金融工程讲义t

d m dt s dz Imagine two derivatives dependent on
with prices ?1 and ?2. Suppose
d ?1 ƒ1

1
dt
1
dz
d ƒ2 ƒ2
2
dt 2
dz
19.4
Forming a Riskless Portfolio
19.5
Market Price of Risk (Page 500)
Since the portfolio is riskless: = r t
This gives: 12 21 r 2 r 1
or 1 r 2 r
1
2
• This shows that ( – r )/ is the same for all derivatives dependent on the same underlying variable,
We can set up a riskless portfolio , consisting of + 2 ƒ2 of the 1st derivative and 1ƒ1 of the 2nd derivative
(2 ƒ2 )?1(1ƒ1)?2 = (12 ƒ1ƒ2 21ƒ1ƒ2 )t
• We refer to ( – r )/ as the market price of risk for and denote it by l
19.6
Differential Equation for ƒ
(Equation 19.10, page 501)
Using Ito’s lemma to obtain expressions for and in terms of m and s. The equation
Chapter 19
19.3
Derivatives Dependent on a Single
Underlying Variable
Consider a variable, , (not necessarily the price
of a traded security) that follows the process
19.1
OPTIONS、FUTURES & OTHER DERIVATIVES
STUDENT PRESENTATION TUTOR：PROF YE 指导老师：叶永刚教授
STUDENT NAME：HEHAO 学生姓名：贺昊
NO.：200321050773
19.2
Extension of the Theoretical Framework for Pricing Derivatives; Martingales and Measures
19.13
Forward Risk Neutrality
We refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g.
In a world where the market price of risk is l
df (r l) fdt fdz
19.12
A Key Result (Page 509)
If we set l equal to the volatility of a security g, then Ito's lemma shows that f g is a martingale for all derivative security prices f ( f and g are assumed to provide no income during the period under consideration)
• This analogy shows that we can value ƒ in a risk-neutral world providing the drift rate of is reduced from m to m – ls
Extension of the Analysis
19.8
l=r
becomes
ƒ ƒ (m ls) ? t
s22
2ƒ 2
r?
19.7
Risk-Neutral Valuation
• The differential equation shows that is like a stock price paying a dividend yield of r – m + ls
• A martingale has the property that its expected future value equals its value today
19.11
Alternative Worlds
In the traditional risk - neutral world df rfdt fdz
to Several Underlying Variables
(Equations 19.12 and 19.13, page 503)
d ?
n
ƒBiblioteka Baidu
dt i dzi
i 1
n
r l ii
i 1
19.9
Derivatives Dependent on Commodity Prices (Page 506)
For a commodity the futures price gives the expected value in the traditional risk-neutral world
19.10
Martingales (Page 507)
• A martingale is a stochastic process with zero drfit