General American Derivatives

102 4 American Derivative Securities

[{r� N} ( 1 + r)T- N GT 1 = [{ r=N} GN .

In order to make this as large a.'i possible, we should choose r(^w1 . . . WN) = ^N if GN (WI . . . wN ) > 0 and r(w1 . . . wN) = oo if GN (WJ . . . wN ) � 0. With this choice of _r, we have ll { r=N } GN = max{GN , 0} , and (4.4.2) is established.

Before working out other consequences of Definition 4 .4 . 1 , we rework Ex­

ample 4.2. 1 to verify that this definition is consistent with the American put prices obtained in that example.

Go = 1

Fig. 4.4. 1 . Intrinsic value.

Example 4.2. 1 continued For the American put with strike 5 on the stock given in Figure 4 . 2 . 1 , the intrinsic value is given in Figure 4.4. 1 . In this example, N = 2 and ( 4.4.2) becomes

We next apply ( 4.4. 1 ) with n = 1 , first considering the case of head on the first toss. Then

(4.4.3) To make the conditional expectation on the right-hand side of (4.4.3) as large as possible, we should take r(HH ) = oo (do not exercise in the case of HH) and take r(HT) = 2 (exercise at time one in the case of HT). The decision of whether or not to exercise at time two is based on the information available at time two, so this does not violate the property required of stopping times.

This exercise policy makes (4.4.3) equal to 1 1

(

⁴

)

²

-

¹

V1 (H)

=

^{- ·} 2 ^{0 + -}2 ^{· -}5 ^{G2(HT) = 0.40.}

4.4 General American Derivatives 103 We next apply (4.4.1) with n

= 1

when the first toss results in a tail. In this case,

v.{T) = �� iE. [ ^n{r$2} (�) ^{T-l Gr} ] ^{(T). ( 4.4.4)}

To make this conditional expectation as large as possible, knowing that the first toss results in a tail, we must consider two possibilities: exercise at time one or exercise at time two. It is clear that we want to exercise at one of these two times because the option is in the money regardless of the second coin toss. If we take

r(T H) = r(TT) = 1,

then

_

[ ( ⁴ ) ^r-

^l

l

lEt I(T�2}

₅

GT (T) = Gt(T)

= 3.

If we take

-r(T H) = r(TT) = 2,

^then

We cannot choose

r(T H)

=

1

and

r(TT) = 2

because that would violate the property of stopping times. We see then that

{4.4.4)

yields

V1(T)

⁼ 3.

Finally, when _n = 0, we have

Vo

= max iE

[n{r<2} (�)TaT]

⁰

TESn - ₅ ^{4.4.5)

There are many stopping times to consider (see Exercise

4.5),

but a moment's reflection shows that the one that makes iE

[Htr�2} _(�r Gr]

as large as possi­

ble is

r(HH)

= oo,

r(HT)

=

2, r(TH) = r(TT) = 1. (4.4.6)

With this stopping time,

(4.4.5)

becomes

1 1 ( ⁴ ) ^{2 1 4 1 16 1 4}

^°

V0 = -·0+- - G2{HT)+-·-·G.(T) = -·-·1+-·-·3= 1.36. (4.4 7)

4 4 5 2 5 4 25 2 5

^°

We record the option prices in Figure

4.4.2.

These agree with the prices in Figure

4.2.2,

the only difference being that in Figure

4.2.2

these prices are recorded as functions v,. of the underlying stock prices and here they are recorded as random variables (i.e., functions of the coin tosses). The prices in the two figures are related _by the formula v

..

^{= v}

..

^{(S ..}

⁾

^{. 0}

We now develop the properties of the American derivative security price process of Definition 4.4.1. These properties justify calling

V,.

defined by

( 4.4.1) the _price of the derivative security.

10·1 4 American Derivative Securities

Fo = 1 .36

h ( H ) = 0.-!0

/

/ �

'\!2 (HT) = V2 (TH) = 1

�

_{l 't (T) = 3}

/

�

Fig. 4.4.2. American put prices.

