The Black-Scholes equation for the value of an option V(S,t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x,7) using S = Ee™, t = T − 27/0², k = 2r/o² and - (k-1) x (k+1)² + T V=Eu(x,T) exp 2 4 where r is the risk-free interest rate, σ is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution u(x,T): = 1 TT or L 2√Tπ 1 uo (s) exp 4T : (x - s)²) ds. Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S – E). -
The Black-Scholes equation for the value of an option V(S,t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x,7) using S = Ee™, t = T − 27/0², k = 2r/o² and - (k-1) x (k+1)² + T V=Eu(x,T) exp 2 4 where r is the risk-free interest rate, σ is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution u(x,T): = 1 TT or L 2√Tπ 1 uo (s) exp 4T : (x - s)²) ds. Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S – E). -
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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Transcribed Image Text:The Black-Scholes equation for the value of an option V(S,t), where S
is the value of the underlying asset and t is time, can be transformed
into the diffusion equation for a new variable u(x,7) using S = Ee™,
t = T − 27/0², k = 2r/o² and
-
(k-1) x
(k+1)² +
T
V=Eu(x,T) exp
2
4
where r is the risk-free interest rate, σ is the volatility, E is the strike
price, and T is the expiry date. The diffusion equation has the solution
u(x,T):
=
1
TT or
L
2√Tπ
1
uo (s) exp
4T
: (x - s)²) ds.
Determine, showing all steps, the function uo(s) for a European asset-
or-nothing call with the pay-off A(S) = SH(S – E).
-
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