MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 10
Background: Section 4.4 – American Options in the CRR Model.
Exercise 1 Assume the CRR model M = (B, S) with T = 3, the stock
price S0 = 100, S
u
1 = 120, S
d
1 = 90, and the risk-free interest rate r = 0.1.
Consider the American put option on the stock S with the maturity date
T = 3 and the constant strike price K = 121.
(a) Find the arbitrage price P at of the American put option for t = 0, 1, 2, 3.
(b) Find the rational exercise times τ ∗t , t = 0, 1, 2, 3 for the holder of the
American put option.
(c) Show that there exists an arbitrage opportunity for the issuer if the
option is not rationally exercised by its holder.
Exercise 2 Assume the CRR model M = (B, S) with T = 3, the stock
price S0 = 100, S
u
1 = 120, S
d
1 = 90, and the risk-free interest rate r = 0.
Consider the American call option with the expiration date T = 3 and the
running payoff g(St, t) = (St − Kt)+, where the variable strike price equals
K0 = K1 = 100, K2 = 105 and K3 = 110.
(a) Find the arbitrage price Xat of the American call option for t = 0, 1, 2, 3
and show that it is a strict supermartingale under P̃.
(b) Find the holder’s rational exercise times τ ∗0 for the American call option.
(c) Find the issuer’s replicating strategy for the American call option up
to the rational exercise time τ ∗0
Exercise 3 (MATH3975) Consider the CRR binomial model M = (B, S)
with the initial stock price S0 = 9, the interest rate r = 0.01 and the volatility
equals σ = 0.1 per annum. Use the CRR parametrization for u and d, that is,
u = eσ
√
∆t, d =
1
u
,
with the time increment ∆t = 1.
1
We consider call and put options with the expiration date T = 5 years
and strike K = 10.
(a) Compute the price process Ct, t = 0, 1, . . . , 5 of the European call op-
tion using the binomial lattice method.
(b) Compute the price process Pt, t = 0, 1, . . . , 5 for the European put
option.
(c) Does the put-call parity relationship hold for t = 0?
(d) Compute the price process P at , t = 0, 1, . . . , 5 for the American put
option. Will the American put option be exercised before the expiration
date T = 5 by its rational holder?
Exercise 4 (MATH3975) Consider the game option (See Section 4.5) with the
expiration date T = 12 and the payoff functions h(St) and `(St) where
Ht = h(St) = (K − St)+ + α
and
Lt = `(St) = (K − St)+
where α = 0.02 and K = 27. Assume the CRR model with d = 0.9, u =
1.1, r = 0.05 and S0 = 25.
(a) Compute the arbitrage price process (X
g
t )
T
t=0 for the game option using
the recursive formula, for t = 0, 1, . . . , T − 1,
X
g
t = min
{
h(St), max
[
`(St), (1 + r)
−1(p̃Xgut+1 + (1− p̃)Xgdt+1)]}
with πT (X
g) = `(ST ).
(b) Find the optimal exercise times τ ∗0 and σ
∗
0 for the holder and the issuer
of the game option. Recall that
τ ∗0 = inf
{
t ∈ {0, 1, . . . , T} |Xgt = `(St)
}
and
σ∗0 = inf
{
t ∈ {0, 1, . . . , T} |Xgt = h(St)
}
.
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