a Wiener Chaos approach Pricing the Convexity Adjustment

a Wiener Chaos approach Pricing the Convexity Adjustment Eric Benhamou

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

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)) Pricing the Convexity adjustment. 28 April 1999 Slide 3

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 Pricing the Convexity adjustment. 28 April 1999 Slide 4

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

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

Introduction • Swap Rates (81): – OTC deals – Straightforward computation by noarbitrages: with zero coupons bonds maturing at time – Exponential growth Pricing the Convexity adjustment. 28 April 1999 Slide 7

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 Pricing the Convexity adjustment. 28 April 1999 Slide 8

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) Pricing the Convexity adjustment. 28 April 1999 Slide 9

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 multidimensional Wiener Process Pricing the Convexity adjustment. 28 April 1999 Slide 10

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

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

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 Pricing the Convexity adjustment. 28 April 1999 Slide 13

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 Pricing the Convexity adjustment. 28 April 1999 Slide 14

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

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

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

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 Pricing the Convexity adjustment. 28 April 1999 Slide 18

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

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

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 Pricing the Convexity adjustment. 28 April 1999 Slide 21

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) Pricing the Convexity adjustment. 28 April 1999 Slide 22