Then the estimate of the mean continuously compounded

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Unformatted text preview: e of return of this stock α is less than 3%. ˆ Solution: FALSE α = ln (114/110) ≈ 0.0357. ˆ 9. A caplet is a financial instrument used as protection against the increase in the interest rate for all repayment installments of a loan to be repaid over multiple periods. Solution: FALSE The caplet only applies to one installment; it’s the cap that provides protection over multiple periods. 3 10. Assume the Black-Scholes stock-pricing model is in force. Let E∗ denote the expectation under the risk-neutral probability measure P∗ . Let {S (t), t ≥ 0} denote the price of a continuous-dividend-paying stock. Then, in our usual notation, E∗ [S (T )] = S (0)e(r−δ)T . Solution: TRUE 11. In the Black-Derman-Toy model, the interest rate at any node is the geometric average of the rates at the two nodes at adjacent heights. Solution: TRUE 12. The Black-Scholes option pricing formula can always be used for pricing American-type call options on non-dividend-paying assets. Solution: TRUE Part II. Free-response problems Please, explain carefully all your statements and assumptions. Numerical results or single-word answers without an explanation (even if they’re correct) are worth 0 points. 1. (25 points) Let S (0) = $120, K = $100, σ = 0.3, r = 0 and δ = 0.08. a. (10 pts) Let VC (0, T ) denote the Black-Scholes European call price for the maturity T . Does the limit of VC (0, T ) as T → ∞ exist? If it does, what is it? b. (10 pts) Now, set r = 0.001 and let VC (0, T, r) denote the Black-Scholes European call price for the maturity T . Again, how does VC (0, T, r) behave as T → ∞? c. (5 pts) Interpret in a sentence or two the differences, if any, between your answers to questions in a. and b. Solution: a. By the Black-Scholes pricing formula, the function VC (0, T ) has the form VC (0, T ) = S (0)e−δT N (d1 ) − Ke−rT N (d2 ) = S (0)e−δT N (d1 ) − K N (d2 ), where N denotes the distribution function of the unit normal distribution and 1 S (0) d1 = √ ln K σT...
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