7.5 Asian Options

指導老師：戴天時演講者：鄭凱允

序• An Asian option is one whose payoff

includes a time average of the underlying asset price.

• The average may be over the entire time period between initiation and expiration or may be over some period of time that begins later than the initiation of the option and ends with the option’s expiration.

• The average may be from continuous sampling,

• Or may be form discrete sampling,

• It more difficult for anyone to significantly affect the payoff by manipulation of the underlying asset price.

• The price of Asian options is not known in closed form. Therefore, in this section we discuss two ways to derive partial differential equations for Asian option prices. The first of these was briefly presented in Example 6.6.1. The other method for computing Asian option prices is Monte Carlo simulation.

0

1( ) ,

T

S t dtT

1

1( ),

m

jj

S tm

7.5.1 Fixed-Strike Asian Call

• Once again, we begin with a geometric Brownian motion S(t) given by

where is a Brownian motion under the risk-neutral measure . Consider a Fixed-strike Asian call whose payoff at time T is

where the strike price K is a nonnegative constant. The price at times t prior to the expiration time T of this call is given by the risk-neutral pricing formula

The usual iterated conditioning argument shows

is a martingale under

( ) ( ) ( ) ( ),dS t rS t dt S t dW t (7.5.1)

( ),W t 0 t T P

0

1( ) ( ( ) ) ,

T

V T S t dt KT

(7.5.2)

( )( ) [ ( ) | ( )],r T tV t E e V T F t 0 t T (7.5.3)

( ) [ ( ) | ( )],rt rTe V t E e V T F t 0 t T

P

7.5.2 Augmentation of the State• The Asian option payoff V(T) in (7.5.2) is path-dependent. We cann

ot invoke the Markov property to claim that V(t) is a function of t and S(t) because V(T) is not a function of T and S(T); V(T) depends on the whole path of S.

• To overcome this difficulty, we augment the state S(t) by defining a second process

• Because the pair of processes (S(t),Y(t)) is governed by the pair of stochastic differential equations (7.5.1) and (7.5.5), they constitute a two-dimensional Markov process. Furthermore, the call payoff V(T) is a function of T and the final value (S(T),Y(T)) of this process. Indeed, V(T) depends only on T and Y(T), by the formula

0

( ) ( ) .t

Y t S u du (7.5.4)

1( ) ( ( ) ) .V T Y T k

T (7.5.6)

( ) ( ) .dY t S t dt (7.5.5)

• This implies that there must exist some function v(t,x,y) such that the Asian call price (7.5.3) is given as

• Theorem 7.5.1

the Asian call price function v(t,x,y) of (7.5.7) satisfies the partial differential equation

( ) 1( , ( ), ( )) [ ( ( ) ) | ( )]r T tv t S t Y t E e Y t K F t

T

( )[ ( ) | ( )]r T tE e V T F t (7.5.7)

2 21( , , ) ( , , ) ( , , ) ( , , ) ( , , ),

2t x y x xv t x y rxv t x y xv t x y x v t x y rv t x y

0 ,0 , ,t T x y R (7.5.8)

and the boundary conditions

Proof: Using the stochastic differential equations (7.5.1) and (7.5.5) and noting that dS(t)dY(t)= dY(t) dY(t)= 0 ,we take the differential of the -martingale This differential is

( )( ,0, ) ( ) ,0 ,

lim ( , , ) 0,0 ,0

( , , ) ( ) ,0 ,

r T t

y

yv t y e K t T y R

Tv t x y t T x

yv T x y K x y R

T

(7.5.9)

(7.5.10)

(7.5.11)

P( ) ( , ( ), ( )).rt rte V t e v t S t Y t

2 2

( ( , ( ), ( )))

1[ ]

21

[ ] ( ).2

rt

rtt x y x x

rt rtt x y x x x

d e v t S t Y t

e rvdt v dt v dS v dY v dSdS

e rv v rSv Sv S v dt e Sv dW t

(7.5.12)

In order for this to be a martingale, the dt term must be zero, which implies

Replacing S(t) by the dummy variable x and Y(t) by the dummy variable y, we obtain (7.5.8)

We note that S(t) must always be nonnegative, and so (7.5.8) holds for

. If S(t)=0 and Y(t)=y for some value of t, then S(u)=0 for all

and so Y(u) is constant on [t,T]. Therefore, Y(T)=y, and the value of the Asian call at time t is , discounted from T back to t. this gives us the boundary condition (7.5.9).

