The Volatility Smile

The Black-Scholes-Merton option model was the greatest innovation of twentieth century finance, and remains the most widely applied theory in all of finance. Nevertheless, the model is fundamentally at odds with the observed behavior of option markets: a graph of implied volatility against strike will typically display a curve or smile, which the model cannot explain. Option valuation is not a solved problem, and the past forty years have witnessed an abundance of new ideas and models that try to reconcile theory with markets. Beginning with the principles of financial valuation, The Volatility Smile presents a unique and unified treatment of the Black-Scholes-Merton option model and the more advanced models that have replaced it. Celebrated author, quant, and co-originator of the local volatility model Emanuel Derman and Michael B. Miller explain not just the mathematics but the ideas behind the models. By examining the foundations, the implementation, and the pros and cons of various models, and by carefully exploring their derivations and the consequences of different assumptions, readers will learn not only how to handle the volatility smile but how to evaluate and build their own financial models. Key features: The principles of valuation The Black-Scholes-Merton model Hedging strategies and transaction costs The behavior of the volatility smile Static and dynamic replication of standard and exotic options New models: their origin, implementation, and consequences Local volatility Stochastic volatility Jump-diffusion