Theorem 4.4.2. The American derivative security price process given by Definition 4.4. 1 has the following properties:

(i) v�, 2 max{

G, ,

0 } for· all n ;

(ii} the discounted process _{( l}

�

^{r)" \�, i8 a} suw-:rmar·tingale;

(iii) if }�, is another process satisfying }�, 2 max{

Gn .

0} for all n and for which

(1+\)" Yn

is a supermartingale. then }�, 2

Vn

for all n .

We summarize property (iii) b y saying that V,, is the smallest pmcess satisfy­

ing (i) and (ii}.

We shall see that property (ii) in Theorem 4.4.2 guarantees that an agent beginning with initial capital Vo can construct a hedging portfolio whose value at each time n is V,, . Property ( i ) guarantees that if an agent does this, he has hedged a short position in the derivativc security; no matter when it is exercised, the agent's hedging portfolio value is sufficient to pay off the derivative security. Thus, (i) and ( ii) guarantee that the derivative security price is acceptable to the seller. Condition (iii) says that t he price is no higher than necessary in order to be acceptable to the seller. This condition ensures that the price is fair for the buyer.

P ROOF: We first establish (i) . Let 11 be given, and consider the stopping time 7 in

Sn

that takes the value n, regardless of the coin tossing. Then

Since V,, is the largest possible value we can obtain for

En [n{,.SN}

^{( l +r}

^\

^T

^{" G,.} ]

when we consider all stopping times T E

S11,

we must have V, 2

Gn .

^{On the}

other hand, if we take 1' to he the stopping time in

S.,

that takes the value oo^, regardless of the coin tossing, then

I

4.4 General American Derivatives 105 En

[n{r�N} (1

₊

� ^{)r -}

ⁿ

^cT]

^{= 0.}

Again, Vn is the maximum of expressions of this type, and hence Vn � 0. We conclude that ( i ) holds.

We next prove (^{ii). Let n} be given, and suppose T* attains the maximum in the definition of Vn+l ; i.e., T* E Sn+ l and

Vn+l = En +l

[n{r·�N} (1 + r)1r•-n-! Gr·]

^·

But T* E Sn also, which together with iterated conditioning implies V., � En

[n{r·�N} (1

+

: )r

^•

_-

n

Gr•]

= En

[

^En+!

[n{r· �N} (1 + :)r•-n Gr• ]]

= En

L:

^{r En+I}

[n{r·�N} (1 + _{r):• -n-l} Gr·] ]

= En

[-

_{1 + r}

1

Vn+!

]

^.

(

4.4.8)

( 4.4.9) Dividing hoth sides hy

(1 + r)",

we obtain the supermartingale property for the discounted price process:

(1 : r)"

Vn � En

[(1 + �

) n+ l Vn+ I

]

^.

Finally, we prove (iii ) . Let

Yn

be another process satisfying conditions (i) and ( i i ) . Let n S N be given and let ^T be a stopping time in _{S11 •} Because Yk _� max{Gk , O} for all k, we have

IT{r�N}Gr

S

IT{r�N}

max{

Gr

_, 0}

:::; K{r�N)

max{

GNAT,

0 }

+ n{r=oo}

max{

GN

II n 0 }

=

_max{

GN

^{A T}, O} s YNIIT ·

We next use the Optional Sampling Theorem 4.3.2 and the supermartingale property for

(l_;r)k Yk

to write

En

[n{r�N) (1 : r)r Gr]

^{= En}

[n{r�N) (1 + �)l'lllr Gr]

:S En

[(1 + �)NAT YN��r]

<

1

- (1 + r)nllr Ynllr

=

(l + r)" "'

1 Y.

- - - - - - - - - - - ----�

106 4 American Derivative Securities

the equality at the end being a consequence of the fact that r E

Sn

is greater than or equal to n on every path. Multiplying by

( 1

^{+ r )}

n ,

^{we obtain}

En [rr{r<N} ^{- (} ₁

+ ^{1 )} 1"

r-n cT]

