5.2Risk-Neutral MeasurePart 2

報告者：陳政岳

5.2.2 Stock Under the Risk-Neutral Measure

• is a Brownian motion on a probability space , and is a filtration for this Brownian motion. T is a fixed final time.

• A stock price process whose differential is

where :the mean rate of return and :the volatility are adapted processes. Assume that, is almost surely

not zero.

( ), 0W t t T ( , , )F , 0 ,F t t T

,0dS t t S t dt t S t dW t t T t t

0, ,t T t

• The stock price is a generalized Brown motion (see Example 4.4.8), and an equivalent way of writhing is

• Supposed we have an adapted interest rate process R(t). We define the discount process

and

2

0 0

10 exp ( ) .

2

t tS t S s dW s s s ds

0

tR s ds

D t e

.dD t R t D t dt

• Define so that and .

• First, we introduction the function for which and use the Ito-Doeblin formula to write

0

tI t R s ds dI t R t dt

0dI t dI t ,xf x e

, ,f x f x f x f x

1

2

dD t df I t

f I t dI t f I t dI t dI t

f I t R t dt

R t D t dt

• Although D(t) is random, it has zero quadratic variation. This is because it is “smooth.” Namely, one does not need stochastic calculus to do this computation.

• The stock price S(t) is random and has nonzero quadratic variation. If we invest in the stock, we have no way of knowing whether the next move of the driving Brownian motion will be up or down, and this move directly affects the stock price.

( ) ( ) ( )D t R t D t

• Considering a money market account with variable interest rate R(t), where money is rolled over at this interest rate. If the price of a share of this money market account at time zero is 1, then the price of a share of this money market account at time t is

0

1.

( )

tR s ds

eD t

• If we invest in this account, over short period of time we know the interest rate at the time of the investment and have a high degree of certainty about what the return.

• Over longer periods, we are less certain because the interest rate is variable, and at the time of investment, we do not know the future interest rates that will be applied.

• The randomness in the model affect the money market account only indirectly by affecting the interest.

• Changes in the interest rate do not affect the money market account instantaneously but only when they act over time.( Warning: . For a bond, a change in the interest rate can have an instantaneous effect on price.)

• Unlike the price of the money market account, the stock price is susceptible to instantaneous unpredictable changes and is, in this sense, “more random” than D(t). Because S(t) has nonzero quadratic variation, D(t) has zero quadratic variation.

• The discounted stock price process is

(5.2.19) and its differential is

where we define the market price of risk to be

2

0 0

10 exp

2

t tD t S t S s dW s s R s s ds

(5.2.20)

d D t S t t R t D t S t dt t D t S t dW t

t R tt D t S t dt dW t

t

t D t S t t dt dW t

t R tt

t

• (5.2.20) can derive either by applying the Ito-Doblin formula or by using Ito product rule.

• The first line of (5.2.20), compare with

shows that the mean rate of return of the discounted stock price is , which is the mean rate of the undiscounted stock price, reduced by the interest rate R(t).

• The volatility of the discounted stock price is the same as the volatility of the undiscounted stock price.

,dS t t S t dt t S t dW t

t R t t

• The probability measure defined in Girsanov’s Theorem, Theorem 5.2.3, which uses the market price of risk

In terms of the Brownian motion of that theorem, we rewrite (5.2.20) as

• We call , the measure defined in Girsanov’s Theorem, the risk-neutral measure because it is equivalent to the original measure P

P

.t R t

tt

. (5.2.22)d D t S t t D t S t dW t

P

• According to (5.2.22),

and under the process is an Ito-integral and is a martingale.

• The undiscounted stock price S(t) has mean rate of return equal to the interest rate under , as one can verify by making the replacement in

0

0 , t

D t S t S u D u S u dW u

P 0

tu D u S u dW u

P dW t t dt dW t

.dS t t S t dt t S t dW t

.dS t R t S t dt t S t dW t

• We can solve this equation for S(t) by simply replacing the Ito integral by its equivalent in

to obtain the formula

0

ts ds

0 0

t ts dW s s R s ds

2

0 0

10 exp

2

t tS t S s dW s s s ds

2

0 0

10 exp

2

t tS t S s dW s R s s ds

• In discrete time: the change of measure does not change the binomial tree, only the probabilities on the branches of the tree.

• In continuous time, the change from the actual measure P to the risk-neutral measure change the mean rate of return of the stock but not the volatility.

P

• After the change of measure , we are still considering the same set of stock price paths, but we shifted the probability on them.

• If , the change of measure puts more probability on the paths with lower return so return so that the overall mean rate of return is reduced from to R(t)

t R t

t

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure

• Initial capital X(0) and at each time t, holds shares of stock, investing or borrowing at the R(t).

• The differential of this portfolio value is given by the analogue of (4.5.2)

0 ,t T t

dX t rX t dt t r S t dt t S t dW t

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure

•

• By Ito product rule, (5.2.18) and

(5.2.20) imply

dX t t dS t R t X t t S t dt

t t S t dt t S t dW t R t X t t S t dt

R t X t dt t t R t S t dt t t S t dW t

R t X t dt t t S t t dt dW t

dD t R t D t dt d D t S t t D t S t t dt dW t

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure

•

d D t X t X t dD t D t dX t dD t dX t

X t D t R t dt

D t R t X t dt t t S t t dt dW t

D t R t dt R t X t dt t t S t t dt dW t

t t D t S t t

dt dW t

t d D t S t

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure

• Changes in the discounted value of an agent’s portfolio are entirely due to fluctuations in the discounted stock price. We may use

(5.2.22)to rewrite as

d D t S t t D t S t dW t

.d D t X t t t D t S t dW t

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure

• We has two investment options: (1) a money market account with rate of ret

urn R(t), (2) a stock with mean rate of return R(t) un

der • Regardless of how the agent invests, the

mean rate of return for his portfolio will be R(t) under P, and the discounted value of his portfolio, D(t)X(t), will be a martingale.

