Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Asymptotics for local volatility and Sabr models Peter Laurence Facoltá di Statistica, Università di Roma I Kolmogorov Equations in Finance and Physics, Modena, Italy September 8, 2010 Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Collaborators in work presented today Gérard Ben Arous, Courant Institute Jim Gatheral, Baruch College Elton Hsu, Northwestern University Cheng Ouyang, Purdue University Tai-Ho Wang, Baruch College, CUNY Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Outline of Work to be discussed Overview Background Our Approach and Results Outline Collaborators in various aspects of work presented today 1 Outline of Work to be discussed 2 Overview Background 3 Models Local Volatility Models Stochastic Volatility models Methodology to be used Curvature Our Approach and Results 4 Local volatility Models revisited Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Outline of Work to be discussed Overview Background Our Approach and Results Outline of Results Two lines: Contributions of a Theoretical nature Provide rigorous proofs of short time to maturity expansion formulas for i) call prices and ii) implied volatility in local volatility setting. Practical Nature New expansion formulas for call prices and implied volatility. I.e. expansion up to second order with optimal (in a certain sense) coefficients. Already order 1 more accurate for several models tested than earlier expansions tested. BS ( t ) + σ ( 1 ) BS ( t )( T − t ) + σ ( 2 ) σ BS ( t , T ) = σ 0 BS ( t )( T − t ) 2 � �� � second order coeffts + o ( T − t ) 2 Dimensionless parameters σ 2 ( T − t ) should be small. In practice expansion highly accurate even when this parameter is of order Peter Laurence Asymptotics for local volatility and Sabr models 1.
Collaborators in various aspects of work presented today Outline of Work to be discussed Overview Background Our Approach and Results Regimes Historically, in mathematical finance, as time to maturity τ → 0,several regimes have been considered: ATM regime: S 0 = K , τ → 0 τ = T − t Megvedev-Scaillet Regime: S 0 − K = C √ τ → 0 τ C is constant Can be viewed as a close to the money regime. Standard heat kernel Regime (Henry-Labordère) S 0 , K fixed , τ → 0 In this case, when τ gets small, the deviation of call prices from their intrinsic value, decays exponentially (like τ 3 / 2 e − d 2 2 τ ). But the rate of exponential decay is a key tool to derive asymptotic formulas for the implied volatility. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Outline of Work to be discussed Overview Background Our Approach and Results Regimes In practice, since the heat kernel leads to a representation for the transition probability density, one can try to use it to calculate all three regimes. Once can also show, using L ’Hospital’s rule, that limit of away from the money expansion for small time, yields ATM expansion. Similarly, introducing S 0 ( τ ) = K + C √ τ , into the heat kernel expansion, one can recover Megvedev-Scaillet results (diffusion case). General Principle: Heat kernel expansion was devised to be accurate for small (dimensionless time σ 2 τ ), but is often accurate for moderate values of σ 2 τ . To make expansion work for longer maturities, can combine heat kernel method with Levi Parametrix approach (A. Friedman). Use Heat kernel exp. as first iterate, rather traditional constant coefft solution. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Outline Collaborators in various aspects of work presented today 1 Outline of Work to be discussed 2 Overview Background 3 Models Local Volatility Models Stochastic Volatility models Methodology to be used Curvature Our Approach and Results 4 Local volatility Models revisited Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Local volatility The Local volatility model dS t = b ( t ) S t dt + a ( S t , t ) dW t where { S t } t ≥ 0 is price process for the stock { W t } t ≥ 0 is a Brownian motion. Important contributions by Bruno Dupire. Still popular model today, in certain (e.g. French) banks. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Sabr type models Sabr Model in its original form (Hagan and Woodward, Hagan, Kumar, Lesniewski and Woodward, Andreasen-Andersen) dF t = F β t y t dW 1 t dy t = α y t dW 2 t < dW 1 t , dW 2 t > = ρ dt Calibrates well to smile, but for only one maturity. "Dynamic Sabr Model" dF t = γ ( t ) C ( F t ) y t dW 1 t dy t = ν ( t ) y t dW 2 t < dW 1 t , dW 2 t > = ρ ( t ) dt , time dependent parameters. Can be calibrated to implied volatility surface for several maturities. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Heston Heston Model � dS t = µ S t dt + V t S t dW t � dV t = κ ( θ − V t ) dt + σ V t dZ t dW t dZ t = ρ dt where { S t } t ≥ 0 and { V t } t ≥ 0 are price and variance processes. { W t } t ≥ 0 and { Z t } t ≥ 0 are Wiener processes with constant instantaneous correlation ρ . θ is long-run mean, κ is the rate of reversion and σ is volatility of variance. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Heston + local vol The Heston Model with local vol � dS t = µ S t dt + V t σ ( S t , t ) dW t � dV t = κ ( θ − V t ) dt + ¯ σ V t dZ t < dW t , dZ t > = ρ dt where { S t } t ≥ 0 and { V t } t ≥ 0 are price and volatility processes { W t } t ≥ 0 and { Z t } t ≥ 0 are Wiener processes with correlation ρ θ is long-run mean, κ is the rate of reversion and ¯ σ is volatility of variance. Andreasen and others. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Lipton-Andersen Quadratic SV Model Lipton-Andersen Model dS ( t ) � � � b ( t ) S ( t ) + ( 1 − b ( t )) S 0 + 1 c ( t ) ( S ( t ) − S 0 ) 2 = λ ( t ) z ( t ) dW t 2 S 0 � dz ( t ) = κ ( 1 − z ( t )) dt + η ( t ) z ( t ) dZ ( t ) z ( 0 ) = 1 where < dW ( t ) , dZ ( t ) > = ρ dt Needs adjustment at the wings, since local martingale but not a martingale in general. Can be seen as special case of Heston-local vol model. Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Outline Collaborators in various aspects of work presented today 1 Outline of Work to be discussed 2 Overview Background 3 Models Local Volatility Models Stochastic Volatility models Methodology to be used Curvature Our Approach and Results 4 Local volatility Models revisited Peter Laurence Asymptotics for local volatility and Sabr models
Collaborators in various aspects of work presented today Models Outline of Work to be discussed Methodology to be used Background Curvature Our Approach and Results Methods Passage from stochastic volatility model to local vol model: Gyongy-Dupire-Derman and Britten-Jones and Neuberger method for reducing the computation of call prices in stochastic volatility model to computation of an effective local volatility in a local volatility model. Combine with: Peter Laurence Asymptotics for local volatility and Sabr models
Recommend
More recommend
