Jump-Diffusion Option (Merton)

The night before an earnings release, a trader on the options book knows the stock can gap open — and that plain Black-Scholes, with its smooth Brownian path, prices that gap at zero. This calculator values a European call under Merton's 1976 jump-diffusion model, which layers random jumps on top of the usual diffusion. It is what you reach for when the smile steepens and the continuous model leaves the wings too cheap: scheduled announcements, rate decisions, credit shocks. The premium it returns respects fat tails and discontinuous moves rather than pretending prices drift quietly.

The math rests on a clean identity. Conditional on exactly n jumps over the life of the option, the call is worth an ordinary Black-Scholes value with an adjusted variance and drift; the full price is the average of those values, weighted by the Poisson probability that n jumps arrive. Each jump is lognormal — you supply the mean and standard deviation of the jump's log — and the variance of the n-th term becomes σ² + nδ²/T. The intensity λ is the expected number of jumps per year. One assumption deserves a flag: Merton treats jump risk as diversifiable and therefore unpriced, and λ is held constant, so there is no regime-dependent hazard rate and no correlated, market-wide jump.

Enter the spot, strike, time to maturity in years, risk-free rate, and diffusion volatility; then give λ along with the mean μ and deviation δ of the log-jump. The output is the European call premium, in price units. To sanity-check, set λ to zero: with no jumps the sum collapses to its first term and you recover the exact Black-Scholes price — any gap there points to a bad input. Pushing μ away from zero or raising δ thickens the tails and, as a rule, makes the option dearer.