Non-Path-Dependent American Derivatives

In this section, we develop a. pricing algori�hm for American derivative se­

curities when the payoff is not path dependent. We first review the priciug algorithm for European derivative securities when the payoff is not path de­

pendent. In an N-period binomial model with up factor -u, down factor d, and interest rate r satisfying the no-arbitrage condition 0 < d < 1 + r < u, con­

sider a derivative security that pays off g(SN) at time N for some function g.

Because the stock price is Markov, we can write the value

Vn

of this derivative security at each time _{n as a} function Vn of the stock price a.t that t.ime i.e.,

V, = 1111(511), ^{n =} 0, 1, ^{. . .} , N (Theorem 2.5.8 of Chapter 2). The risk-neutral pricing formula (sec (2.4.12) and its Markov simplification (2.5.2) of Chapter 2) implies that, for 0 ::::; n ::::;

N,

the function Vn is defined by the European algorithm:

VN(s)=max{g(s), 0},

v, (s )=- 1 - [iivn+1

(us) + ijv,+l (ds)

]

, n

= N

-1, N -2, . . . , 0.

l + r

(4.2.1) (4.2.2)

where

p = _1t:'dd

and ij ₌

";:-:;tr

arc the risk-neutral probabilities that the stock goes up and down, respectively. The replicating portfolio (which hedges a short position in the option) is given by (see (1.2.17) of Chapter I)

A ^_ Vn+l(uSn) - Vn+t(dS,,) _ O l

N Lln

-(u ^_ d)S, , 7t -

,

, · · · , · (4.2.3) Now consider an American derivative security. Again, a payoff function g

is specified. ln any period n :5 N, the holder of the derivative security can

4.2 Non-Path-Dependent American Derivatives

91

exercise and receive payment

g(Sn)·

(In this section, the payoff depends only on the current stock price

Sn

at the time of exercise, not on the stock price path.) Thus, the portfolio that hedges a short position should always have value X11 satisfying

X

n

_2::

g(Sn),

_n = 0, 1,

. . .

, N.

(4.2.4)

The value of the derivative security at each time

n

is at least as much as the so-called

intrinsic value g(Sn),

and the value of the replicating portfolio at that time must equal the value of the derivative security.

This suggests that to price an American derivative security, we should replace the European algorithm

(4.2.2)

by the

American algorithm:

VN(s)

= max{g(s), O},

Vn(s)

= max

{ ^g(s),

¹

� _r [vvn+J(us)

+

QVn+l(ds)] } ^,

n = N

-

1, N

-

2, ^{. . .} , 0.

(4.2.5) (4.2.6)

Then

Vn

=

Vn(Sn)

would be the price of the derivative security at time

n.

So = 4

s.(H) _{= s}

/

�

Sz(HT) = Sz(TH) = 4

�

_{s.(T) = 2}

/

�

Fig. 4.2.1. A two-period model.

Example 4.2.1. In the two-period model of Figure

4.2.1,

let the interest rate be

1' =

i,

so the risk-neutral probabilities are p :::: q =

�.

Consider an American put option, expiring at time two, with strike price

5.

In other words, if the owner of the option exercises at time

n,

she receives 5 -Sn. ^{We take}

g(s)

=

5 - s,

and the American algorithm

(4.2.5), (4.2.6)

becomes v2(.s) = max{5 -

s,

0},

Vn(s)

= max

{ 5 -

^s,

_� [vn+l (2s)

+

Vn+J G)]},

^{n = 1,0.}

92 4 American Derivative Securities

In particular,

v2 ( 16)

= 0, 1'2 (4)

=

1 ,

t'2 ( 1 )

= 4 ,

v, (8) = max { (5 - 8 ) , � (0 + 1)} max{ -3. 0.40}

=

0.40, v1 (2) = max { (5 - 2 ) . � ( 1 + 4) } max{3. 2} = 3,

vo (4) = max { (5 - 4 ) . � (0.40 + 3) } = max { 1 , 1.36} = 1 .36.

This algorithm gives

a

different result than the European algorithm in the

!I] (8) ⁼ 0.40

/

vo(4) = 1 .36

�

t'J (2) = 3

Fig. 4.2.2. American put prices.

computation of

v1

(2) , where the discounted expectation of the time two option price, _{g (l} + 4) , is strictly smaller than the intrinsic value. Because vi (2) is strictly greater than the price of a comparable European put, the initial price v0 (4) for the American put is also strictly greater than the initial price of a comparable European put. Figure 4.2.2 shows that American put prices.

Let us now construct the replicating portfolio. We begin with initial capital 1 .36 and compute Ll0 so that the value of the hedging portfolio at time one agrees with the option value. If the first toss results in a head, this requires that.

0.40 =

v1 (SI (H ) )

= S1

(

H

) Ll

o + ( 1 + r ) ( Xo - LloSo)

4.2 Non-Path-Dependent American Derivatives 93

= 8Llo + 5 4 (1.36-4Llo)

= 3..10 + ^{1. 70,}

which implies that ..10 = -0.43. On the other hand, if the first toss results in a tail, we must have

3 ₌ vi(St(T))

= S1 (T)Llo + (1 + r)(Xo - LloSo)

= 2Llo + 4(1.36 - 4Llo) 5

= -3Llo ₊ 1. 70,

which also implies that ..10 ₌ -0.43. We also could have- found this value of ..10 by substituting into (4.2.3):

Llo ₌ V! (8) -8 - 2 V) (2) ₌ 0.408 - 2 -3 = -0.43.

