Pricing the Convexity Adjustment

1. a Wiener Chaos approach Pricing the ConvexityAdjustment Eric Benhamou

2. Convexity and CMS Coherence and consistence Wiener Chaos Results Conclusion Framework 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

3. Introduction • 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))

4. Introduction • 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 formula

5. Introduction • Bullish market Euribor

6. Introduction • Bullish market US

7. Introduction • Swap Rates (81): • OTC deals • Straightforward computation by no-arbitrages: with zero coupons bonds maturing at time • Exponential growth

8. Pricing problem • 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

9. Consistency and coherence • 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)

10. Coherence • 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 Process

11. Coherence • Assumption • models on Zero coupons HJM framework is a p-dim. Brownian motion Novikov Condition

12. Coherence Ito lemma A CMS rate defined by

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. Wiener Chaos • 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

15. Wiener Chaos • Completeness of Wiener Chaos Definition Result

16. Wiener Chaos • Getting Wiener Chaos Expansion See Lacoste (96) enables to get the convexity adjustment for a CMS product

17. Results • Applying this result to our pricing problem leads to: Expansion in the volatility up to the second order

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

19. Extension • For vanilla contract • Result holds for any type of deterministic volatility within the HJM framework

20. Market Data • Market data justifies approximation:

21. Conclusion 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 convexity

22. Conclusion 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)