(a) Express the 10,000 simulated underlying price paths under risk-neutral measure, denoted by {S} for i=1,..., 10000, i.e. what is S? (b) What is the payoff function in terms of one

price path {S? Express it mathematically. (c) Express the simulated price of this product at the initial time with standard MC. (d) Express the simulated price of this product at the initial time with antithetic variates. (e) Express the simulated price of this product at the initial time with control vari- ates (e.g., using the final level of underlying asset as control). (f) Express the simulated price of this product at the initial time with empirical martingale correction. (g) Express the simulated Gamma of this product with finite-difference method./n3. A structured product (with one underlying asset) with the barrier event, defined as the underlying asset price once ≤ 16.4, has the following payoff: • If no barrier event occurred, Final Level of underlying asset 25.23 Payoff = 1000 x max (1.2, 1+ (1.2,1+! • If a barrier event occurred, Payoff = 39.6354x Final Level of underlying. Suppose that we model the underlying asset's price with GBM(u, o), where u and o are considered as known constants, and the initial price is So. In our simulations, we simulate the stock price paths with frequency At and the maturity of the product is T = mat. We also assume that the risk-free interest rate is r. Suppose also that we have already generated 10000m standard normal random variates, denoted by 2 for i=1,..., 10000 and j = 1,..., m. Using the notations above,