Exercises 2.9 P&L Theta-Gamma We study the variation of a trader's self-nancing portfolio composed of an asset with price St at time t and a European option with payo (ST ) at the maturity date T. The fair value of the option is C(t; St) at time t. The portfolio's value is at time t:

t = C(t; St) ????
tSt We take zero interest rate.

1. We assume that on the market the asset follows a Black-Scholes lognormal process with a volatility real. By using It^o's formula, prove that the innitesimal variation of the self-nancing delta-hedge portfolio between t and t + dt is dt = @tCdt +

1 2S2@2SC2 realdt The term @tC (resp. @2S C) is called the Theta (resp. the Gamma) of the option.

2. We assume that the option was priced using a Black-Scholes model with a volatility model. Prove that dt =

1 2S2@2S C

????

2 real ???? 2 model

dt 3. Deduce that the expectation value EP[
T ] of the variation of the portfolio between t = 0 to t = T is EP[
T ] =

1 2

Z T 0

EP[S2@2S C]

????

2 real ???? 2 model

dt This expression is called the prot and loss (P&L) Theta-Gamma. For a European call option, the Gamma is positive and therefore, if real >

model (resp. real < model) , the P&L is positive (resp. negative).

When the market realized volatility real equals the model volatility

model, the P&L vanishes. In this context, model is called the breakeven volatility.