这个作业是完成股票利率收益相关的数学金融问题

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。
考虑奖励程序g(St
,t)=(St-Kt)
+
当t = 0、1、2时,随机行使价满足K0 = 40,则K1(ω)= 35.5
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
独特的mar度量Pe。
(b)计算价格过程C
一种
对于此选项,使用递归关系
C
一种
t =最大n
(St-Kt)
+,(1 + r)
-1 EPe

C
一种
t + 1 |英尺
Ø
终止条件为C
一种
2 =(S2-K2)
+。
(c)找出合理的运动时间τ

0
此选项的持有者。
(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
在时间T的选择权。
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