Stochastic Calculus and the Black-Scholes PDE, the delta-neutral and gamma-neutral hedging strategy We consider the holder...

 

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Stochastic Calculus and the Black-Scholes PDE, the delta-neutral and gamma-neutral hedging strategy We consider the holder of a European call with strike K> 0 and maturity date T > 0, who needs to hedge the long position. The agent is able to invest in a money market account that pays the rate r, the underlying asset with price per share S(t), at time t, and in a European call with a different strike price K' K and the same maturity date T. To this end, we are going to use the PDE approach seen in class with the goal of replicating the European call with strike K with the hedging portfolio. We denote by c(t, s) the price, at time t, of the European call with strike K, when the current price of the underlying asset is s, by c'(t, s), the price, at time t, of the European call with strike K', and by X(t), the value, at time t, of the hedging portfolio. Finally, we denote by a, the number of shares of underlying asset held in the hedging portfolio and by the number of European calls with strike K' held in the portfolio. Finally, note that we are not specifying the model for the dynamics of S(t). 1. Assuming the linearity of all the portfolios, write the dynamics of the self-financing hedging portfolio invested in the money market account, the underlying asset and the European call with strike K'. 2. 3. 4. 5. 6. 7. Apply Ito's formula to differentiate c(t, S(t)) and c'(t, S(t)). Deduce the differential of the profit and loss c(t, S(t)) - X(t) Determine the value of 3 that cancels the second derivative terms. Determine the value of a that cancels the Brownian motion terms. Finally, we consider the portfolio which is long the call with strike K and short the hedge. Its value, at time t, is given by H(t, s) = c(t, s) -as - Be(t, s). Compute its delta, d, and gamma H by differentiating H twice with respect to s. Interpret the above result. Stochastic Calculus and the Black-Scholes PDE, the delta-neutral and gamma-neutral hedging strategy We consider the holder of a European call with strike K> 0 and maturity date T > 0, who needs to hedge the long position. The agent is able to invest in a money market account that pays the rate r, the underlying asset with price per share S(t), at time t, and in a European call with a different strike price K' K and the same maturity date T. To this end, we are going to use the PDE approach seen in class with the goal of replicating the European call with strike K with the hedging portfolio. We denote by c(t, s) the price, at time t, of the European call with strike K, when the current price of the underlying asset is s, by c'(t, s), the price, at time t, of the European call with strike K', and by X(t), the value, at time t, of the hedging portfolio. Finally, we denote by a, the number of shares of underlying asset held in the hedging portfolio and by the number of European calls with strike K' held in the portfolio. Finally, note that we are not specifying the model for the dynamics of S(t). 1. Assuming the linearity of all the portfolios, write the dynamics of the self-financing hedging portfolio invested in the money market account, the underlying asset and the European call with strike K'. 2. 3. 4. 5. 6. 7. Apply Ito's formula to differentiate c(t, S(t)) and c'(t, S(t)). Deduce the differential of the profit and loss c(t, S(t)) - X(t) Determine the value of 3 that cancels the second derivative terms. Determine the value of a that cancels the Brownian motion terms. Finally, we consider the portfolio which is long the call with strike K and short the hedge. Its value, at time t, is given by H(t, s) = c(t, s) -as - Be(t, s). Compute its delta, d, and gamma H by differentiating H twice with respect to s. Interpret the above result.

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1 Assuming the linearity of all the portfolios write the dynamics of the selffinancing hedging portfolio invested in the money market account the unde... View the full answer