If at time t we set Y(t)=y, then Y(T) is defined by (7.5.5). In integrated form, this formula is

( , ( ), ( )) ( ) ( , ( ), ( )) ( ) ( , ( ), ( ))t x yv t S t Y t rS t v t S t Y t S t v t S t Y t

2 21( ) ( , ( ), ( )) ( , ( ), ( )).

2 x xS t v t S t Y t rv t S t Y t

0 x[ , ],u t T

( )y

KT

( ) ( )T

tY T y S u du (7.5.13)

Remark 7.5.2

After we set the dt term in (7.5.12) equal to zero, we see that

The discounted value of a portfolio that at each time t holds shares of the underlying asset is given by (see (5.2.27))

To hedge a short position in the Asian call, an agent should equate these two differentials, which leads to the delta-hedging formula

( ( , ( ), ( ))) ( ) ( , ( ), ( )) ( ).rt rtxd e v t S t Y t e S t v t S t Y t dW t (7.5.14)

( )t

( ( )) ( ) ( ) ( ).rt rtd e X t e S t t dW t (7.5.15)

( ) ( , ( ), ( )).xt v t S t Y t

7.5.3 Change of Numeraire• In this subsection we present a partial differential equation whose so

lution leads to Asian option prices. We work this out for both continuous and discrete averaging. The derivation of this equation involves a change of numeraire, a concept discussed systematically in Chapter 9. in the section, we derive formulas under the assumption that the interest rate r is not zero. The case r=0 is treated in Exercise 7.8

• We first consider the case of an Asian call with payoff

where c is a constant satisfying and K is a nonnegative constant. If c=T, this is the Asian call (7.5.2) considered in Subsection 7.5.2. Here we also admit the possibility that the averaging is over less than the full time between initiation and expiration of the call.

1( ) ( ( ) ) ,

T

T cV T S t dt K

c

(7.5.16)

0 c T

To price this call, we create a portfolio process whose value at time T is

We begin with a nonrandom function of time which will be the number of shares of the risky asset held by our portfolio. There will be no Brownian motion term in , and because of this it will satisfy

This implies that

which further implies

Rearranging terms in (7.5.18), we obtain

1( ) ( )

T

T cX T S u du K

c

( ),0 ,t t T

( )t

( ) ( ) ( ) ( ) 0.d t d t d t dS t

( ( ) ( )) ( ) ( ) ( ) ( ),d t S t t dS t S t d t (7.5.17)

( ) ( ) ( )

( ) ( ) ( )

( ( ) ( )) ( ( ) ( )) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

r T t r T t r T t

r T t r T t r T t

d e t S t e d t S t re t S t dt

e t dS t e S t d t re t S t dt

(7.5.18)

( ) ( ) ( )( )( ( ) ( ) ) ( ( ) ( )) ( ) ( )r T t r T t r T te t dS t rS t dt d e t S t e S t d t

(7.5.19)

an agent who holds shares of the risky asset at each time t and finances his by investing or borrowing at the interest rate r will have a portfolio whose value evolves according to the equation

using this equation and (7.5.19), we obtain

To study the Asian call with payoff (7.5.16), we take to be

( )t

( ) ( ) ( ) ( ( ) ( ) ( ))dX t t dS t r X t t S t dt ( ) ( )( ( ) ( ) ).rX t dt t dS t rS t dt (7.5.20)

( ) ( ) ( )( ( )) ( ) ( )r T t r T t r T td e X t re X t dt e dX t ( )

( ) ( )

( )( ( ) ( ) )

( ( ) ( )) ( ) ( ).

r T t

r T t r T t

e t dS t rS t dt

d e t S t e S t d t

(7.5.21)

( )t

( )

