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Pricing the Convexity Adjustment

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a Wiener Chaos approach. Pricing the Convexity Adjustment. Eric Benhamou. 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. - PowerPoint PPT Presentation

Text of Pricing the Convexity Adjustment

  • Pricing the ConvexityAdjustment

    Eric Benhamoua Wiener Chaos approach

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Framework Convexity and CMS Coherence and consistence Wiener Chaos

    Results

    ConclusionThe 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

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    IntroductionTwo intriguing and juicy facts for options market:Volatility smileConvexity

    ConvexityDifferent meaningsBut one mathematical senseMany rules of thumb (Dean Witter (94))

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    IntroductionCMS/CMT productsDefinitionOTC dealsIncreasing popularity

    Actual way to price the convexityNumerical Computation (MC)Black Scholes Adjustment (Ratcliffe Iben (93))Approximation with Taylor formula

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    IntroductionBullish market Euribor

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    IntroductionBullish market US

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    IntroductionSwap Rates (81):OTC dealsStraightforward computation by no-arbitrages:

    with zero coupons bonds maturing at time Exponential growth

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Pricing problemCMS rate defined as Assuming a unique risk neutral probability measure (Harrison Pliska [79])

    risk free interest rate

    Problem non trivial with specific assumptionsBlack-Scholes adjustment incoherent

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Consistency and coherenceInterest rates modelsEquilibrium modelsVasicek (77)Cox Ingersoll Ross (85)Brennan and Schwartz (92)No-arbitrage modelsBlack Derman Toy (90)Heath Jarrow Morton (93) Hull &white (94)Brace Gatarek Musiela (95)Jamshidian (95)

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    CoherenceAssumptions (See Duffie (94))= Classical assumption in Assets pricing:Market completenessNo-Arbitrage OpportunityContinuous time economy represented by a probability space Uncertainty modelled by a multi-dimensional Wiener Process

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    CoherenceAssumptionmodels on Zero coupons HJM framework

    is a p-dim. Brownian motion

    Novikov Condition

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    CoherenceIto lemma

    A CMS rate defined by

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    General FormulaEven for one factor model, no CFUsual techniques:Monte-Carlo and Quasi-Monte-CarloTree computing (very slow)Taylor expansion

    Surprisingly, little literature (Hull (97), Rebonato (95)) Our methodology: Wiener Chaos

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Wiener ChaosHistorical factsIntuitively, 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 *

    Wiener ChaosCompleteness of Wiener Chaos

    Definition

    Result

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    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 *

    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 *

    General Formula: the stochastic expansion Notation:correlation term

    T- forward volatility

    Payment datesensitivity of the swap

    Forward Zero coupons

    Convexity adjustmentsmall quantityregular contracts positive : real convexitycorrelation tradingStrongly depending on our model assumptions

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Extension For vanilla contract

    Result holds for any type of deterministic volatility within the HJM framework

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Market DataMarket data justifies approximation:

    Maturity

    CMS Rate

    Forward swap

    Spread in bp

    (Convexity

    Adjustment

    1 year

    4.737

    4.719

    1.8

    2 year

    5.061

    5.026

    3.5

    5 year

    5.795

    5.711

    8.4

    7 year

    6.086

    5.977

    10.9

    10 year

    6.302

    6.145

    15.7

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Conclusion INTERESTS:Methodology could be applied to other intractable optionsVery interesting for multi-factor models where numerical procedures time-consumingEnables to price convexity consistent with yield curve modelsDemystify convexity

    Pricing the Convexity adjustment. 28 April 1999 Slide *

    Conclusion

    LIMITATIONS:Need Market completenessNo stochastic volatilityNeed model given by its zero coupons diffusionsWiener Chaos only useful for small correction (Swaptions, Asiatic should not work)