^{:::; Yn .}

Since

V ⁿ

i s the maximum value we can obtain for

En [rr{r:<:;N} (l+r\T n eT]

^as

r ranges over

Sn , a

ⁿd all these values arc less than or equal to }�, , ^{Wf' must}

have

Vn

^{:::; Yn . 0}

We now generalize the American pricing algorithm given by

(4.2.5)

^and

(4.2.6)

to path-dependent securities

Theorem 4.4.3. We have the following Am erican pricing algorithm for the path-dependent derivative security price process given by Definition 4 . 4 . 1 :

(4.4. 10) V,,(w1

^{. . •}

w, ) = max { _{G, (w1}

• • •

w, ) , (4.4. 1 1 )

- 1- [_pv;, + 1 (w1 . . . w,H)

⁺

qVn+l (wl . . . w,T)] } ^,

1 + 1"

for n = N -

1 , .. . , 0.

PROOF: We shall prove that

V ⁿ

defined recursively by

(4.4. 10)

^and

(4.4.1 1 )

satisfies properties (i) and (ii) of Theorem

4.4.2

and is the smallest process with these properties. According to Theorem

4.4.2,

the process

V,,

^{given by}

( 4.4. 1 )

is the smallest process with these properties, and hence the algorithm

(4.4. 10), (4.4. 1 1 )

must generate the same process as formula

(4.4.1 ) .

We first establish property ( i ) of Theorem

4.4.2.

It is clear that

VN

^de­

fined by

(4.4. 10)

satisfies property (i) with n = N . We proceed by induction backward in time. Suppose that for some n between

0

^{and N} -

1

^{we have}

Vn + l 2': max{Gn+l · O} .

Then, from

(4.4. 1 1 ) ,

we see that

V, (w1

^{. . • w,}₎ 2':

max {Gn (WJ . . . w, ) . O} .

This completes the induction step and shows that V,, defined recursively by

(4.4. 10). (4.4. 1 1 )

satisfies property (i) of Theorem

4.4.2.

We next verify

that (l.;r)"

V, , is a supermartingale. From

(4.4.1 1) ,

^{we see}

immediately that

V,,(w1

^{. . •}

w11 ) 2': -1- [_pv;,+ , (w, . . . W11 H) 1

^{+ ,. +}

q\�,+1 (w, . . . w,T)]

=

E

^,

[ - 1- V,, + 1 ] (w1 . . . w,). (4.4. 12) 1

^{+ ,.}

·Multiplying bot h sides by

(l.;rJ" ,

we obtain the desired superrnartingale prop­

erty.

4.4 General American Derivatives ₁₀₇ Finally, we must show that

Vn

defined by (4.4.10), (4.4.11) and satisfying (i) and (ii) of Theorem 4.4.2 is the smallest process satisfying (i) and (ii). It is immediately clear from (4.4.10) that

VN

is the smallest random variable satisfying

VN

_;::: ma.x{GN,O}. We proceed by induction backward in time.

Suppose that, for some n between 0 _{and N -1,}

Vn+

1 is as small as possible. The supermartingale property (ii) implies that

Vn

must satisfy (4.4.12). In order to satisfy property (i),

Vn

must be greater than or equal to

Gn.

Therefore, properties (i) and (ii) of Theorem 4.4.2 imply

Vn(Wt . . . wn) ;:::

max

{ ^Gn(Wt

^{. . . wn), (4.4.13)}

1

�

^r

[iWn+ l (wl

^{· · · WnH) +}

qVn+l (wl . . . WnT)] }

for n = N - 1, .. . ,0. But (4.4.11) defines

Vn(w, . . . wn)

to be equal _to the right-hand side of (4.4.13), which means

Vn(w1

• • .

wn)

is as small as possible.