1(1 ),

1(1 ),

( ) {rc

r T t

erc

erc

t

0 ,

,

t T c

T c t T

(7.5.22)

and we take the initial capital to be

In the time interval [0,T-c], the process mandates a buy-and-hold strategy. At time zero, we buy shares of the risky asset, which costs . Our in initial capital is insufficient to do this, and we must borrow from the money market account. For ,the value of our holdings in the risky asset is and we owe to the money market account. Therefore,

In particular,

1(0) (1 ) (0) .rc rTX e S e K

rc (7.5.23)

( )t1

(1 )rcerc

1

(1 ) (0)rce Src

rTe K

0 t T c 1

(1 ) ( )rce S trc

( )r T te K

( )1( ) (1 ) ( ) .rc r T tX t e S t e K

rc (7.5.24)

1( ) (1 ) ( ) .rc rcX T c e S T c e K

rc (7.5.25)

For we have and we compute X(t) by first integrating (7.5.21) form T-c to t and using (7.5.25) and (7.5.22) to obtain

therefore,

,T c t T ( )1( ) r T td t e

c

( )

( ) ( )

( )

( )

( )

( ) ( ( ) ( )) ( ) ( )

1(1 ) ( ) ( ) ( )

1 1(1 ) ( ) ( )

1( ) ( ) ( )

r T t

t trc r T u r T u

T c T c

rc rc r T t

trc rc

T c

tr T t

T c

e X t

e X T c d e u S u e S t d u

e e S T c K e t S trc

e e S T c S u durc c

K e t S t S u duc

( ) ( ) ( )1 1( ) (1 ) ( ) ( )

tr T t r T t r T t

T cX t e S t e S u du e K

rc c

,T c t T (7.5.26)

In particular,

As desired, and

The price of the Asian call at time t prior to expiration is

The calculation of the right-hand side of (7.5.29) uses a change-of-numeraire argument, which we now exlain. Let us define

1( ) ( ) ,

T

T cX T S u du K

c (7.5.27)

( ) ( ) max{ ( ),0}.V T X T X T (7.5.28)

( ) ( )( ) [ ( ) | ( )] [ ( ) | ( )]r T t r T tV t E e V T F t E e X T F t (7.5.29)

( ) ( )( )

( ) ( )

rt

rt

X t e X tY t

S t e S t

This is the value of the portfolio denominated in units of the risky asset rather than in dollars. We have changed the numeraire, the unit of account, from dollars to the risky asset.

we work out the differential of Y(t). Note first that

Therefore,

( ( )) ( ) ( ) ( ) ( )rt rt rt rtd e S t re S t dt e dS t e S t dW t (7.5.30)

1

2 3

2 3 2 2

1 2 1

[( ( )) ]

( ( )) ( ( )) ( ( )) ( ( )) ( ( ))

( ( )) ( ( )) ( ) ( ( )) ( ( ))

( ( )) ( ) ( ( ))

rt

rt rt rt rt rt

rt rt rt rt

rt rt

d e S t

e S t d e S t e S t d e S t d e S t

e S t e S t dW t e S t e S t dt

e S t dW t e S t dt

• On the other hand, (7.5.20) and (7.5.30) imply

• product rule implies

( ( )) ( ) ( )rt rt rtd e X t re X t dt e dX t

( ) ( ( ) ( ))

( ) ( ) ( )

rt

rt

t e dS t rS t dt

t e S t dW t

ˆ 'Ito s

1( ) [( ( ))( ( )) ]rt rtdY t d e X t e S t 1 1( ) [( ( )) ] ( ( )) ( ( ))rt rt rt rte X t d e S t e S t d e X t

1( ( )) [( ( )) ]rt rtd e X t d e S t 2 2( ) ( ) ( ) ( ) ( ) ( )

[ ( ) ( )][ ( ) ].

Y t dW t Y t dt t dW t t dt

t Y t dW t dt

(7.5.31)

• The process Y(t) is not a martingale under because its differential (7.5.31) has a dt term. However, we can change measure so that Y(t) is a martingale , and this will simplify (7.5.31). We set

• And then have

• According to Girsanov’s Theorem, Theorem 5.2.3, we can change the measure so that , ,is a Brownian motion. In this situation, plays the role of in Theorem 5.2.3, and and play the roles of W and P. the Radon-Nikodym derivative process of (5.2.11) is

( ) ( )SW t W t t (7.5.32)

( ) [ ( ) ( )] ( ).SdY t t Y t dW t (7.5.33)

P

( )SW t 0 t T W P

21( ) exp{ ( ) }.

