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in T 1 year with terminal payoff g(ST), where the function 0 if S < 10; s 10 if 10 < s < 15; 2s25 if 15 < s. This problem guides you to calculate the time-0 price of this derivative securities. (a) (4 points) Simulate 100, 000 independent paths of the pseudo-price process ST under (ISt rStdt + agtdBt, (b) (3 points) Calculate the time-0 price of this derivative securities using the risk-neutral pricing formula erTE [g(ST)] (c) (3 points) Consider a different derivative securities that matures in T different terminal payoff (ST), where the function 1 year with a 0 if S < 8; 2 -IS-IOI if 8 < S s 12; 0 if 12 < S. Calculate the time-0 price of this new derivative securities. (Note, I I is the absolute value function.) (d) (3 points) Now change the volatility a to 80%, while keeping other parameters con- stant. Calculate the time-0 price of these two derivative securities again using this new volatility parameter a 80%. (e) (3 points) Compare your answer of the time-0 price of these two derivative securities under the two assumptions a 40% and a 80%. Explain intuitively why when the volatility a is higher, the price of the first derivative securities increases and the price of the second derivative securities decreases.