In any case, if we begin with initial capital Xo = 1.36 and take a position of Llo shares of stock at time zero, then at time one we will have X1 ₌ V1 = v1 (S1 ), regardless of the outcome of the coin toss.

Let us assume that the first coin toss results in a tail. It may be that the owner of the option exercises at time 1, in which case we deliver to her the $3 value of our hedging portfolio and no further hedging is necessary. However, the owner may decline to exercise, in which case the option is still alive and we must continue hedging.

We consider in more detail the case where the owner does not exercise at time 1 after a first toss resulting in tail. We note that next period the option will be worth v2(4) ₌ 1 if the second toss results in head and worth v2(l) = 4 if the second toss results in tail. The risk-neutral pricing formula says that to construct a hedge against these two possibilities, at time 1 we need to have a hedging portfolio valued at

but we have a hedging portfolio valued at v1 (2) ⁼ 3. Thus, we may consume

$1 and continue the hedge with the remaining $2 value in our portfolio. As this suggests, the option holder has let an optimal exercise time go by.

More specifically, we consume $1 and change our position to ..11 (T) shares of stock. If the second coin toss results in head, we want

1 = V2(S2(TH))

= 4Lll(T) + 5 4(2 - 2Llt(T))

= l.SLl1 (T) + 2.50,

94 4 American Derivative Securit,ies

and this implies Ll1 (T) = -1. I f the second coin toss results in tail, we want

4 = v2(S2(TT))

= Ll1(T) + ^{4 (2 -}5 2Ll1(T))

= -1.5Ll1 (T) + 2.50.

and this also implies Ll1 (T) = - 1 . We could also have gotten this result direct!�' from formula (4.2.3):

Llt (T) ⁼ v2(4) - v2(l) ⁼ l -4 ⁼ _-1.

4 - 1 4 - l

For the sake of completeness, we cousider finally t.he case where the first t.oss results in head. At. ^time 1, we will have a portfolio valued at X1 (H) = 0.40.

We choose

Ll (H) = v2(l6)-v2(4) =

�

^{= _}

_!_.

I 16 - 4 16 -4 _{1 2}

If the second toss results in head, at ^time 2 the value of our hedging portfolio

is 5

X2(H H) = 16Ll1 (H) + ^-4 (0.40 -_{8Ll1 (H))} = 0 = v2(16).

If the second toss results in tail, at ^time 2 the value of our hedging portfolio

i^s 5

X2(HT) ⁼ 4L\t (H) + 4 {0.40 -8dl (H)) = 1 = V2(4).

� ^Ut (8) = 0.32

/

�

�1.12(16) _{= 0}

uo(4) ⁼ 1.36 � u2(4) = 0.64

�

_{�VI (2) = 2.40}

/

�

^{�1.12(1) = 2.56}

Fig. 4.2.3. Discounted American put prices.

Finally. we consider the discounted American _put prices in Fignre 4.2.3.

These constitute a supermartingale under the risk-neutral probabilities p =

4.2 Non-Path-Dependent American Derivatives 95 q =

�.

At each node, the discounted American put price is greater than or equal to the average of the discounted prices at the two subsequent nodes.

This price process is not a martingale because the inequality is strict at the time-one node corresponding to a tail on the first toss. ₀ The following theorem formalizes what we have seen in Example 4.2.1 and justifies the American algorithm (4.2.5), (4.2.6). We shall eventually prove the more general Theorems 4.4.3 and 4.4.4, which cover the case of path dependence as well as path independence, and thus do not pause to prove Theorem 4.2.2 below.

Theorem 4.2.2. (Replication of path-independent American deriva­

tives) Consider an N -period binomial asset-pricing model with 0 < ^{d <}

1 + r < u and with

_ l+r

-

^d

p = u - d '

u - 1 - r

ij =

_{u - d}

.

Let a payoff function g( s) be given, and define recursively backward in time the sequence of functions

V

N(s),

V

N-

t

(s), . . . , vo(s) by (4.2.5}, (4.2.6}. Next define

Ll

n -

^_

Vn+t(USn) - Vn+I (dSn)

_{(u - d)}

_Sn

'

Cn = v,.(S

^{.. ) - -}_{l + r} ^1-I.Pv

n

+ I(u

S n

₎ +

QVn+I(dSn)] ,

(4.2.7) ( 4.2.8) where n ranges between 0 and N - 1. We have C,. _2: 0 for all n. If we set

Xo

=

vo(So)

and define recursively forward in-time the portfolio values

X1, X2, . . . ,XN

by

Xn+t = ^LlnSn+t

^{+ ( 1 + r}

⁾

⁽

^{X n -}

^{C ..}

^{- Ll}

^.. S .. )^, then we will have

for all n and all

w1 . . . Wn·

In particular,

Xn

_2:

g(Sn)

for all n.

(4.2.9)

(4.2.10)

Equation (4.2.9) is the same as the wealth equation {1.2.14

)

of Chapter 1, except that we have included the possibility of consumption. Theorem 4.2.2 guarantees that we can hedge a short position in the American derivative security with intrinsic value

g(Sn)

at each time n. In fact, we can do _so and perhaps still consume at certain times. The value of our hedging portfolio

Xn

is always at least as great as the intrinsic value of the derivative security because of

(

⁴.2.10) and the fact, guaranteed by (4.2.6), that v ..

(Sn)

2:

g(Sn)·

The nonnegativity of

Cn

also follows from (4.2.6), which implies that Vn

(Sn)

^2: 1

�

^,.

^{!Pvn+l (}

^u

^S

^{,.) +}

^QVn+l (dS,. )J .

96 4 American Derivative Securities

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