0

In order to justify Definition 4.4.1 for American derivative security prices, we must show that a short position can be hedged using these prices. This requires a generalization of Theorem 4.2.2 to the path-dependent case.

Theorem 4.4.4 (Replication of path-dependent American deriva­

tives).

Consider an N -period binomial asset-pricing model with 0 _< d _<

1 + r _< u and with

_ 1 + r - d

p = ---­_{u - d '} u - 1 - r

q = .

u - d

For each n, n ₌ 0, 1 , . .. ^{, N, let} G .. be a random variable depending on the first n coin tosses. With

Vr.,

n _{= 0,} 1 , . . . , N, given by Definition 4.4.1, we

define

(4.4.14)

where n ranges between 0 and N -1. We have

Cn ;:::

0 for all n. If we set X0 =

Vo

and define recursively forward in time the portfolio values X1, X2,

.

. .

, XN

by

(4.4.16) then we have

Xn(WJ . . . w,.)

₌

Vn(wl . . . wn)

(4.4.17) for all n and all

Wt . . . w,..

In particular,

Xn

_;::: G,. for all n.

108 4 American Derivatiye Securities

PROOF: The nonnegativity of

Cn

is a consequence of property (ⁱⁱ) of Theorem 4.4.2 or. equivalently, (4.4. 1 2 ) .

To prove (4.4. 1 7 ) , we proceed h�· induction o n n . This part o f the proof is the same as the proof of Theorem 2.4.8. The induction hypothesis is that x1l

(wl . . . wn ) = Vn (W1 . . . Wn )

^{for some n E} {0, 1 , . . .. ^N - 1 } and all

w1 . . . '-'-''n·

We need to show that

X.,+J (w1 . . . w11 H) = Fn+dw1 . . . w" H).

Xn+ 1 ('-'-''1 . . . u.,•,.T) = V,.+ J ("-·1 . . . wnT).

We prove (4.4. 18); the proof o f (4.4.19) i s analogous.

Note first that

Vn ("-'1 . . . w,. ) - C, (Wt

^.

.

^. :...111 )

=

₁

:

^r

[iH�•+1 (wt . . . w,H) + qVn+1 (:...�1 . . . W11T) ] _.

(

^{4.4. 18)}

(4.4. 19)

Since

Wt . . . '-'-''n

will be fixed for the rest of the proof, we will suppress these symbols. For example, the last equation will be written simply as

We compute

Xn+1 (H) = L1n 8n+dH)

^{+ ( 1}

+ r)(Xn - Cn - L1n8n)

=

Vn+t (H) - Fn+1 (T) (Sn+t (H) -

^{( 1}

+ r)Sn) Sn+dH) - Sn+t (T)

+ ( 1 + r)(V,, -

Cn)

V,,+t (H) - V,,+ 1 (T) ₍

_{· S}

_-

_{( 1 +}

·)8 ) (

^{U. _}

d)Sn

^{ll 11 7}

n +pVn+1 (H) + qV,+ J (T)

u. - 1 - r

= (Fn+ t (H) - "�• +dT)) u. - d + fiVn+ 1 (H) + qV,,+I (T)

= (l�•+ J (H) - \�•+t (T))q

⁺

pV,, + t (H) + q\�,+1 (T)

= (p + q) Vn+1 (H) = l'n+ 1 ( H).

This is (4.4. 18).

The final claim of the theorem. that

X11

^{2 G, for all n ,} follows from

(4.4. 17) and property (i) of Theorem 4. 1.2. 0

Theorem 4.4.4 shows that t he American derivative security price given by (4.4. 1 ) is acceptable to the seller because he can construct a hedge for the short position. We next argue that it is also acceptable to the buyer. Let us fix n , imagine we have gotten to time n without the derivative security being

4.4 General American Derivatives 109

exercised, and denote by r" E Sn the stopping time that attains the maximum in (4.4.1), so that

(4.4.20)

For k = n, n + 1, .. . , N, define

If the owner of the derivative security exercises it according to the stopping .

time r* , then she will receive the cash flows Cn, Cn+l , . . . , C N at times n, n + 1, . . . , N, respectively. Actually, at most one of these Ck values is non-zero.

