1. Carlos.veiga@millenniumbcp.pt 1 MathFinance Colloquium Frankfurt, June 1 st, 2006 Exploring the Limits of Closed Pricing Formulas in the Black and Scholes Framework by Carlos Veiga

2. 2 Part I Exotic Options

3. Carlos.veiga@millenniumbcp.pt 3 Development of a Closed Pricing Formula for a Generic Exotic Option. Forecast The Generic Exotic Option includes, among others: Options on maximum of several assets, options on the 2 nd best of several assets, target redemption options, and more …

4. Carlos.veiga@millenniumbcp.pt 4 Outline Background The Result Applications Summary

5. Carlos.veiga@millenniumbcp.pt 5 Background Motivation New exotic options appear constantly Monte Carlo methods are slow with several underlying assets Hedge parameters are unstable using Monte Carlo Closed Pricing Formulas Benefits Calculation time Easy implementation Analytical tractability Closed Pricing Formulas Shortcomings Few underlying diffusion models Small range of payoff profiles Unable to handle early exercise features

6. Carlos.veiga@millenniumbcp.pt 6 Result Consider the contract where k i, S i T is the underlying asset price at maturity, I i is the indicator of C i. It has the following pricing formula where ℮ -q i (T-t) is the usual discount factor, S i t is the underlying asset price at time t, P S i (C i ) is the probability of C i on the risk-neutral measure of S i.

7. Carlos.veiga@millenniumbcp.pt 7 Result Proof Using the risk free asset as the numeraire, with Q its risk-neutral measure, With the Radon-Nikodym derivative

8. Carlos.veiga@millenniumbcp.pt 8 Result: Result: P S i (C i ) In general  t,  T ) is not a closed pricing formula. Though, if C i is of the form Then can be evaluated with the cumulative function of the Multivariate Normal distribution Then P S i (C i ) can be evaluated with the cumulative function of the Multivariate Normal distribution where h m, M, u (m), v (m) {1,…,n}, T u(m),T v(m) T.

9. Carlos.veiga@millenniumbcp.pt 9 Applications European Call – Black and Scholes Exchange Option – William Magrabe

10. Carlos.veiga@millenniumbcp.pt 10 Applications Options on the Maximum or the Minimum of Several Assets – Herb Johnson 2 nd Best of Several Assets

11. Carlos.veiga@millenniumbcp.pt 11 Applications Target Redemption Option

12. Carlos.veiga@millenniumbcp.pt 12 Appl. Tests The performance of the method is compared with a standard Monte Carlo valuation. The performance is affected by several factors, namely: The option payoff profile (kinks, number of underlyings, quanto) Context Market Data (asset prices, dividends, volatilities and correlations) Technical Factors (random number generator, Multivariate Normal calculation algorithm)

13. Carlos.veiga@millenniumbcp.pt 13 Appl. Tests Option on the Maximum of Several Assets ATM Maximum of 10 underlying assets Linear combination of 11 terms (1 for each asset + strike) Thus 11 ten-dimensional multivariate normals to evaluate No correlations Price  13.36 If we allow 100 seconds of calculation time, we have the following precision, with 99% confidence New Method 0.52% Monte Carlo 0.88%

14. Carlos.veiga@millenniumbcp.pt 14 Appl. Tests 2 nd Best of Several Assets ATM 2 nd best of 4 underlying assets Linear combination of 22 terms (20 for the underlyings and 6 for the strike) Thus 22 four-dimensional multivariate normals to evaluate Random volatilities and correlations Price  10.17 If we allow 100 seconds of calculation time, we have the following precision, with 99% confidence New Method 0.20% Monte Carlo 0.48%

15. Carlos.veiga@millenniumbcp.pt 15 We developed a Closed Pricing Formulafor a Generic Exotic Option: We developed a Closed Pricing Formula for a Generic Exotic Option: Calls and Puts (Black and Scholes) Maximum of Several Assets (Herb Johnson) Options on the 2 nd Best Asset or N th Best Asset Worst of Calls Target Redemption Options … We developed an Efficient Pricing Method In the 1 st example, it achieved an error 40% less than Monte Carlo In the 2 nd example, it achieved an error 58% smaller than Monte Carlo Summary

16. Carlos.veiga@millenniumbcp.pt 16 Future Work Develop a Closed Formula for the Greeks Deltas, Gammas and Crossed Deltas, Vega and Correlation Deriv. Include other features in the payoff profile, like Compound options. May lead to an American Option Closed Formula

17. Carlos.veiga@millenniumbcp.pt 17 Background Related Work Fischer Black and Myron Scholes “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (May/June 1973) Robert Merton “A Rational Theory of Option Pricing.” Bell Journal of Economics and Management Science, 4 (Spring 1973) William Magrabe “The Value of an Option to Exchange One Asset for Another.” Journal of Finance, 33 (March 1978) Herb Johnson “Options on the Maximum or the Minimum of Several Assets.” Journal of Financial and Quantitative Analysis, Vol.22, NO.3, September 1987 Peter Carr “Deriving Derivatives of Derivative Securities.” Journal of Computational Finance, Vol. 4/Nr. 2, Winter 2000/2001

18. 18 Part II European Options with Discrete Dividends

19. Carlos.veiga@millenniumbcp.pt 19 Background Motivation The same benefits referred in Part I Dividends can have a significant weight on option prices and on hedging strategies. The existing solutions rely heavily on heuristics Performance of the hedging strategy is not available in most cases Existing alternatives Converting the Discrete Dividend in a Dividend Yield Discounting the NPV of the Dividend to the Stock Price Modified Binomial Model (nonrecombining tree)

20. Carlos.veiga@millenniumbcp.pt 20 Proposition Forward This proposition also relies heavily on heuristics This is work in progress. No proofs will be provided here. Presented for discussion purposes Heuristics Consider a Call Option on an no-dividend underlying. If the underlying pays a surprising dividend, the option will lose some of its value. The price of a Call option on an underlying with discrete dividends can be expressed as the sum of: Price of a Call Option with no dividends Expected Value of the Option Price drop on the dividend ex-date

21. Carlos.veiga@millenniumbcp.pt 21 Calculation

22. Carlos.veiga@millenniumbcp.pt 22 Results

23. Carlos.veiga@millenniumbcp.pt 23 Future Work Comparative Performance of the Hedging Strategy Generalize the result for several dividends payments