金融工程讲义(上海财经大学)_3

According to the results obtained in chapter 2, (2.4.11), the time zero value of this derivative asset is
~ V0 E 4 ~ ( ) 3 V3 ( ) P ( ) 5 (1 r ) 1 1 1 1 1 1 1 1 0.512[0 * 8 * 0 * 6 * 0 * 2 * 2 * 3.50 * ] 8 8 8 8 8 8 8 8
(3.1.9)
and ( ) P( ) is called the state price corresponding to .
Remark: the state price ( ) P( ) gives us the time-zero price of a contract that pays 1 at time N when occurs. The state price density price of this contract per unit of actual probability,
~ =1/2. The risk-neutral probability measure is the risk-neutral probability of a tail is q
3
1 ~ 1 ~ 1 ~ 1 ~ P ( HHH ) P ( HHT ) P ( HTH ) P ( HTT ) 8, 8, 8, 8 1 ~ 1 ~ 1 ~ 1 ~ P (THH ) P (THT ) P (TTH ) P (TTT ) (3.1.4) 8, 8, 8, 8
S3 TTT 0.5
{HHH, HHT, HTH, HTT , THH, THT , TTH , HHH}
If we take p=2/3 and q=1/3 as the actual probability of a head and a tail respectively, then the actual probability measure is P(HHH)=
Definition 3.1.3 state price ＆ state price density
In the N-period binomial model with actual probability measure P and risk-neutral probability measure P , let Z denote the Radon-Nikodym derivative of P with respect to P, i.e,
1 ,... N and ＃T (1 ,... N ) denotes the number of tails appearing in this sequence.
The state price density random variable is ( )
Z ( ) (1 r ) N
Such probability measures often times are obtained by monitoring the price movement over a period of time. From the practical decision making point of view, it is this “actual” probability measure that is really relevant, not the risk -neutral probability distribution. In this chapter we’ll demonstrate how to use the properties of the risk-neutral probabilities to compute the expected values band on the “actual” probability measures.
S2 HH 16 S1 H 8
S2 H T 4 S2 T H
S3 H H T S3 H T H 8 S3 T H H
S0 4
S1 T 2 S2 TT 1
S3 HTT S3 THT 2 S3 TTH
Z (1 , 2 ,
, N )
P(1 , 2 , P(1 , 2 ,
, N ) p ( ) # H (1 ,2 , , N ) p
,N )
q ( ) #T (1 ,2 , q
,N )
(3.1.8)
Where ＃ H (1 ,... N ) denotes the number of heads appearing in the sequence.
8 4 4 2 , P( HHT ) P( HTH ) P( HTT ) 27 27, 27, 27
P(THH )
4 2 2 1 P(THT ) P(TTH ) P(TTT ) (3.1.3) 27, 27, 27, 27
Let the interest rate be r=1/4 and the risk-neutral probability of a head is ~ p =1/2 and
Chapter 3 State Prices In previous chapters, we have derived the price of a derivative asset by solving system of equations that are constraints by the no-arbitrage assumption. Symbolically the expressions of the prices of the stock and the derivative asset at time n can be expressed as a weighted sum of those at time (n+1). If we define those weights as risk-neutral probability (those weights actually satisfy the probability conditions), the stock prices and the price of derivative assets carry several nice properties, such as martingale and Markov. In practice, the actual prices may not evolve based on the possibility of the risk-neutral distribution. Notice that the risk-neutral probabilities
~ (iii) for any random variable Y, E (Y ) E (ZY ) .
2
Consider a three-period model Figure 3.1.1 A three-period binomial Asset-price model
S3 HHH 32
Z ( ) tells us the time zero (1 r ) N
5
Using the linearity property of expectation, we have the time zero price of any derivative asset is
Z ( )
~ P ( ) P( )
(3.1.1)
Z ( ) is called the Radon-Nikodym derivate of P with respect to P .
Theorem 3.1.1 let P and P be probability measures on a finite sample space ,
Therefore the Radm-Nikodym derivative of P with respect to p is Z(HHH)=
27 27 27 27 , Z ( HHT ) Z ( HTH ) Z ( HTT ) 64 32, 32, 16
Z (THH )
27 27 27 27 Z (THT ) Z (TTH ) Z (TTT ) 32, 16, 16, 8
4
=1.376 The advantages of using (3.1.7) are (i) the actual probability is used and (ii) a weight Z is applied to the payoff using the variable Z.
(3.1.7)
3
V3
=1.376
(3.1.6)
If we apply the theorem (3.1.1) to this example, we have
V0 E
V3 Z 4 ~ ( ) 3 V3 ( ) Z ( ) P ( ) 3 5 (1 r ) 27 8 27 4 27 4 27 2 0.512[0 * * 8* * 0* * 6* * 64 27 32 27 32 27 16 27 27 4 27 2 27 2 27 1 0* * 2* * 2* * 3.50 * * ] 32 27 16 27 16 27 8 27
1Leabharlann Baidu
Definition : Radon-Nikodym derivative
Consider a finite sample space on which we have two probability measures P and P . Let’s assume that P and P both give positive probability to every element of . Define a random variable Z
(3.1.5)
The lookback option defined in Example 1.2.4 of Chapter 1 has payoff at time three given below:
V3 ( HHH ) O,V3 ( HHT ) 8,V3 ( HTH ) 0,V3 ( HTT ) 6 V3 (THH ) 0,V3 (THT ) 2,V3 (TTH ) 2,V3 (TTT ) 3.50
p
1 r d u 1 r are determined by the range of price changes and the ,q ud ud
risk-free rate. They don’t have anything to do with the “actual” chances that describe how likely the price would move up or down at each time period.
~ assuming that P( ) >0 and P ( ) >0 for every , and define the random
variable Z by (3.1.1). Then we have the following : (i) P(Z>0)=1 ; (ii)E(Z)=1 ;