Deriving Derivatives from Vanilla Options

Deriving Derivatives from Vanilla Options

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 It is actually based on Wiener Chaos expansion. The methodology developed here could be applied to other financial products 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

Recently Viewed Presentations

  • Acid-Base Imbalance - Metropolitan Community College

    Acid-Base Imbalance - Metropolitan Community College

    Acid-Base Imbalance. Metropolitan Community College. Fall 2013. Acid Base Balance. Hydrogen ions - Low concentrations but highly reactive ... RR x alveolar volume = minute alveolar ventilation. ... Assessment of ACID BASE. Arterial Blood Gases (ABG) most often and the...
  • The 2006 Forte Corporate Best Practices Summit Cathy

    The 2006 Forte Corporate Best Practices Summit Cathy

    The 2006 Forte Corporate Best Practices Summit Cathy Frierson GE Energy Financial Services May 3, 2006 GE… Intense Performance Culture Shared Values and Growth Traits Disciplined and Rigorous Process Key Development Process …
  • Big Question: - scottsboro.org

    Big Question: - scottsboro.org

    Big Question: How can people adapt to a new school? Monday. Tuesday. ... notice how I use tone of voice to show Fiona's sadness at being in a new place and missing her old friends. ... It is important to...
  • Federal Documents at UF - Center for Research Libraries

    Federal Documents at UF - Center for Research Libraries

    Federal Documents at UF. The University of Florida (UF) is the Regional Federal Depository Library for Florida, Puerto Rico and the U.S. Virgin Islands. It is also an active participant in the ASERL Collaborative Federal Depository Program with over 35...
  • ME 114 Lab Introduction - San Jose State University

    ME 114 Lab Introduction - San Jose State University

    If the temperature and relative humidity is right, liquid will condense as the air passes through the evaporator. Air is then reheated near the end. A normal air conditioning unit would not include all of these processes. Air flow rate...
  • (23) Evolution of PopulationsMicroevolution  Natural selection acts on

    (23) Evolution of PopulationsMicroevolution Natural selection acts on

    Consider, for example, a population of medium ground finches on Daphne Major Island During a drought, large-beaked birds were more likely to crack large seeds and survive The finch population evolved by natural selection
  • Chapter Eighteen - UCSB's Department of Economics

    Chapter Eighteen - UCSB's Department of Economics

    Chapter Nineteen Profit-Maximization Economic Profit A firm uses inputs j = 1…,m to make products i = 1,…n. Output levels are y1,…,yn. Input levels are x1,…,xm.
  • The Effects of Cold Working & Annealing on Resistivity

    The Effects of Cold Working & Annealing on Resistivity

    "Influence of Cold Working on the Pitting Corrosion Resistance of Stainless Steels."Corrosion Science. Isbergues:Elsevier, April 2007. 1933-1948. "Materials Engineering; Study data from M. Gonzalez and colleagues update understanding of materials engineering.