In this project you will investigate hedging in discrete time within the Black-Scholes model.

You are to assume that an asset S = (St)t0 follows the Black-Scholes model with S0 = 100,  = 20%,  = 10% and the risk-free rate is constant at 2%.
You have just sold an at-the-money 1 4 year put written on this asset and you wish to hedge it. Assume that, if needed, you may also trade in a call option (on the same asset) struck at K = 100 (maturity 12 year), the stock, and the bank account. As well, you will account for transaction costs by assuming you are charged 0:005\$ on every one unit of equity traded, and 0:01\$ on every unit of options traded.

[NOTE: the call that you trade when Gamma hedging has a FIXED maturity date, but that implies
its time to maturity keeps reducing as time ows forward { just like the put that you sold.]

1. Compare the move-based with the time-based hedging strategy with delta hedging. Assume a base band of 0:05.

2. Compare the move-based with the time-based hedging strategy with delta and gamma hedging.

3. What happens if the real-world P volatility is  = 15% but the risk-neutral Q volatility is  =20%?

4. Investigate the role that the rebalancing-band in  plays on the hedge.
Comment on any observations.