PDE, Linear algebra, Probability.
Partial di?erential equations
1. Consider the Black-Scholes problem
“2 S2 @2V
@S2 + r S
@S – r V = 0, S>0, t
(i) Given that V (S, t), the solution of (1), is infinitely di?erentiable in S > 0 and
t < T, show that V1(S, t) = S @V @S (S, t) also satisfies the partial di?erential equation in (1). (ii) Assume that V (S, t) and all its t and S-partial derivatives are di?erentiable with respect to the parameter r, for all S > 0 and t < T, and any value of r. Deduce that ?(S, t) = @V @r (S, t) satisfies the problem @? @t + 12 “2 S2 @2? @S2 + r S @? @S – r ? = V – S @V @S , S>0, t
?(S, t) = (T – t)
(S, t) – V (S, t)
(iv) You may assume that if K >0 is a constant and
log(S/K) + (r – 12
“2)(T – t) p
“2(T – t)
, “(x) =
V (S, t) = e-r(T-t)”(d)
is a solution of the Black-Scholes equation in (1), for S > 0, t < T, ” > 0. For
this particular V , and assuming S > 0 and t < T, find f(S) = lim t!T PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT ?
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