MATH3075 Financial Derivatives (Mainstream)

1. [10分] CRR模型：美式看涨期权。假设CRR模型为
T = 2，股票价格S0 = 45，Su
1 = 49.5，Sd
1 = 40.5，利率r = -0.05。

，t）=（St-Kt）
+

ω∈{ω1，ω2}，对于ω∈{ω3，ω4}且K2 = 36.45，K1（ω）= 38.5。
（a）找到参数u和d，计算时间t = 2时的股价并找到

（b）计算价格过程C

C

t =最大n
（St-Kt）
+，（1 + r）
-1 EPe

C

t + 1 |英尺
Ø

2 =（S2-K2）
+。
（c）找出合理的运动时间τ

0

（d）找到发行人的复制策略ϕ，直至合理行使时间τ

0并表明复制策略的财富与
（b）部分中计算的价格。
（e）如果持有人决定行使发行人在时刻T的利润，则计算

2. [10 marks] Black-Scholes model: European claim. We place ourselves within
the setup of the Black-Scholes market model M = (B, S) with a unique martingale
measure Pe. Consider a European contingent claim X with maturity T and the
following payoff
X = max (K, ST ) − LST
where K = e
rT S0 and L > 0 is an arbitrary constant. We take for granted the
Black-Scholes pricing formulae for the call and put options.
(a) Sketch the profile of the payoff X as a function of the stock price ST at time T
and show that X admits the following representation
X = K + CT (K) − LST
where CT (K) denotes the payoff at time T of the European call option with
strike K.
(b) Find an explicit expression for the arbitrage price πt(X) at time 0 ≤ t < T in
terms of Ft
:= e
rtS0, St and S0. Then compute the price π0(X) in terms of S0
and use the equality N(x) − N(−x) = 2N(x) − 1 to simplify your result.
(c) Find the limit limT→0 π0(X).
(d) Find the limit limσ→∞ π0(X).
(e) Explain why the price of π0(X) is positive when L = 1 by analysing the payoff
X when L = 1