Consider an asset with a constant dividend yield q. Assume the price S of the asset satisfies
Question:
Consider an asset with a constant dividend yield q. Assume the price S of the asset satisfies dS S = (μ −q)dt +σ dB, where B is a Brownian motion under the physical probability, and μ and σ are constants. Consider a European call and a European put with strike K on the asset. Assume the risk-free rate is constant, and adopt the assumptions of Section 16.2.
(a) Let A denote the event ST > K. Show that E[ST1A] = e(μ−q)TS0 N(d∗
1), where d∗
1 = log(S0/K) +
μ −q + 1 2σ2
T
σ
√T .
Hint: This can be computed directly under the physical probability or by changing probabilities using e(q−μ)TST/S0 as the Radon-Nikodym derivative.
(b) Show that E[K1A] = K N(d∗
2), where d∗
2 = d∗
1 −σ
√T.
(c) It follows from the previous parts that the expected return of the European call under the physical probability, if held to maturity, is e(μ−q)TS0 N(d∗
1)− K N(d∗
2)
e−qTS0 N(d1) − e−rTK N(d2)
, where d1 and d2 are defined in (16.28). Assuming T = 1, μ = 0.12, r = 0.04, q = 0.02, and σ = 0.20, show that the expected rate of return of a European call that is 20% out of the money (S0/K = 0.8)
exceeds 100%.
(d) Show that the expected return of the European put under the physical probability, if held to maturity, is K N(−d∗
2)− e(μ−q)TS0 N(−d∗
1)
e−rTK N(−d2)− e−qTS0 N(−d1)
.
Assuming T = 1, μ = 0.12, r = 0.04, q = 0.02, and σ = 0.20, show that the expected rate of return of a European put that is 20% out of the money (S0/K = 1.2) is less than −50%.
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