ACST888 Quantitative Asset and Liability Modelling 2

Question

The stock price of  rm XYZ is currently $50. Let ln St denote the log of stock price at time t, assumed to follow the stochastic di erential equation (SDE) of d ln St = 0:03dt + 0:2dWt

under the real-world probability measure P, where Wt is a standard Brownian motion. The constant continuous dividend yield of
rm XYZ is 1% per annum and the continuously com- pounded risk-free rate is 4% per annum.

(a) Using 1000 simulation scenarios and equal sub-intervals of
t = 1 12 year, estimate the

mean of S5 (i.e., the stock price at the end of 5 years from now) for  rm XYZ. Compare the estimated value with the expected value of S5 under measure P.

(b) Suppose that you want to estimate the current value of a European cash-or-nothing put option written on stock XYZ with a time to expiry of 5 years from now and a strike price of EP (S5) from part (a). This binary option pays $50 when it is exercised.

(i) Using the same set of random numbers in the 1000 simulation scenarios as used previously, perform the necessary simulations to estimate the current price of the binary option.

(ii) Following from part (i), estimate the risk-neutral probability of exercising the binary option at expiry.

(iii) Calculate the theoretical price for the binary option and also the corresponding risk- neutral probability of exercising the option at expiry. Compare these theoretical values with the estimated values in parts (i) and (ii).

(iv) Suppose that the estimated price in part (i) is the market observed price for the

binary option, determine the implied volatility of stock XYZ and compare the answer with the assumed value given above.

(v) Estimate the delta of the binary option by performing simulations using the same set of random numbers in part as used previously and relevant parameters except

that the current stock price is changed slightly to $50:10. Compare your answers to the theoretical value of the binary option's delta.

(c) Suppose that you want to construct a portfolio consisting of the cash-or-nothing put option and the underlying stock XYZ.

(i) Determine the composition of this portfolio such that it is delta-neutral at time 0  t < T.

(ii) Consider a scenario where the current (i.e., at time 0) stock price has jumped (you may treat this as a sudden change) from $50 to $50.10. Using the estimated delta 2

in part (b) (v), calculate the value of the portfolio before and after the jump and show that the portfolio is indeed delta neutral at time 0.

(d) Consider the numerical holdings of the portfolio at time 0 as used in part (c) (ii). Assume that the current stock price is $50 (not $50.10 as described in part (c) (ii)) and the value of the cash-or-nothing put option exactly equals to the theoretical price at all times.

Using the 1000 simulation scenarios as used previously, calculate the value of this delta-neutral portfolio shortly before time t = 1 12 under each scenario.

Are the values of the delta-neutral portfolio the same for all scenarios? Explain your Fndings.