课程名称︰数理金融导论
课程性质︰数学系选修
课程教师︰韩传祥
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰103/11/10
考试时限（分钟）：9:10 ~ 12:10
试题 :
(20%) Multiple Choices
1. Who is considered as the "father" of mathematical finance?
(A) C.Gauss (B) L.Bachelier (C) F.Black
2. What is the first option exchange?
(A) CBOE (B) NYSE (C) LSE
3. Which contract is nonlinear?
(A) forward (B) futures (C) option
4. Which financial contract is not a credit derivative?
(A) IRS (B) CDS (C) CDO
5. The put-call parity exists because of
(A) no arbitrage (B) supple and demand (C) excess return
6. Which method can construct Brownian motion?
(A) cumulative normal random variables (B) rescaled symmetric random walk
(C) both
7. Which mathematical tool is suitable for solving problems of option pricing
and hedging?
(A) calculus (B) differential equation (C) stochastic calculus
8. Which process can be treated by Ito's formula?
(A) Brownian motion (B) random walk (C) Binomial tree model
9. When an asset price is "oscillating," it can be modeled by
(A) Brownian motion (B) geometric Brownian motion
(C) mean-reverting process
10.When an asset price is "trend," it can be modeled by
(A) Brownian motion (B) geometric Brownian motion
(C) mean-reverting process
(20%) Fill in blanks, and line up one box in the left column and another box in
the right colunm when their concepts are relevant.
(Ⅰ) Financial markets and their risks
Five major financial markets Their associated risks are
given in this column. given in this column.
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ equity ∣ ∣ ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ ∣ ∣ weather ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ credit ∣ ∣ ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ ∣ ∣ default ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ FX ∣ ∣ ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
(Ⅱ) Properties of stochastic processes and asset pricing
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ Markov property ∣ ∣ up trend ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ martingale ∣ ∣ down trend ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ supermartingale ∣ ∣ buy and hold strategy ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ ∣ ∣ trading portfolio ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ arbitrage ∣ ∣drift and martingale terms ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣ stochastic integral ∣ ∣ memoryless ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ ∣ ∣ ∣
∣stochastic differential eqn∣ ∣ ∣
∣ ∣ ∣ ∣
——————————————— ———————————————
——————————————— ———————————————
∣ t ∣ ∣ ∣
∣ ∫1 dS ∣ ∣ free lunch ∣
∣ 0 u ∣ ∣ ∣
——————————————— ———————————————
(60%) Calculation and proof
1.[买卖权价平的偏微分方程解] 假设一个欧式选择权的价格，记为P(t,x)，它会满足以下
Black-Scholes PDE
L P(t,x) = 0, P(T,x) = h(x)，
BS
其中的偏微分算子(parital differential operator) 是
2
∂ 1 2 2 ∂ ∂
L (‧) = ( —— + ——σ x ———— + rx —— - r ) (‧) ，
2
BS ∂t 2 ∂x ∂x
T 是到期日，h(x) 是选择权的报酬函数。
(a) 若K是履约价，分别写下买(卖)权的报酬函数，以及画出其图形。
(b) 在相同的契约条件下，买入一买权并卖出一卖权。写出此投资组合的报酬函数，并画
出其图形。
(c) 证明 L (‧) 是线性算子
BS
(d) 验证以下 BS PDE L P(t,x) = 0 ， P(T,x) = h(x) = x - K
BS
-r(T-t)
的解是 P(t,x) = x - Ke 。
(e) 由上推论出买卖权价平关系(put-call parity)如下
-r(T-t)
C (t,x) - P (t,x) = x - Ke 。
BS BS
2.[买卖权价平]若选择权契约的标的股票恰好在选择权到期时支付一笔现金股利D，使用
“无套利评价法”证明买卖权价平的关系式为
-r(T-t) -r(T-t)
C - P = S - De - Ke 。
t t t
3.令柏努利随机变量(Bernoulli random variable) X 取值为1或-1 (X∈{1,-1}) 且发生
∞
的机率各为 p(0≦p≦1) 与 1-p。给定了一序列独立同分配的柏努利随机变量{X } ，且
j j=1
∞
p = 0.5，对称随机漫步(symmetric random walk, SRW) 记为 {M } 的定义如下：起始
k k=0
k
值 M = 0 且 M = Σ X , k = 1,2,...。缩放对称随机漫步(scaled SRW)的定义如下：
0 k j=1 j
(n) 1 +
固定正整数n，W (t ) = ——— M ，nt ∈Z 。
i √n nt i
i
(n)
(a) 证明对于任何一个 0 < T，var(W (T)) = T。
(n)
(b) 在期间[0,T]中，随机过程 W (t) 的二次变分(quadratic variation)的定义为
n-1 (n) (n) 2 i
Σ (W (t ) - W (t )) ，t = ——。计算出此量。
i=0 i+1 i i n
(c) 评论变异数与二次变分之不同。
T
4.假设资产S服从 dS = σS dW 且θ = 1/S ，考虑 I =∫θdS
t t t t t T 0 t t
(1) 解释此随机积分的金融意义。
(2) 证明此策略的总损益I 的性质 (a) I = σW ， (b) I 的均值是0，标准差是σ√T。
T T T T
~
5.[Correlation Estimation]给定一常数▕ ρ▕ ≦ 1，W 和 W 是独立的布朗运动，
t t
2 ~
(a) 证明 Z =ρW + √(1-ρ ) W 是布朗运动，
t t t
(b) 计算dW dZ = ρdt，
t t
(c) 利用(b)导出对相关系数ρ的(一致，consistent)估计式。
6.Let Z is a standard Brownian motion. You are given:
t
(i) U = 2 Z - 2
t t
2
(ii) V = Z - t
t t
2 t
(iii) W = t Z - 2 ∫s Z ds
t t 0 s
Determine which of the processes defined above has/have zero drift.
7.The price of a stock is governed by the stochastic differential equation:
dS
t
—— = 0.03 dt + 0.2 dZ ,
S t
t
where Z is a standard Brownian motion. Consider the geometric average
t
1/3
G = [S ×S ×S ] .
1 2 3
(a) Prove that cov(Z , Z ) = min{s,t}.
s t
(b) Find the variance of ln(G).
8.Consider the stochastic differential equation dX = -3 X dt + 2 dZ , where
t t t
Z is a standard Brownian motion. You are given that a solution is
t
-At t Ds
X = e [B + C∫ e dz ], where A,B,C and D are constants. Solve for these
t 0 s
constants.
9.(Historial Volatility) Given the Black-Scholes model dS = μS dt + σS dW ,
t t t t
(a) Calculate d ln(S )
t
(b) Calculate d ln(S )‧d ln(S )
t t
(c) Write out (b)'s integral from and its discretization
(d) use (c) to construct an estimator for the volatility σ.