Calls on assets with different drifts or jumps
Question
Two stocks each follow Black-Scholes dynamics with the same volatility but different expected returns. (a) How do the prices of identical European calls on them compare? (b) Now suppose one stock additionally suffers random downward jumps. How does that change the comparison?
Short answer
(a) The calls cost the same; (b) the jumping stock's call is generally worth more. Under the risk-neutral measure both drifts are replaced by , so the real drift never enters. Jumps add risk that cannot be hedged away with the stock; with the jump compensated in the risk-neutral drift, the price becomes a Poisson-weighted mix of Black-Scholes prices, and by convexity of the call in spot this exceeds the no-jump price.
Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q2.4, p.16; question and answer rewritten by SelfMadeQuant.com.
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Open the worked solutionLessons behind this question
- The Black-Scholes PDE, Formula and Implied Volatility
- Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
- The Black-Scholes Formula Derived via Change of Measure
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