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Individual Question for Student with k-number k23019917

Suppose that a stock price St evolves according to the stochastic differential equation

dSt = µSt dt + σ(t)St dWt. (1) Here S0 = 150.00, µ = 0.10 is a constant, σ(t) is given by the deterministic function

σ(t) = {0(0).2(.1)4(7) + 0.07 × 0.60(t)

t < 0.60 otherwise ,

and Wt is a Brownian motion. Suppose that there is also a risk-free asset which grows at the continuously compounded rate r = 0.02.

If σ was constant, a call option with strike K and maturity T at a time

0 < t < T would have its price given by the Black–Scholes formula

price = BScall (St ,K, T − t,σ, r).

We will similarly write BSput (St ,K, T − t,σ, r) for the Black–Scholes formula for put options.

1. The price of an option at time t in this time-dependent volatility model is given by

BScall (St ,K, T − t,st , r)

where

(st )2 = T σ(t)2 dt

Show numerically using a simulation that it is possible to replicate a call option with strike K = 201.00 and maturity T = 1.70 by charging this price and then following the delta-hedging strategy.

Make sure that you describe how you simulated the stock prices, and give a clear mathematical description of how you performed the simulation of delta-hedging. Give the price, P0 of the option at time 0 in your table of results.

2. Show that the delta-hedging strategy does not work if the trader is mis- taken about the volatility. To do this, suppose a trader believes the stock has constant volatility σ(T), where T is the maturity, and decides to fol- low the delta-hedging strategy with all prices and deltas computed on this basis. However, suppose that in reality the stock price process is still gen- erated according to the equation (1). Calculate numerically the expected profit (which may be negative), p∆ , of the trader if:

(i) they sell the option at time 0 for the price BScall (S0 ,K,T,σ(T), r);

(ii) they then follow the delta-hedging strategy assuming constant volatil-

(iii) they then buy back the option for the price BScall and calculate their profit.

Use 1000 simulations and 1000 time-steps to compute this value. Give the 95% confidence interval for p∆ in your table of results.

3. Suppose it is possible at all times t, 0 ≤ t < T, to buy or sell a put option with strike KH = S0 for a price BSput (St ,KH , T − t,σ(T), r). In the gamma-hedging strategy the trader assumes that the volatility is constant and equal to σ(T). At all times they hold qS units of stock and qH units of the hedging option with strike KH in such a way as to ensure that the total delta of their portfolio is equal to the delta of the option with strike K and the total gamma of their portfolio is equal to the gamma of the option with strike K. Any remaining funds (or debts) grow at the risk-free rate.

Show through a simulation that, if the stock price is generated using a model where volatility is constant and equal to σ(T), then a trader can charge BScall (S0 ,K,T,σ(T), r) at time 0, and can then guarantee to have a portfolio with market value BScall - lowing the gamma-hedging strategy.

Since continuous time trading is not possible, perfect replication using the delta-hedging strategy cannot be achieved in practice. Show using appropriate charts that one advantage of the gamma-hedging strategy is that with gamma-hedging you do not need to rehedge so often to replicate the price BScall

(The reason you are only being asked to consider hedging up to time T/2 is to ensure that the gamma remains bounded throughout your simulation. One difficulty with gamma hedging is that the gamma can become very large if the stock price is near the strike price close to maturity and the question is designed to avoid this issue.)

4. Another feature of the gamma-hedging strategy is that so long as the price of the hedging option with strike KH is given by BSput (St ,KH , T − t,σ(T), r) it is possible to replicate the payoff BScall (ST/2,K, T/2,σ(T), r) price is generated using equation (1). Calculate the expected profit pΓ of following the gamma hedging strategy in this case using 1000 scenarios and 1000 time steps. Give a 95% confidence interval for your result.