Question 3. a. Let 9t be a deterministic function of t, i.e. g : [0, T] → R, satisfying the integrability condition f (9+)² dt <∞ Show that the Itô integral f9dW₁ is normally distributed with mean zero and variance ſ (9+)² dt Hint: Consider a partition of [0, T] having n sub-interval and write the Itô integral f9dWt as a random sum. Then, consider taking limit of the random sum as n → ∞0. b. Let YT be an Fr-m measurable random variable. Suppose that under the risk-neutral probability measure, the random variable YŢ is normally distributed with mean μ and variance o². We consider a European call option whose payoff at time T is given by (YT-K), where K> 0 is the strike. Let Co be the time-0 price of this option. Furthermore, let r be the constant risk-free interest rate. Use the risk-neutral pricing approach to show that Co is given by Co = σ -TT-(K-μ)²/20² ·e √2π + eπT (μ- K) N ( " - K) -T σ
Question 3. a. Let 9t be a deterministic function of t, i.e. g : [0, T] → R, satisfying the integrability condition f (9+)² dt <∞ Show that the Itô integral f9dW₁ is normally distributed with mean zero and variance ſ (9+)² dt Hint: Consider a partition of [0, T] having n sub-interval and write the Itô integral f9dWt as a random sum. Then, consider taking limit of the random sum as n → ∞0. b. Let YT be an Fr-m measurable random variable. Suppose that under the risk-neutral probability measure, the random variable YŢ is normally distributed with mean μ and variance o². We consider a European call option whose payoff at time T is given by (YT-K), where K> 0 is the strike. Let Co be the time-0 price of this option. Furthermore, let r be the constant risk-free interest rate. Use the risk-neutral pricing approach to show that Co is given by Co = σ -TT-(K-μ)²/20² ·e √2π + eπT (μ- K) N ( " - K) -T σ
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![Question 3.
a. Let tgt be a deterministic function of t, i.e. g : [0, T] → R, satisfying the integrability condition f (gt)²dt <∞
Show that the Itô integral f9dWt is normally distributed with mean zero and variance f (9+)² dt
Hint: Consider a partition of [0, T] having n sub-interval and write the Itô integral f g+dWt as a random sum. Then,
consider taking limit of the random sum as n → ∞.
b. Let YT be an FT-m measurable random variable. Suppose that under the risk-neutral probability measure, the random
variable YT is normally distributed with mean μ and variance o².
We consider a European call option whose payoff at time T is given by (Yr – K), where K > 0 is the strike. Let Co be
the time-0 price of this option. Furthermore, let r be the constant risk-free interest rate.
Use the risk-neutral pricing approach to show that Co is given by
σ
Co
2πT
=
-rT-(K-μ)2/20²
()N (μ-K)
+ e-T (μ-K)N](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F882ce437-6984-48a2-b1c1-0c70a26630a5%2Fadced48a-d2c4-4ea2-a0cb-4916e966c57c%2Fbparg5z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 3.
a. Let tgt be a deterministic function of t, i.e. g : [0, T] → R, satisfying the integrability condition f (gt)²dt <∞
Show that the Itô integral f9dWt is normally distributed with mean zero and variance f (9+)² dt
Hint: Consider a partition of [0, T] having n sub-interval and write the Itô integral f g+dWt as a random sum. Then,
consider taking limit of the random sum as n → ∞.
b. Let YT be an FT-m measurable random variable. Suppose that under the risk-neutral probability measure, the random
variable YT is normally distributed with mean μ and variance o².
We consider a European call option whose payoff at time T is given by (Yr – K), where K > 0 is the strike. Let Co be
the time-0 price of this option. Furthermore, let r be the constant risk-free interest rate.
Use the risk-neutral pricing approach to show that Co is given by
σ
Co
2πT
=
-rT-(K-μ)2/20²
()N (μ-K)
+ e-T (μ-K)N
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