The Value of Being American

Anthony Neuberger

University of WarwickNewton Institute, Cambridge, 4 July 2005

Objective

• What is the value of American as opposed to European-style rights?– given a complete set of European options for all

relevant maturities, how cheap/dear can the American option be without permitting arbitrage?

– these are arbitrage bounds, no assumptions about nature of price path

Motivation

• How close are American options to European options?

• What are the determinants of the value of being American?

• Is conventional valuation biased downwards?– if holder is required to pre-specify exercise

strategy, value is not diminished• How to hedge American options?

American options

• The commonest, and most complex of exotics– pay-off depends not only on path but on strategy, and

strategy depends on beliefs about possible parths?

• Discrete time framework– for much of the seminar, just times 0, 1 and 2

Outline

• The General Set-up• A two period world

– an upper bound– testing for rational bounds– the supremum and the bounding process– some numerics– the lower bound

• A multi-period world

CAUTION: results are preliminary and some are mere conjectures

The Model

• Discrete time t = 0, 1, …, T• Risky underlying, price St

– no transaction costs, frictions– S unrestricted (allow negative)

• Risk free asset– constant, equal to zero

• American put A(K1, …, KT)– can be exercised once only– if exercised at t, pay-off is Kt – St

– Kt’s strictly positive and strictly decreasing

European Puts

• There are European puts P(K, t)– for all real K– for all t from 1 to T

• P denotes both the claim and its time 0 price• Y is a portfolio of European puts

– it pays y(St, t) at time t

– define

, ,T

u t

Y x t y x u

Pay-off Diagram in a two period world

S

Pa

y-o

ff (Y

) t1

t2

K2 K1

Bounds

• Write American, buy Y where:

Y = P(K2, 2) + P(K1, 1) - P(K2, 1)

• If A exercised at time 1 and S1 < K2 buy the underlying to lock in the intrinsic value

• Strategy at least breaks even, and makes money if:– S1 < K2 and S2 > K2

– or S1 (K2, K1) and S2 < K2

S

Pay

-off

t1

t2

K2 K1

Tightest bound

• Is Y the best we can do?• If there is a martingale process for

S such that:– the expected pay-off to every

European option is equal to its price– and there is zero probability of

money-making paths

then it must be the best we can do

• But if picture as on right, paths with{S1 < K2 and S2 > K2} have finite probability

K2

A Family of Dominating Strategies

S

Pa

y-o

ff (Y

) t1

t2

Z2 Z1

1-a

a

X

2 1 2

2

, 1 ,2 ,2 ,1 ,1

where 1 ; [0,1) and i i

Y a X a P X aP Z aP Z aP Z

K a X aZ a X K

K2 K1

More Bounds

• Write American, buy Y(a, X)

• If A exercised at time 1 and S1 < X buy (1-a) or 1 of underlying to lock in intrinsic value

• Strategy at least breaks even, and makes money if:– S1 < Z2 and S2 > Z2

– or S1 (Z2, Z1) and {S2 < Z2 or S2 > X}

– or S1 > X and S1 (Z2, X)

– American option exercised prematurely (S1 > Z1) or too late

S

Pa

y-o

ff (Y

) t1

t2

Z2 Z1 X

Rational Bounds

• Cheapest bounding strategy found by choosing a, X to minimise Y(a, X)– foc’s are

– if foc’s are satisfied (and subject to regularity conditions), there is a martingale process with no weight on money-making paths

– then the corresponding Y must be the least upper bound on the American option

1 1

2 2 2 21 2 1 2;

where ( , ) /

Z X Z X

Z Z Z Z

t

dF x dF x xdF x xdF x

F x P x t x

A Bounding Process

X

Z1

Z2

Time: 0 1 2

Exerc I se

Intuition

• The European option prices determine the marginal distributions at times 1 and 2, but not the paths

• Europeans determine the average volatility, but not distribution across paths

• Volatility is wasted if American option already exercised

• So seek to find process that puts maximum volatility on high paths where option has not been exercised

Does it matter?

• Look at Bermudan options (2 dates) which are “atm” in sense that P(K1, t1) = P(K2, t2)

• Take S0 = 100, t1 = 1 year, t2 = 2 years, all European options trading on BS implied vol of 10%– implicit interest rate 1½ - 6½%

0

2

4

6

8

10

12

14

90 95 100 105 110

Strike (K1)

Eur putsBS Price

Naïve BoundRatl Bound

Suggestive Implications

• Bounds are wide– early exercise premium (Am-Eur) could be

worth twice the Black-Scholes value– but naïve bounds very close to rational

bounds

• Very preliminary – need to test over range of parameters

Lower Bound

• Buy American, sell Y where:

Y = P(K1, 1)

• If S1 < K1 exercise the American and pay off Y

• Strategy at least breaks even, and makes money if:– S1 > K1 and S2 < K2

– (or if value of P(K2, 2) at time 1 exceeds K1 – S1)

S

Pay-o

ff t1

t2

K2 K1

S

Pa

y-o

ff (Y

)

t1t2

A Family of Dominated Strategies

1 2 2

2

, 1 ,1 ,1 1 ,2 ,2 ,2

where 1 ; [0,1) and i i

Y a X a P X aP Z a P X P K aP Z

K a X aZ a X K

XK2 Z1Z2

S

Pay-

off (

Y)

t1t2

The Strategy

1

2 2

, 1 ,1 ,1

1 ,2 ,2 ,2

Y a X a P X aP Z

a P X P K aP Z

• Buy American, sell Y; at time 1:– if S1 < Z1 exercise the American and pay off

maturing options, receive from Y at t=2

– if S1 > X, do nothing and receive at t=2

– otherwise, buy 1-a of underlying, exercise American if in the money at t=2

• Strategy at least breaks even, and makes money if:

1 1 2 2 2

1 1 2 2

1 2

and or

or , and ,

or and

S Z S Z S X

S Z X S Z X

S X S X

Rational Lower Bound

• Cheapest bounding strategy found by choosing a, X to maximise Y(a, X)– foc’s are

– if foc’s are satisfied (and subject to regularity conditions), there is a martingale process with no weight on money-making paths

– then the corresponding Y must be the greatest lower bound on the American option

1 2 1 21 2 1 2;

where ( , ) /

X X X X

Z Z Z Z

t

dF x dF x xdF x xdF x

F x P x t x

A Bounding Process

X

Z1

Z2

Time: 0 1 2

Exerc I se

Dominating Strategies in aMulti-period world

S

Pay

-off

(Y

)

t1

t2

t3

t4

Extreme Process

Zombie zone

Diffusion zone

Jump Zone

Time

Ass

et P

rice