ACST3007 : Quantitative Asset and Liability Modelling - Real-World Probability Measure - Accounting and Finance Assignment Help

Assignment Task :

Question 

The stock price of firm XYZ is currently $50. Let us denote St as the stock price at time t, assumed to follow a stochastic differential equation (SDE) of dSt = 0.04Stdt + 0.12StdWt under the real-world probability measure P, where Wtis a standard Brownian motion. The constant continuous dividend yield of firm XYZ is 1% per annum and the continuously com pounded risk-free rate is 2.5% per annum. 

(a) Using 1000 simulation scenarios and equal sub-intervals of ?t =12year, estimate the mean of S5 (i.e., the stock price at the end of 5 years from now) for firm XYZ by discretizing the above SDE. Compare the estimated value with the theoretical expected value of S5 under measure P. 

(b) Suppose that you want to estimate the current value of the following pair of binary options: 

a European cash-or-nothing put option which pays a fixed amount of $1 a European asset-or-nothing put option both of which are written on stock XYZ with a time to expiry of 5 years and a strike price of EP(S5) from part (a). 

(i) Using the same set of random numbers in the 1000 simulation scenarios as used previously, perform the necessary simulations to estimate the current value of these two binary options. Describe and explain the procedures clearly 

(ii) Following from part (i), estimate (1) the current value of the cash-or-nothing put option and (2) the risk-neutral probability of exercising the asset-or-nothing put option at expiry, and compare these estimated values with the theoretical values under the Black-Scholes framework. Explain.  

(iii) Following from part (i), estimate the corresponding current value of a cash-or nothing call option with identical features as the above cash-or-nothing put option. Explain.

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