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Reflection principle
创建日期：2024-06-08 14:31:03
所属分类：数学mathematics
标签： function
Reflection principle

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Question 1

a) Assume an underlying Black–Scholes model with σ > 0, r > 0, S0 > 0 and without dividend
payments. Formulate a partial differential equation that is satisfied by the price V of the
European down-and-in barrier call option with barrier B and strike K, where K > B, and
where V (t, S) is a function of current time t and stock-value S. Assume S0 > B. Indicate
clearly the solution domain and the boundary conditions V (t, B) and V (T, S).
b) Assume an underlying Black–Scholes model with σ > 0, r > 0, S0 > 0 and without dividend
payments. Determine the linear complementarity problem for the American up-and-out put
option with Barrier B and strike K, K < B, S0 < B. Indicate clearly the solution domain
and the boundary condition at time T .
c) Find the probability density function of Mt := sup0≤s≤tWs where (Wt)t≥0 is a standard
Wiener process.
Hint: Reflection principle.
d) Let cfix(S,M, t,K) and pfl(S,M, t) denote the price function of a (European) fixed strike
lookback call option and a (European) floating strike lookback put option respectively, where
S is the time-t price of the underlying stock, M is the running maximum at time t and K is
the strike price. Suppose M > K. Compute cfix(S,M, t,K)− pfl(S,M, t).
Hint: Consider a portfolio long in the call option and short in the put option.
e) Compare the time-0 price of an up-and-out-call option with the price of a regular call option
with same strike and same maturity.
Hint: In-out-parity.
1
Question 2 (25 marks)
Assume a Black–Scholes model with drift µ ∈ R, risk-free rate r > 0, initial value S0 > 0 and
volatility σ > 0 and assume the stock does not pay any dividends. Suppose the arbitrage-free price
at time t with 0 ≤ t ≤ T of an up-and-out barrier call option with barrier B > S0 and strike K > 0
is given by V (t, St), where St denotes the value of the underlying asset at time t.
(a) Formulate (without proof) a partial differential equation that is satisfied by V and explicitly
state boundary conditions and the solution domain.
(b) Express the arbitrage-free price at time 0 of an up-and-in barrier put option (with the same
maturity T , same strike K and same barrier B as the up-and-out barrier call option) as a
function of the following four quantities:
• V (0, S0),
• p: the price of a regular put option with strike K and maturity T ,
• buo: the price of an up-and-out option with payoff 1, barrier B and maturity T ,
• suo: the price of an up-and-out option with payoff ST , barrier B and maturity T .
Question 3 (25 marks)
Suppose we have two traded risky assets S
(1)
t and S
(2)
t , and one risk-free asset with Bt = e
rt for
r > 0. The dynamics of the risky assets under the real-world probability measure are given by
dS
(1)
t = S
(1)
t µ1dt+ S
(1)
t σ1dW1,t
dS
(2)
t = S
(2)
t µ2dt+ S
(2)
t σ2dW2,t
dW1,tdW2,t = ρdt
where W1,t and W2,t are two correlated Wiener processes, µ1, µ2 ∈ R denote the drifts of the
processes, σ1, σ2 > 0 denote the volatilities of the first and second asset and where ρ ∈ [−1, 1].
Find the arbitrage-free price at time 0 of a financial derivative with payoff φ(S
(1)
T , S
(2)
T ) at maturity
T , where
φ(S
(1)
T , S
(2)
T ) := S
(1)
T 1
{
S
(2)
T ≤KS
(1)
T
}
for K > 0.
Question 4 (25 marks)
A down-and-in cash call is a claim that pays 1 at maturity date T > 0 provided the barrier B is
hit and that ST ≥ K. Suppose B ≤ K. Assume an underlying Black–Scholes model with σ > 0,
r > 0, S0 > 0 and without dividend payments.
Compute the price at time 0 of a down-and-in cash call.
Hint: Apply the reflection principle to the log-price process, similar to the approach from Lecture
10. You may use the results from the lecture without proof.
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