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IEOR 4706: Homework
创建日期：2024-04-23 15:12:14
标签： IEOR 4706
Homework

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IEOR 4706: Homework 

1. European pricing. Given a standard Black-Scholes model. Fix the time of maturity
T and consider the following European type of options whose payo↵ is K if S(T )  A;
the payo↵ is K + A S(T ) if A < S(T ) < K + A, and the payo↵ is 0 if S(T ) > K + A.
Determine the arbitrage free price of this contract.
2. Pricing a forward. Assume that the market contains one single risky asset with
price process governed by the Black-Scholes model
dSt = µStdt+ StdBt,
and the interest rate r is constant. There is a forward contract on the risky asset with
maturity T 0 > 0. What is the price of a European call option on the forward contract,
with maturity T < T 0 and strike K ?
3. Implied volatility. Recall the expression of the Black-Scholes formula from Theorem
5.5 in the lecture notes. We write the European call option price CBS() as a function
of the volatility .
(a) Compute dCBS/d. What is the range of ! CBS()? Justify the implied volatility
model.
(b) Compute d2CBS/d2. On which domain is ! CBS() convex? Plot the graph of
! CBS().
(c) Provide an algorithm to compute the implied volatility given the market price Cmkt(t).
Explain your choice of the parameters in the algorithm.
4. Feynman-Kac. Show that the solution to the PDE
@v
@t
r(t, x)v + µ(t, x)@v
@x
+
1
2
2(t, x)
@2v
@x2
= 0
can be written as v(t, x) = E
⇣
e
R T
t r(s,Xs)dsG(XT )|Xt = x
⌘
.
Hint: Apply Itoˆ’s formula to f(u,Xu) = e
R u
t r(s,Xs)dsv(u,Xu) (t fixed).
5. Dividend. Given a Black-Scholes financial market. Consider an underlying asset
S paying a continuous stream of dividend (St, t 0) for some given constant rate > 0.
(a) Show that the position of the security holder at time t is Stet.
Hint: the holder of the option can immediately re-invest the dividend into the asset.
(b) Compute in closed-form the call option price of this asset with continuous dividend.
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