Exercise 5.4 (Black-Scholes-Merton formula for time-varying, non- random interest rate and volatility). Consider a stock whose price dif- ferential is dS(t) = r(t)s(t) dt + o(t) dŵ (t), where r(t) and r(t) are nonrandom functions of t and W is a Brownian motion under the risk-neutral measure P. Let T> 0 be given, and consider a European call, whose value at time zero is E [exp{- √² r(1) dt} (S(T) — K)+] · c(0, S(0)) = E (i) Show that S(T) is of the form S(0)ex, where X is a normal random variable, and determine the mean and variance of X. (ii) Let BSM(T, T, K, R, E) = IN I K (5√/7 [108 1/2 + (R + log ² + E²/2)T]) (7 [108 / +(R - 2²/2)T]) √T -e -RT KN denote the value at time zero of a European call expiring at time T when the underlying stock has constant volatility and the interest rate Ris constant. Show that c(0, S(0)) = BSM S(0), T, 1 T₁ = = [/ ²^r (1) d²₁ √/ / / "²0¹² (1) ² r(t)dt, (t)dt
Exercise 5.4 (Black-Scholes-Merton formula for time-varying, non- random interest rate and volatility). Consider a stock whose price dif- ferential is dS(t) = r(t)s(t) dt + o(t) dŵ (t), where r(t) and r(t) are nonrandom functions of t and W is a Brownian motion under the risk-neutral measure P. Let T> 0 be given, and consider a European call, whose value at time zero is E [exp{- √² r(1) dt} (S(T) — K)+] · c(0, S(0)) = E (i) Show that S(T) is of the form S(0)ex, where X is a normal random variable, and determine the mean and variance of X. (ii) Let BSM(T, T, K, R, E) = IN I K (5√/7 [108 1/2 + (R + log ² + E²/2)T]) (7 [108 / +(R - 2²/2)T]) √T -e -RT KN denote the value at time zero of a European call expiring at time T when the underlying stock has constant volatility and the interest rate Ris constant. Show that c(0, S(0)) = BSM S(0), T, 1 T₁ = = [/ ²^r (1) d²₁ √/ / / "²0¹² (1) ² r(t)dt, (t)dt
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 5CR
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Math Finance
![Exercise 5.4 (Black-Scholes-Merton formula for time-varying, non-
random interest rate and volatility). Consider a stock whose price dif-
ferential is
dS(t) = r(t)S(t) dt + o(t) dŵ(t),
where r(t) and r(t) are nonrandom functions of t and W is a Brownian motion
under the risk-neutral measure P. Let T> 0 be given, and consider a European
call, whose value at time zero is
E [exp{- [² r(t)dt} (5(T) - K)+].
c(0, S(0)) = E
(i) Show that S(T) is of the form S(0)ex, where X is a normal random
variable, and determine the mean and variance of X.
(ii) Let
BSM(T, z; K, R, E) = IN
(5√/T [108 / 7 + (R +
K
-e
-RT KN
² + E²/2)T])
(7 [108 7/2 + (R - 2²/2)T])
√T
denote the value at time zero of a European call expiring at time T when
the underlying stock has constant volatility and the interest rate Ris
constant. Show that
c(0, S(0)) = BSM S(0), T,
1 T₁ = / ²^r(t) d²₁ √/ / / ²^ 0² (1) ²
r(t)dt,
(t)dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffc34a14-336d-4879-a115-3e436a77b08b%2Fd2b5193b-fc21-41d6-8835-95c7a9861383%2F42qgnyh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 5.4 (Black-Scholes-Merton formula for time-varying, non-
random interest rate and volatility). Consider a stock whose price dif-
ferential is
dS(t) = r(t)S(t) dt + o(t) dŵ(t),
where r(t) and r(t) are nonrandom functions of t and W is a Brownian motion
under the risk-neutral measure P. Let T> 0 be given, and consider a European
call, whose value at time zero is
E [exp{- [² r(t)dt} (5(T) - K)+].
c(0, S(0)) = E
(i) Show that S(T) is of the form S(0)ex, where X is a normal random
variable, and determine the mean and variance of X.
(ii) Let
BSM(T, z; K, R, E) = IN
(5√/T [108 / 7 + (R +
K
-e
-RT KN
² + E²/2)T])
(7 [108 7/2 + (R - 2²/2)T])
√T
denote the value at time zero of a European call expiring at time T when
the underlying stock has constant volatility and the interest rate Ris
constant. Show that
c(0, S(0)) = BSM S(0), T,
1 T₁ = / ²^r(t) d²₁ √/ / / ²^ 0² (1) ²
r(t)dt,
(t)dt
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