The Value of Being American Anthony Neuberger University of Warwick Newton 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 S t –no transaction costs, frictions –S unrestricted (allow negative) Risk free asset –constant, equal to zero American put A(K 1, …, K T ) –can be exercised once only –if exercised at t, pay-off is K t – S t –K t ’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(S t, t) at time t –define

Pay-off Diagram in a two period world K2K2 K1K1

Bounds Write American, buy Y where: Y = P(K 2, 2) + P(K 1, 1) - P(K 2, 1) If A exercised at time 1 and S 1 < K 2 buy the underlying to lock in the intrinsic value Strategy at least breaks even, and makes money if: –S 1 K 2 –or S 1  (K 2, K 1 ) and S 2 < K 2 K2K2 K1K1

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{ S 1 K 2 } have finite probability K2K2

A Family of Dominating Strategies Z2Z2 Z1Z1 1-a a X K2K2 K1K1

More Bounds Write American, buy Y(a, X) If A exercised at time 1 and S 1 < X buy (1-a) or 1 of underlying to lock in intrinsic value Strategy at least breaks even, and makes money if: –S 1 Z 2 –or S 1  (Z 2, Z 1 ) and {S 2 X} –or S 1 > X and S 1  (Z 2, X) –American option exercised prematurely (S 1 > Z 1 ) or too late Z2Z2 Z1Z1 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

A Bounding Process X Z 1 Z 2 Time: ExercIseExercIse

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(K 1, t 1 ) = P(K 2, t 2 ) Take S 0 = 100, t 1 = 1 year, t 2 = 2 years, all European options trading on BS implied vol of 10% –implicit interest rate 1½ - 6½%

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(K 1, 1) If S 1 < K 1 exercise the American and pay off Y Strategy at least breaks even, and makes money if: –S 1 > K 1 and S 2 < K 2 –(or if value of P(K 2, 2) at time 1 exceeds K 1 – S 1 ) K2K2 K1K1

A Family of Dominated Strategies XK2K2 Z1Z1 Z2Z2

The Strategy Buy American, sell Y; at time 1: –if S 1 < Z 1 exercise the American and pay off maturing options, receive from Y at t=2 –if S 1 > 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:

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

A Bounding Process X Z 1 Z 2 Time: ExercIseExercIse

Dominating Strategies in a Multi-period world

Extreme Process