2Z t W t t

• In other words,

• Under the probability measure defined by

is a Brownian motion and Y(t) is a martingale.

• Under the probability measure , the process Y(t) is Markov. It is given by the stochastic differential equation (7.5.33) is a function of t and Y(t) and has no source of randomness other than Y(t). Equation (7.5.33) is a stochastic differential equation of the type (6.2.1), and solutions to such equations are Markov (see Corollary 6.3.2).

( )( ) .

(0)

rte S tZ t

S

(7.5.34)

( ) ( )S

AP A Z T dP

SP

,A Ffor all

( )SW t

SP

• We return to the option price V(t) of (7.5.29) and use Lemma 5.2.2 to write (7.5.29) as

where denotes conditional expectation under the probability measure . Because Y is Markov under , there must be some function g(t,y) such that

( ) [ ( ) | ( )]rt rTV t e E e X T F t

( ) ( )[ ( )( ) | ( )]

( ) ( )

( )[ ( ) ( ) | ( )]

( )

( ) [ ( ) | ( )]

rtrT

rt rt

S

S t e X TE e S T F t

e S t e S T

S tE Z T Y T F t

Z t

S t E Y T F t

[ | ( )]SE F t SP SP

( , ( )) [ ( ) | ( )]Sg t Y t E Y T F t (7.5.36)

• From (7.5.36), we see that

• We note that can take any value since the numerator X(T), given by (7.5.27), can be either positive or negative , and the denominator S(T) can be any positive number. Therefore, (7.5.37) leads to the boundary condition

• The usual iterated conditioning argument shows that the right-hand side of (7.5.36) is a martingale under , and so the differential of g(t,Y(t)) should gave only a term. This differential is

( , ( )) [ ( ) | ( )] ( )Sg T Y T E Y T F T Y T (7.5.37)

( , ) ,g T y y y (7.5.38)

( , ( )) ( , ( )) ( , ( )) ( )t ydg t Y t g t Y t dt g t Y t dY t

1( , ( )) ( ) ( )

2 yyg t Y t dY t dY t

2 21[ ( , ( )) ( ( ) ( )) ( , ( ))]dt

2t yyg t Y t t Y t g t Y t

+ ( ( ) ( )) ( , ( )) ( )Syt Y t g t Y t dW t

( )( )

( )

X tY t

S t

SP( )SdW t

• We conclude that g(t,y) satisfies the partial differential equation

• Theorem 7.5.3 ( ).

For , the price V(t) at time t of the continuously averaged Asian call with payoff (7.5.16) at time T is

where g(t,y) satisfies (7.5.39) and X(t) is given by (7.5.24) and (7.5.26). The boundary conditions for g(t,y) are (7.5.38) and

2 21( , y) ( ( ) y) ( , y)=0,

2t yyg t t g t 0 t<T,y . (7.5.39)

Vecer

0 t T

X(t)V(t)=S(t)g(t, ),

S(t)(7.5.40)

y -lim g(t,y)=0, y

lim[g(t,y)-y]=0,

0 t T (7.5.41)

• We adapt the arguments just given to treat a discretely sampled Asian call. Assume we are given times and the Asian call payoff is

• We wish to create a portfolio process so that

• In place of (7.5.22), we define

m+

jj=1

1V(T)=( S(t )-K) .

m (7.5.42)

m

jj=1

1X(T)= S(t )-K.

m

i

m-r(T-t )

ji=j

(t )= e , j=0,1, ,m. (7.5.43)

• Then

and We complete the definition of by setting

• This defines for all In this situation, (7,5,21) still holds, but now in each subinterval Integrating (7.5.21) from to and using (7.5.44) and the fact that for we obtain

j-1-r(T-t )

j j-1

1(t )= (t )- e ,

m j=1, ,m. (7.5.44)

m

1(T)= (t )= .

