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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 BlackScholes stockpricing model is in force. Let E∗ denote the expectation
under the riskneutral probability measure P∗ . Let {S (t), t ≥ 0} denote the price of a
continuousdividendpaying stock. Then, in our usual notation,
E∗ [S (T )] = S (0)e(r−δ)T .
Solution: TRUE
11. In the BlackDermanToy 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 BlackScholes option pricing formula can always be used for pricing Americantype
call options on nondividendpaying assets.
Solution: TRUE
Part II. Freeresponse problems
Please, explain carefully all your statements and assumptions. Numerical results or
singleword 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 BlackScholes 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 BlackScholes 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 BlackScholes 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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 Fall '12
 CUDINA
 Math

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