[試題] 101上 韓傳祥 數理金融導論 小考
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︰2012年12月10日
考試時限：12:20 - 13:00，共40分鐘
是否需發放獎勵金：是
背景介紹：因為期中考大家考得不甚理想，所以教授就出了這份小考題目，
題目和期中考的題目很類似。
其中，期中考第二題答對15分以上的同學需要做第二頁題目，
其他同學做第一頁題目。
試題 :
Page 1
Quiz: Introduction to Mathematical Finance
2012/12/10
[Notation:W_(t≧0) is an one-dimensional Browian motion.]
Total points:35.
1. Prove that (W_t)^2 is a submartingale by two approaches:
(a) Definition: Show that E{(W_t)^2 | W_(0≦u≦s)} ≧(W_s)^2.
(b) Ito's Lemma: Compute d(W_t)^2 and use that its drift term coeficiente is
positive.
2. (Stochastic Differential Equation, SDE)
(a) If the SDE d(S_t) = (α_t)dt + (β_t)d(W_t) =0 for each t≧0, show that
α_t = β_t = 0 for each t≧0.
(b) Solve the Black-Scholes SDE
d(S_t) = μ(S_t)dt + σ(S_t)d(W_t), S_0 = x.
And what are the mean and variance of S_t?
(c) Solve the following sde,
d(S_t) = (μ_t)(S_t)dt + (σ_t)(S_t)d(W_t), S_0 = x.
(d) Solve the following mean-reverting SDE,
d(r_t) = α(m - r_t)dt + σd(W_t), r_0 = x.
And what are the mean and variance of r_t?
(e) Solve the following sde,
d(r_t) = α(m_t - r_t)dt + (σ_t)d(W_t), r_0=x.
Page 2
Quiz: Introduction to Mathematical Finance
2012/12/10
[Notation: W_(t≧0) is an one-dimension Brownaion motion.]
Total Points: 35.
1. Prove that (W_t)^2 is a submartingale by two approaches:
(a) Definition: Show that E{(W_t)^2 | W_(0≦u≦s) } ≧(W_s)^2
(b) Ito's Lemma: Compute d(W_t)^2 and use that its drift term coeficiente is
postiive.
2. If the SDE d(S_t) = (α_t)dt+(β_t)d(W_t) = 0 for each t≧0, show that α_t=
β_t = 0 for each t≧0.
3. 若X_t是下列隨機微分方程的解
d(X_t) = (μ(X_t) + μ')dt + (σ(X_t) + σ')d(W_t), X_0 = 0
且令S_t = exp{[μ-(σ^2)/2]t +σ(W_t)}。
1. 推導出(S_t)^(-1)所滿足的隨機微分方程式。
2. 證明d{(X_t)[(S_t)^(-1)]} = [(S_t)^(-1)](μ'-σ'σ)dt + σ'd(W_t).
3. 推導出X_t的解。
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