American and European options in multi-factor jump-diffusion models, near expiry

Abstract

We derive a general formula for the time decay θ for out-of-the-money European options on stocks and bonds at expiry, in terms of the density of jumps F(x,dy) and the payoff g+: −θ(x)=∫g(x+y)+F(x,dy). Explicit formulas are derived for the standard put and call options, exchange options in stochastic volatility and local volatility models, and options on bonds in ATSMs. Using these formulas, we show that in the presence of jumps, the limit of the no-exercise region for the American option with the payoff (−g)+ as time to expiry τ tends to 0 may be larger than in the pure Gaussian case. In particular, for many families of non-Gaussian processes used in empirical studies of financial markets, the early exercise boundary for the American put without dividends is separated from the strike price by a nonvanishing margin on the interval [0,T), where T is the maturity date.