.P

5.2.4 Pricing Under the Risk-Neutral Measure

• In Section 4.5, Black-Scholes-Merton equation for the value the European call have initial capital X(0) and portfolio process an agent would need in order hedge a short position in the call.

• In this section, we generalize the question.

t

5.2.4 Pricing Under the Risk-Neutral Measure

• Let V(T) be an F(T)-measurable random variable. This payoff is path-dependent which is F(T)-measurability.

• Initial capital X(0) and portfolio process

we wish to know that an agent would need in order to hedge a short position, i.e., in order to have

X(T) = V(T) almost surely.• We shall see in the next section that this can be

done.

, 0 ,t t T

5.2.4 Pricing Under the Risk-Neutral Measure

• In section 4.5, the mean rate of return, volatility, and interest rate were constant.

• In this section, we do not assume a constant mean rate of return, volatility, and interest rate.

• D(T)X(T) is a martingale under implies

.

D t X t E D T X T F t

E D T V T F t

5.2.4 Pricing Under the Risk-Neutral Measure

• The value X(t) of the hedging portfolio is the capital needed at time t in order to complete the hedge of the short position in the derivative security with payoff V(T).

• We call the price V(t) of the derivative security at time t, and the continuous-time of the risk-neutral pricing formula is

, 0 . (5.2.30)D t X t E D T V T F t t T

5.2.4 Pricing Under the Risk-Neutral Measure

• Dividing (5.2.30) by D(t), which is F(t)-measurable. We may write (5.2.30) as

• We shall refer to both (5.2.30) and (5.2.31) as the risk-neutral pricing formula for the continuous-time model.

,0 . (5.2.31)T

tR u du

V t E e V t F t t T

5.2.5 Deriving the Black-Scholes-Merton Formula

• To obtain the Black-Scholes-Merton price of a European call, we assume constant volatility constant interest rate r, and take the derivative security payoff to be

• The right-hand side of

becomes

,

.V T S T K

T

tR u du

V t E e V t F t

.T

tR u du

V t E e S T K F t

5.2.5 Deriving the Black-Scholes-Merton Formula

• Because geometric Brownian motion is a Markov process, this expression depends on the stock price S(t) and on the time t at which the conditional expectation is computed, but not on the stock price prior to time t.

• There is a function c(t,x) such that , .r T tc t S t E e S T K F t

5.2.5 Deriving the Black-Scholes-Merton Formula

• Computing c(t,x) using the Independence Lemma, Lemma 2.3.4.

• Lemma 2.3.4 (Independence) Let be a probability space, and let G

be a sub- -algebra of F. Suppose the random variables are G-measurable and the random variables are independent of G. Let be a function of the dummy variables and and define

Then

, ,F P

1, , KX X

1, , LY Y 1 1, , , , ,K Lf x x y y

1, , Kx x1, , ,Ly y

1 1 1, , , , , , , .K K Lg x x Ef x x Y Y

1 1 1, , , , , , , .K L KE f X X Y Y G g X X

5.2.5 Deriving the Black-Scholes-Merton Formula

• With constant and r, equation

becomes

and then rewrite

2

0 0

10 exp

2

t tS t S s dW s R s s ds

210 exp ,

2S t S W t r t

2

2

1exp

2

1 exp ,

2

S T S t W T W t r

S t Y r

• where Y is the standard normal random

variable , and is the “time

to expiration” T-t.

• S(T) is the product of the F(t)-measurable random variable S(t) and the random variablewhich is independent of F(t).

W T W tY

T t

• Therefore,

2

2

12 2

,

1 exp

2

1 1 exp

22

r T t

r

yr

c t S t E e S T K F t

E e x Y r K

e x y r K e dy

5.2.5 Deriving the Black-Scholes-Merton Formula

• The integrand

is positive if and only if

21exp

2x Y r K

21 1, log .

2

xy d x r

K

5.2.5 Deriving the Black-Scholes-Merton Formula

• Therefore,

2

2

1, 2 2

2, 2

1,

2

2,

1 1, exp

22

1 1 exp

2 22

1

2

1 exp

22

d x yr

d x

d x yr

d x

c t S t e x y r K e dy

x y y dy

e Ke dy

xy

,

, ,

r

r

dy e KN d x

xN d x e KN d x

5.2.5 Deriving the Black-Scholes-Merton Formula

• where

• So, the notation

• where Y is a standard normal random variable under

2

, ,

1 1 log .

2

d x d x

xr

K

2

, ; , ,

1 exp ,

2r

BSM x K r

E e x Y r K

.P

5.2.5 Deriving the Black-Scholes-Merton Formula

• We have shown that

• Here we have derived the solution by device of switching to the risk-neutral measure.

( , ; , , )

, , .r

BSM x K r

xN d x e KN d x