If the option is exercised at or before the expiration time N, then the Ck corresponding to the exercise time is the only nonzero payment among them.

However, on different paths, this payment comes at difrerent times. In any case, (4.4.20) becomes

We saw in Theorem 2.4.8 of Chapter 2 that this is just the value at time n of the cash flows Cn, Cn+l• . . . , CN, received at times n, _n+l, .. . , N, respectively.

Once the option holder decides on the exercise strategy ^{r" ,} this is exactly the contract she holds. Thus, the American derivative security price Vn is acceptable to her.

It remains to provide a method for the American derivative security owner to choose an optimal exercise time. We shall consider this problem with n ₌ 0 (i.e., seek a stopping time T* E So that achieves the maximum in (4.4.1) when n ₌ 0).

Theorem 4.4.5 (Optimal exercise). The stopping time

T. = min{n; Vn = Gn}

maximizes the right-hand side of (4.4.1) when n = 0; i.e.,

{4.4.21)

(4.4.22) The value of an American derivative security is always greater than or equal to its intrinsic value. The stopping time r* of (4.4.21) is the first time these two are equal. In may be that they are never equal. For example, the value of an American put is always greater than or equal to zero, but the put can always be out of the money (i.e., with negative _intrinsic value). In this case, the minimum in (4.4.21) is over the empty set (the _set of integers _n for which Vn = Gn is the empty set), and we follow the mathematical convention

1 1 0 4 American Derivative Securities

that the minimum over the empty set is oc . For us, r* ^{= oo} is synonymous with the derivative security expiring unexercised.

PROOF OF THEOREM 4.4.5: We first observe that the stopped process 1 ^v.

{1 ^{+ r})n r.-r • nl\ r • ( 4.4.23)

is a martingale under the risk-neutral probability memmre. This is a conse­

quence of

(

_{4.4.11 )}

.

Indeed, if the first n coin tosses result in

w1 . . . Wn

and along this path r• 2: n + 1, then we know that

Vn (wt . . . wn )

_>

Gn (w1 . . . wn )

and (4.4.11) implies

Vnr.-r• (wi

.

.

· U..'n)

=

Vn (W J . . . wn )

= -_1-

[fiV,, + l (wl . . . w,. H)

₊

qVn + J (wJ . . . WnT)]

l + r

= -l + r

1- [fi\J(n+ l )r.-r • (wJ . . . WnH)

₊

q\J(n+J)I\r• (wl

_... WnT)

]

^.

This is the martingale property for the process

(

4.4.23). On the other hand, if along the path

w1

. • . Wn we have r• ::; n, ^then

V,,r.-r· (wJ

..

.

_Wn)

=

Vr• (wt . . . Wr• )

= p\1:,.. (u..'l ^{· · ·} Wr•) +

q\fr• (u..'l

· · · Wr• )

=

P\l(n+J)I\T• (wl .

_.

.

_{WnH) +} q\J(n+I)I\T" ⁽wl . . . WnT). Again we have the martingale property.

Since the stopped process (4.4.23) is a martingale, we have Vo ⁼IE

[

^{(l +}

1\N/\T·

^VNI\T·

]

=IE

[

^{nfr• $N} (1 +1r)r•}

^Gr· ]

^{+ lE}

[rr{r·=oo}

⁽¹ ₊

\

^{·)N VN}

]

. (4.4.24) But on those paths for which

r*

^{= oc.} we must have Vn ^{> Gn for} all n and, in particular, VN ^>

GN.

In light of (4.4.10), this can only happen if

GN

_< 0 and VN ⁼ 0. Therefore,

ll{r· =ex>}

_VN = 0 and (4.4.24) can be simplified to

( 4.4.25)

This is (4.4.22). ⁰

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