(a) 设 W(t)； 0吨吨；是一维标准维纳过程

;F; fftg; P);其中 fFtg 是自然的
W(t) 的 Öltration：假设股票价格 S(t) 被建模为
dS = (t) Sdt + (t) SdW(t); (1)

(i) 证明 (1) 的解可以明确写成
S(t) = S(0) exp

[看过] [5 分]
(ii) 查找 (t)；其中 S(t) 是 P-鞅。提供解释。
[类似于看到] [4 分]
(b) 解释“局部波动率模型”的含义以及“局部波动率模型”的含义
‘随机波动率模型’。本地卷模型的缺点是什么？

[4 分]
(c) 假设利率 r = 0：考虑一般的局部波动率

dS = (t; S)SdWQ;

X：

(i) 推导 Dupire 公式：
2
(T;K) = 2@C
@T (K; T)
K2@2
@K2C(K; T)
;

T，并用价格过程 S(t) 写在底层证券上。 [书本] [6

(ii) 解释如何在实践中使用 Dupire 公式。 [书-

(d) 给出一个是马尔可夫但不是鞅的过程的例子，并且

(a) (i) 介绍满足 SDE 的过程 Z

dt + (t) dW(t)：

S(t) = S(0) exp(Z(t))：

dS = S(0) exp(Z(t))
S(0) exp(Z(t))dt
+S(0) exp(Z(t)) (t) dW(t)
= (t) Sdt + (t) SdW(t)

(ii) 如果 (t) = 0；然后
dS = (t) SdW(t)

Question 1.
(a) Let W(t); 0  t  T; be a one-dimensional standard Wiener process
on the Öltered probability space (
;F; fFtg; P); where fFtg is the natural
Öltration for W(t): Assume that the stock price S(t) is modelled as
dS = (t) Sdt + (t) SdW(t); (1)
where the rate of return (t) and the volatility (t) are Ft-adapted stochastic
processes with ëgoodíproperties and (t) is almost surely not zero.
(i) Show that the solution to (1) can be written explicitly as
S(t) = S(0) exp

[seen] [5 Marks]
(ii) Find (t); for which S(t) is a P-martingale. Provide an explanation.
[analogous to seen] [4 Marks]
(b) Explain what is meant by ëlocal volatility modelsíand what is meant by
ëstochastic volatility modelsí. What are drawbacks for local vol models and
what are they for stochastic volatility models? [bookwork + ësoftískills]
[4 Marks]
(c) Assume that the interest rate r = 0: Consider a generic local volatility
model written under an EMM Q :
dS = (t; S)SdWQ;
where (t; x) being a deterministic function of time t and the underlierís price
x:

(i) Derive Dupireís formula:
2
(T;K) = 2@C
@T (K; T)
K2 @2
@K2C(K; T)
;
where C(K; T) is is a price of plain-vanilla call option with strike K, maturity
T, and written on an underlier with the price process S(t). [bookwork] [6
Marks]
(ii) Explain how the Dupire formula can be used in practice. [book-
work+ ësoftískills] [2 Marks]
(d) Give an example of a process which is Markov but not a martingale and
an example of a process which is a martingale but not Markov. [unseen] [4
Marks]
Total: 25 marks
Ans. [di§erent expositions of the correct solution will be also ac-
cepted]
(a) (i) Introduce the process Z which satisÖes the SDE

dt + (t) dW(t):
It is clear that
S(t) = S(0) exp(Z(t)):
By the Ito formula, we get
dS = S(0) exp(Z(t))
S(0) exp(Z(t))dt
+S(0) exp(Z(t))(t) dW(t)
= (t) Sdt + (t) SdW(t)
as required.
(ii) If (t) = 0; then
dS = (t) SdW(t)