課程名稱︰數理金融導論
課程性質︰數學系選修
課程教師︰韓傳祥
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰103/11/10
考試時限（分鐘）：9:10 ~ 12:10
試題 :
(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.[買賣權價平的偏微分方程解] 假設一個歐式選擇權的價格，記為P(t,x)，它會滿足以下
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) 是選擇權的報酬函數。
(a) 若K是履約價，分別寫下買(賣)權的報酬函數，以及畫出其圖形。
(b) 在相同的契約條件下，買入一買權並賣出一賣權。寫出此投資組合的報酬函數，並畫
出其圖形。
(c) 證明 L (‧) 是線性算子
BS
(d) 驗證以下 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) 由上推論出買賣權價平關係(put-call parity)如下
-r(T-t)
C (t,x) - P (t,x) = x - Ke 。
BS BS
2.[買賣權價平]若選擇權契約的標的股票恰好在選擇權到期時支付一筆現金股利D，使用
「無套利評價法」證明買賣權價平的關係式為
-r(T-t) -r(T-t)
C - P = S - De - Ke 。
t t t
3.令柏努利隨機變數(Bernoulli random variable) X 取值為1或-1 (X∈{1,-1}) 且發生
∞
的機率各為 p(0≦p≦1) 與 1-p。給定了一序列獨立同分配的柏努利隨機變數{X } ，且
j j=1
∞
p = 0.5，對稱隨機漫步(symmetric random walk, SRW) 記為 {M } 的定義如下：起始
k k=0
k
值 M = 0 且 M = Σ X , k = 1,2,...。縮放對稱隨機漫步(scaled SRW)的定義如下：
0 k j=1 j
(n) 1 +
固定正整數n，W (t ) = ——— M ，nt ∈Z 。
i √n nt i
i
(n)
(a) 證明對於任何一個 0 < T，var(W (T)) = T。
(n)
(b) 在期間[0,T]中，隨機過程 W (t) 的二次變分(quadratic variation)的定義為
n-1 (n) (n) 2 i
Σ (W (t ) - W (t )) ，t = ——。計算出此量。
i=0 i+1 i i n
(c) 評論變異數與二次變分之不同。
T
4.假設資產S服從 dS = σS dW 且θ = 1/S ，考慮 I =∫θdS
t t t t t T 0 t t
(1) 解釋此隨機積分的金融意義。
(2) 證明此策略的總損益I 的性質 (a) I = σW ， (b) I 的均值是0，標準差是σ√T。
T T T T
~
5.[Correlation Estimation]給定一常數▕ ρ▕ ≦ 1，W 和 W 是獨立的布朗運動，
t t
2 ~
(a) 證明 Z =ρW + √(1-ρ ) W 是布朗運動，
t t t
(b) 計算dW dZ = ρdt，
t t
(c) 利用(b)導出對相關係數ρ的(一致，consistent)估計式。
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.