's jump-diffusion model where jumps always reduce the asset price to zero. Assume that the average number of jumps per year is $\lambda$. Show that the price of a European call option is the same as in a world with no jumps except that the risk-free rate is $r+\lambda$ rather than $r$. Does the possibility of jumps increase or reduce the value of the call option in this case? (Hint: Value the option assuming no jumps and assuming one or more jumps. The probability of no jumps in time $T$ is $e^{-\lambda T}$ ).

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