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The Value of Being American

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?

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The Value of Being American

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  1. The Value of Being American Anthony Neuberger University of Warwick Newton Institute, Cambridge, 4 July 2005

  2. 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

  3. 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?

  4. 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

  5. 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

  6. 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

  7. 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

  8. Pay-off Diagram in a two period world K2 K1

  9. 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 K2 K1

  10. 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

  11. a 1-a Z2 Z1 X A Family of Dominating Strategies K2 K1

  12. 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 Z2 Z1 X

  13. 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

  14. A Bounding Process X Z1 Z2 Time: 0 1 2 Exerc I se

  15. 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

  16. 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%

  17. 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

  18. 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) K2 K1

  19. A Family of Dominated Strategies Z2 K2 Z1 X

  20. The Strategy • 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:

  21. 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

  22. A Bounding Process X Z1 Z2 Time: 0 1 2 Exerc I se

  23. Dominating Strategies in aMulti-period world

  24. Extreme Process

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