m (t)

j(t)= (t ), j-1 jt <t t . (7.5.45)

(t) t [0,T].

d (t)=0 j-1 j(t ,t ) j-1t

jtj(t)= (t ) j-1 jt (t ,t ],

j j-1

j j-1

j j-1 j-1

j j-1

r(T-t ) r(T-t )

j j-1

r(T-t ) r(T-t )

j j j-1

r(T-t ) -r(T-t ) r(T-t )

j j j-1 j-1

r(T-t ) r(T-t )

j j j-1 j-1 j-1

e X(t )-e X(t )

= (t )[e S(t )-e S(t )]

1= (t )e S(t )-( (t )- e )e S(t )

m1

= (t )e S(t )- (t )e S(t )+ S(t ).m

• Summing this equation from j=1 to j=k, we see that

• We set

so this equation becomes

k

k

k

r(T-t ) r(T)k

kr(T-t ) r(T)

k k j-1j=1

k-1r(T-t ) r(T)

k k j-1j=1

e X(t )-e X(0)

1= (t )e S(t )- (0)e S(0)+ S(t ).

m

1 1= (t )e S(t )+ S(t )+(- (0)e + )S(0)

m m

-rT r(T) -rT1X(0)=e [ (0)e - ]S(0)-e K,

m

k k k

k-1r(T-t ) -r(T-t ) -r(T-t )

k k k i-1i=1

1e X(t )= (t )S(t )+e S(t )-e K.

m (7.5.46)

• In particular,

as desired.

• To determine X(t) for we integrate (7.5.21) from to t to obtain

m

m ii=1

1X(t)=X(t )= S(t )-K

m (7.5.47)

k kr(T-t ) r(T-t )r(T-t) r(T-t)k k+1 ke X(t)=e X(t )+ (t )[e S(t)-e S(t )]

k

k-1r(T-t ) r(T-t)

k k i k+1i=1

1= (t )e S(t )+ S(t )-K+ (t )e S(t)

m

k k-r(T-t ) r(T-t )k k

1-( (t )- e )e S(t )

m

kr(T-t)

k+1 ii=1

1= (t )e S(t)+ S(t )-K.

m

k k+1t t t , kt

• Therefore,

We now proceed with the change of numeraire as before. This leads again to Theorem 7.5.3 for the discretely sampled Asian call with payoff (7.5.42). The price at time t is given by (7.5.40), where g(t,x) satisfies (7,5,39) with boundary conditions (7.5.38) and (7.5.41). The only difference is that now the nonrandom function appearing in (7.5.39) is given by (7.5.43) and (7.5.45) and the process X(t) in (7.5.40) is given by (7.5.48).

k-r(T-t) -r(T-t)

k+1 ii=1

1X(t)= (t )S(t)+e S(t )-e K,

m k k+1t t t

(t)

5.2 Risk-Neutral Measure5.2.1 Girsanov’s Theorem for a Single Browni

an Motion

• Thm1.6.1:probability space Z>0；E[Z]=1

We defined new probability measure

(5.2.1)

Any random variable X has two expectations:

( , , )F P

( ) ( ) ( )

AA Z dP for all A F

EX=E[XZ] (5.2.2)

5.2.1 Girsanov’s Theorem for a Single Brownian Motion

• If P{Z>0}=1， then P and agree which sets have probability zero and (5.2.2) has the companion formula

Z is the Radon-Nikody’m derivative of

w.r.t. P, and we write

P

[ ] (5.2.3)X

EX EZ

P dPZ

dP

5.2.1 Girsanov’s Theorem for a Single Brownian Motion

• Suppose further that Z is an almost surely positive random variable satisfying E[Z]=1, and we define by (5.2.1). We can then define the

• We perform a similar change of measure in order to change mean, but this time for a whole process.

