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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
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Pricing the ConvexityPricing the ConvexityAdjustmentAdjustment
Eric Benhamou
a Wiener Chaos approacha Wiener Chaos approach
Pricing the Convexity adjustment. 28 April 1999 Slide 2
FrameworkFrameworkThe 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
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))
IntroductionIntroduction
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 formula
IntroductionIntroduction
Pricing the Convexity adjustment. 28 April 1999 Slide 5
IntroductionIntroduction• Bullish market Euribor
Pricing the Convexity adjustment. 28 April 1999 Slide 7
IntroductionIntroduction• Swap Rates (81):
– OTC deals– Straightforward computation by no-
arbitrages:
with zero coupons bonds maturing at time
– Exponential growth
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 problemPricing problem
srQ
Q
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 coherenceConsistency and coherence
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 Process
CoherenceCoherence
Pricing the Convexity adjustment. 28 April 1999 Slide 11
• Assumption– models on Zero coupons HJM framework
is a p-dim. Brownian motion
Novikov Condition
CoherenceCoherence
Pricing the Convexity adjustment. 28 April 1999 Slide 12
Ito lemma
A CMS rate defined by
CoherenceCoherence
Pricing the Convexity adjustment. 28 April 1999 Slide 13
General FormulaGeneral 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 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 ChaosWiener Chaos
Pricing the Convexity adjustment. 28 April 1999 Slide 15
Wiener ChaosWiener Chaos• Completeness of Wiener Chaos
Definition
Result
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 ChaosWiener Chaos
Pricing the Convexity adjustment. 28 April 1999 Slide 17
ResultsResults • 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 18
• 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
General Formula: the General Formula: the stochastic expansionstochastic expansion
Pricing the Convexity adjustment. 28 April 1999 Slide 19
ExtensionExtension • For vanilla contract
• Result holds for any type of deterministic volatility within the HJM framework
Pricing the Convexity adjustment. 28 April 1999 Slide 20
Market DataMarket Data• Market data justifies approximation:
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 convexity
ConclusionConclusion
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