Smile-Consistent Spread Skew

We study the shape of the Bachelier-implied volatility of a spread option on two assets following correlated local volatility models. This includes the limiting case of spread options on two correlated Black–Scholes (BS) assets. We give an analytical result for the at-the-money (ATM) skew of the spread-implied volatility, which depends only on the components’ ATM volatilities and skews. We also compute the ATM convexity of the implied spread option for the case when the assets follow correlated BS models. The results are extracted from the short-maturity asymptotics for basket options obtained previously by Avellaneda, Boyer-Olson, Busca and Friz and, thus, become exact in the short-maturity limit. Numerical testing of the short-maturity analytical results under the Black–Scholes model and in a local volatility model show good agreement for strikes sufficiently close to the ATM point. Numerical experiments suggest that a linear approximation for the spread Bachelier volatility constructed from the ATM spread volatility and skew gives a good approximation for the spread volatility for highly correlated assets.