Radon Nikody m derivative process

( ) [ | ( )], 0 (5.2.6)Z t E Z F t t T

Lemma 5.2.1

• Lemma 5.2.1. Let t satisfying be given and let Y be an -measurable random variable. Then

• Proof: (5.2.2) (unbiasedness) ~ ( ) 5.2.6

Y = [ ] [ [ | ( )]] [Y [ | ( )]] = [ ( )]

Y F t

E E YZ E E YZ F t E E Z F t E YZ t

0 t T ( )F t

Y= [Y ( )] (5.2.8)E E Z t

Lemma 5.2.2

• Lemma 5.2.2. Let s and t satisfying be given and let Y be an -measurable random variable. Then

0 s t T ( )F t

1[Y| ( )]= [Y ( ) | ( )] (5.2.9)

( )E F s E Z t F s

Z s

Lemma 5.2.2

• PROOF :

: measurable

We must check the partial-averaging property (Definition 2.3.1(ii)), which in this case is

the left hand side of(5.2.10)

1[Y ( ) | ( )]

( )E Z t F s

Z s( )F s

A

1[Y ( ) | ( )] ( ) (5.2.10)

( ) AE Z t F s dP YdP for all A F s

Z s

1[ [Y ( ) | ( )]

( )AE I E Z t F sZ s

5.2.1 Girsanov’s Theorem for a Single Brownian Motion ( Lemma 5.2.2)

(Lemma 5.2.1.)

( )

unbiasedness (2.3.25)

5.2.1

1[ [Y ( ) | ( )]

( )

= [ [Y ( ) | ( )]]

= [ [ Y ( ) | ( )]

= [ Y ( )]

= [ Y]

=

A

A

A F s

A

A

lemma

A

A

E I E Z t F sZ s

E I E Z t F s

E E I Z t F s

E I Z t

E I

YdP

Girsanov’s Theorem

5.2.1 Girsanov’s Theorem for a Single Brow

nian Motion( Girsanov, one dimension)

• Using Levy’s Theorem : The process starts at zero at t=0 and is continuous.

• Quadratic variation=t

• It remain to show that is a martingale under

W

2( ) ( ) ( ( ) ( ) ) ( ) ( )dW t dW t dW t t dt dW t dW t dt

( )W t

P

5.2.1 Girsanov’s Theorem for a Single Brownian Motion( Girsanov, one dimension)

• Check Z(t) to change of measure.

• We take

and Z(t)=exp{X(t)} ( f(X)=exp{X})

• By Ito’s lemma( next page):

2

0 0

1( ) ( ) ( ) ( )

2

t tX t u dW u u du

' ''( ) , ( ) , ( )x x xf x e f x e f x e

5.2.1 Girsanov’s Theorem for a Single Brownian Motion( Girsanov, one dimension)

• no drift term>>martingale

' ''

( ) 2 ( ) 2

( ) ( ( ))

1( ( )) ( ) ( ( )) ( ) ( )

21 1

( ( ) ( ) ( ) ) ( )2 2

( ) ( ) ( )

X t X t

dZ t df X t

f X t dX t f X t dX t dX t

e t dW t t dt e t dt

t Z t dW t

5.2.1 Girsanov’s Theorem for a Single Brownian Motion( Girsanov, one dimension)

• Integrating>>

• Z(t) is Ito’ integral >>Z(t)~ martingale

• So ,EZ=EZ(T)=Z(0)=1.

• Z(t) is martingale and Z=Z(T),we have

• Z(t) is a

0( ) (0) ( ) ( ) ( )

tZ t Z u Z u dW u

( ) [ ( ) | ( )] [ | ( )], 0Z t E Z T F t E Z F t t T

Radon Nikody m derivative process

5.2.1 Girsanov’s Theorem for a Single Brownian Motion( Girsanov, one dimension)

• We next show that is martingale under P

• is a martingale under

( ) ( )W t Z t

It'o product

( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ( ) ( ) )( ( ) ( ) ( ))

( ( ) ( ) 1) ( ) ( )

d W t Z t W t dZ t Z t dW t dW t dZ t

W t t Z t C Z t dW t Z t t dt

dW t t dt t Z t dW t

W t t Z t dW t

( )W t P

5.2.21 1

[ ( ) | ( )] [ ( ) ( ) | ( )] ( ) ( ) (s)( ) ( )

martingalelemma

E W t F s E W t Z t F s W s Z s WZ s Z s