1. Pricing the Convexity Adjustment Eric Benhamou a Wiener Chaos approach

2. Pricing the Convexity adjustment. 28 April 1999 Slide 2Framework The major result of this paper is an approximation formula for convexity adjustment for any HJM interest rate model. It is actually based on Wiener Chaos expansion. The methodology developed here could be applied to other financial products Convexity and CMS Coherence and consistence Wiener Chaos Results Conclusion

3. Pricing the Convexity adjustment. 28 April 1999 Slide 3 Two intriguing and juicy facts for options market: –Volatility smile –Convexity Convexity –Different meanings –But one mathematical sense –Many rules of thumb (Dean Witter (94))Introduction

4. Pricing the Convexity adjustment. 28 April 1999 Slide 4 CMS/CMT products –Definition –OTC deals –Increasing popularity Actual way to price the convexity –Numerical Computation (MC) –Black Scholes Adjustment (Ratcliffe Iben (93)) –Approximation with Taylor formulaIntroduction

5. Pricing the Convexity adjustment. 28 April 1999 Slide 5Introduction Bullish market Euribor

6. Pricing the Convexity adjustment. 28 April 1999 Slide 6Introduction Bullish market US

7. Pricing the Convexity adjustment. 28 April 1999 Slide 7Introduction Swap Rates (81): –OTC deals –Straightforward computation by no- arbitrages: with zero coupons bonds maturing at time – Exponential growth

8. Pricing the Convexity adjustment. 28 April 1999 Slide 8 CMS rate defined as Assuming a unique risk neutral probability measure (Harrison Pliska [79]) risk free interest rate Problem non trivial with specific assumptions Black-Scholes adjustment incoherent Pricing problem

9. Pricing the Convexity adjustment. 28 April 1999 Slide 9 Interest rates models –Equilibrium models Vasicek (77) Cox Ingersoll Ross (85) Brennan and Schwartz (92) –No-arbitrage models Black Derman Toy (90) Heath Jarrow Morton (93) Hull &white (94) Brace Gatarek Musiela (95) Jamshidian (95) Consistency and coherence

10. Pricing the Convexity adjustment. 28 April 1999 Slide 10 Assumptions (See Duffie (94)) = Classical assumption in Assets pricing: –Market completeness –No-Arbitrage Opportunity –Continuous time economy represented by a probability space –Uncertainty modelled by a multi- dimensional Wiener ProcessCoherence

11. Pricing the Convexity adjustment. 28 April 1999 Slide 11 Assumption –models on Zero coupons HJM framework is a p-dim. Brownian motion Novikov ConditionCoherence

12. Pricing the Convexity adjustment. 28 April 1999 Slide 12 Ito lemma A CMS rate defined byCoherence

13. Pricing the Convexity adjustment. 28 April 1999 Slide 13 General Formula Even for one factor model, no CF Usual techniques: –Monte-Carlo and Quasi-Monte-Carlo –Tree computing (very slow) –Taylor expansion Surprisingly, little literature (Hull (97), Rebonato (95)) Our methodology: Wiener Chaos

14. Pricing the Convexity adjustment. 28 April 1999 Slide 14 Historical facts –Intuitively, Taylor expansion in Martingale Framework –First introduced in finance by Brace, Musiela (95) Lacoste (96) Idea: –Let be a square-integral continuous Martingale Wiener Chaos

15. Pricing the Convexity adjustment. 28 April 1999 Slide 15 Wiener Chaos Completeness of Wiener Chaos Definition Result

16. Pricing the Convexity adjustment. 28 April 1999 Slide 16 Getting Wiener Chaos Expansion See Lacoste (96) enables to get the convexity adjustment for a CMS product Wiener Chaos

17. Pricing the Convexity adjustment. 28 April 1999 Slide 17Results Applying this result to our pricing problem leads to: Expansion in the volatility up to the second order

18. Pricing the Convexity adjustment. 28 April 1999 Slide 18 Notation: correlation term T- forward volatility Payment datesensitivity of the swap Forward Zero coupons Convexity adjustment small quantity regular contracts positive : real convexity correlation trading Strongly depending on our model assumptions General Formula: the stochastic expansion

19. Pricing the Convexity adjustment. 28 April 1999 Slide 19Extension For vanilla contract Result holds for any type of deterministic volatility within the HJM framework

20. Pricing the Convexity adjustment. 28 April 1999 Slide 20 Market Data Market data justifies approximation:

21. Pricing the Convexity adjustment. 28 April 1999 Slide 21 INTERESTS: Methodology could be applied to other intractable options Very interesting for multi-factor models where numerical procedures time-consuming Enables to price convexity consistent with yield curve models Demystify convexityConclusion

22. Pricing the Convexity adjustment. 28 April 1999 Slide 22Conclusion LIMITATIONS: Need Market completeness –No stochastic volatility –Need model given by its zero coupons diffusions Wiener Chaos only useful for small correction (Swaptions, Asiatic should not work)