The legacy of Dupire
There is an old tradition in applied mathematics of using changes of variable to gain insights into difficult problems. This was the key technical step behind the breakthrough made by Bruno Dupire in understanding the smile, published in Risk in 1994 (available today as part of the Risk Books compilation, Over the rainbow).
The story is well known. Traditional Black-Scholes theory takes as its dependent variable the price of an option of fixed strike and maturity, and leads to a partial differential equation where the independent variables are the underlying asset price and time. Searching for a deterministic volatility process that fitted observed market smiles, Dupire discovered a new equation for the option price – the Dupire equation – with strike and maturity as independent variables, and with time and asset
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