Gamma hedging and rough paths

Abstract We apply rough-path theory to study the discrete-time gamma-hedging strategy. We show that if a trader knows that the market prices of a set of European options are given by a diffusive pricing model, then the discrete-time gamma-hedging strategy enables them to replicate other European options so long as the underlying pricing signal has finite $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -variation for $p&lt;3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> </mml:math> , with the error in the discrete-time replication strategy tending to zero as the length of the largest hedging interval tends to zero. This is a sure result and does not require that the underlying pricing signal has a quadratic variation corresponding to a probabilistic pricing model. We show how to generalise this result to exotic derivatives when the gamma is defined to be the Gubinelli derivative of the delta by deriving rough-path versions of the Clark–Ocone formula. We illustrate our theory by proving that if a stock price path has finite $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -variation for $p&lt;3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> </mml:math> and if the implied volatility process for a European derivative on the stock (with a smooth, convex, nonlinear payoff and maturity $T$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> ) has finite $q$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -variation for $q&lt;2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:math> and $\frac{1}{p}+\frac{1}{q}&gt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>q</mml:mi> </mml:mfrac> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> , one can use the gamma-hedging strategy to replicate any European derivative with smooth payoff and maturity $T$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> . This is a sure result which holds without assuming any probabilistic model for the trajectory of the stock price path.