- Following questions are answered on the case study -
a) Plot the payoff in USD of this structured product as a function of the stock price S(T) at maturity. Consider values for S(T) in the range from 0.00 USD to 200.00 USD.
b) Give a formula for the payoff of the airbag certificate as a function of the stock price S(T) at maturity.
c) The payoff of the airbag certificate can be replicated by taking positions in zero-coupon bonds and/or the underlying asset and/or various options. Propose one such portfolio and show that the payoff function of this portfolio is equal to the payoff function of the airbag certificate.
d) Propose a second portfolio that replicates the payoff of the airbag certificate and again show that its payoff function is equal to that of the airbag certificate.
e) Describe the market view that an investors who buys this structured product today and plans to hold it until the maturity date should have. I.e. what is her expectation for the stock price S(T) at maturity? You need to give details and justify your answer.
f) Assume that all Cox-Ross-Rubinstein assumptions holds. The volatility of FINS3635 Stu- dent Corp. is 20% p.a., the continuously compounded risk-free interest rate is 10% and there are no dividends. Compute the price of one airbag certificate in a two-step Cox-Ross- Rubinstein binomial tree.
Hint: Note that this will only be a very crude approximation to the Black-Scholes price that you compute in part g).
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g) Assume that all Black-Scholes assumptions hold and use the market data from part f). Com- pute the (i) price, (ii) delta, (iii) gamma, (iv) vega and (v) theta of one airbag certificate using a replicating portfolio that contains two call options with different strike prices.
h) How would the price of the airbag certificate change if FINS3635 Student Corp. announced that it would pay a (non-zero) dividend in half a year? Argue without computations.
Hint: As usual, when analyzing sensitivities we make a ceteris paribus assumption. In par- ticular, the current stock price is assumed not to change due to the dividend announcement.
i) Consider the same situation as in part g) but now assume that the stock price instanta- neously jumps to S(0) = 105.00 USD. (i) Explicitly compute the new Black-Scholes price of the airbag certificate and compare it to the result in part g). Explain (ii) the sign of the change and (iii) the magnitude of the change. (iv) Approximate the price change of the airbag certificate using its Black-Scholes delta from part g) and explain why the approxi- mation over-/under-estimates the actual price change.
j) Consider the same situation as in part g) but now assume that the volatility instantaneously jumps to 25% p.a.. (i) Compute the new Black-Scholes price of the airbag certificate and compare it to the result in part g). (ii) Explain the sign of the change.
