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.
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 2
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 Q
(Harrison Pliska [79])
rs
risk free interest rate
Q
• 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