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%.