Models with time- -dependent parameters dependent parameters - PowerPoint PPT Presentation

Models with time- -dependent parameters dependent parameters Models with time using transform methods: using transform methods: Application to Heston’ ’s model s model Application to Heston Alberto Elices Alberto Elices Model Validation Group Model Validation Group Risk Division Risk Division Grupo Santander Grupo Santander Kick- -off off- -Workshop Workshop Kick Special Semester on Stochastics with empphasis on Finance Special Semester on Stochastics with empphasis on Finance Johann ohann Radon Institute for Computational and Applied Radon Institute for Computational and Applied J Mathematics (RICAM) (RICAM) Mathematics September 8- -12th 2008, Linz, Austria 12th 2008, Linz, Austria September 8

Outline of the presentation � Introduction. � Characteristic functions of models with time-dependent parameters. � Application to Heston’s model. � Case study: Calibration to Eurostoxx 50. � Application to Forward start options. � Forward skew of Heston’s model. � Conclusions. 2

Introduction � Exotic valuation: usually carried out with Monte Carlo. � Calibration: fast analytic models are needed for valuation of vanilla products. � Analytic models depend on just a few parameters which cannot fit the whole set of market parameters. � More degrees of freedom are needed in order to calibrate the market across all maturities. � The most natural way of introducing more parameters is to let them depend on time. 3

Characteristic functions of models with time-dependent parameters � Characteristic function methods: � Useful when the characteristic function is analytic. � The Inversion of the characteristic function is carried out through the inverse Fourier transform. � Characteristic function: ( ) ∫ ⋅ ⋅ ϕ = = i X x i X x ( ) ( ) X x E e e f X x d x v v uv u uv u v N R 4

Introduction � Family of characteristic functions for which the methodology can be applied: ( ) ϕ = + ⋅ ( ) exp ( ) ( ) X x C X D X x uv u uv uv u = = L X L ( ) ( ( ), , ( )) ( , , ) x t x t x t X X 1 N 1 N = L ( ) ( ( ), , ( )) D X D X D X , 1 , uv uv uv N � The method proposed introduces time-dependent parameters for a wide variety of models which admit analytic characteristic function: ( ) � Merton jump model. ϕ = + ( ) exp ( ) G g C G iGg uv u uv u g t : sum of all Poisson distributed jumps up to time . u u 5

Introduction � Cox Ingersoll Ross model. ( ) ϕ = + ( ) exp ( ) ( ) R r C R iD R r uv u uv uv u r t : short rate interest rate at time . u u � Heston stochastic volatility model. ( ) ϕ = + + ( , , ) exp ( , ) ( , ) X V x v C X V D X V v iXx uv u u uv uv u u x v : logarithm of underlying. : variance process. u u � Hybrids with jumps, stochastic interest rates and volatility. + + + + ϕ = C D r D v iXx iGg ( , , , , , , ) X V R G x v r g e uv uv , 2 u uv , 1 u u u uv u u u u 6

Characteristic functions of models with time-dependent parameters a ( ) ( ) ϕ ϕ X x X x 0 u 0 uv u τ = τ = t − t t 0 u u uv v u t 0 t v u � All relevant information of a Markov process with t independent increments at an instant is given by the v ϕ ( ) X x joint probability distribution: 0 0 v ϕ ( ) X x 0 0 u ϕ ( ) X x � Objective: Find in terms of 0 0 ϕ v ( ) X x uv u 7

Characteristic functions of models with time-dependent parameters � Characteristic function under search: ( ) ∫ ⋅ ϕ = i X x ( ) X x d x e f x x v 0 0 0 0 v v v v N R t → t → t � Joint density in terms of densities t 0 u 0 v t → t and (independent increments): u v ∫ = ( ) ( ) ( ) f x x d x f x x f x x 0 0 0 0 v v u u u uv v u N ( ) R ϕ ( ) f x v x X x � Subtituting in : 0 0 0 0 v v ∫ ∫ ⋅ ϕ = i X x ( ) ( ) ( ) X x x x x x x x d f d e v f 0 v 0 u 0 u u 0 v uv v u N N 1 4 4 4 2 4 4 4 3 R R ( ) ( ) ϕ = + ⋅ X / x exp ( X ) D ( X ) x C uv u uv uv u 8

Characteristic functions of models with time-dependent parameters ( ) ϕ X x � After substituting : uv u ( ) ∫ ϕ = + ⋅ ( ) ( ) exp ( ) ( ) X x d x f x x C X D X x 0 0 0 0 v u u u uv uv u N R ( ) ( ) ∫ − = ⋅ 1 exp ( ) ( ) exp ( ( )) C X d x f x x i i D X x 0 0 uv u u u uv u N 1 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 3 R ( ) − ϕ 1 i D ( X ) x 0 u uv 0 ⎛ ⎞ ( ) ( ) ⎜ ⎟ − − = + + ⋅ 1 1 exp ( ) ( ) ( ) C X C i D X D i D X x ⎜ ⎟ 1 4 4 4 4 2 4 4 4 4 3 1 4 2 4 4 3 4 ⎜ uv 0 u uv 0 u uv 0 ⎟ ⎝ ( ) ( ) ⎠ C X D X 0 v 0 v 9

Characteristic functions of models with time-dependent parameters � Identifying terms: ( ) ϕ = + ⋅ ( ) exp ( ) ( ) X x C X D X x 0 0 0 0 0 v v v ( ) ( ) ( ) ⎧ − = + 1 ⎪ ( ) X X D X C C C i 0 v uv 0 u uv ⎨ ( ) ( ) − ⎪ = 1 ⎩ ( ) D X D D X i 0 0 v u uv a , 01 , D 12 , D C C D C L − − 12 1 , 1 , 01 M M M M ϕ ϕ ϕ L − 01 1 1 , M M 2 0 t t t t − 1 M 2 1 M 10

Application to Heston´s model � Heston process: ⎧ = μ + ν ⎪ dS S dt S dW 〈 〉 = ρ ⎨ t t t t t , d W Y dt ( ) ν = κ θ − ν + σ ν t t ⎪ d dt dY ⎩ t t t t � The two state variables for Heston’s process are the x = log( ) S logarithm of the stock price and the t t v variance process : t = ν ( ( ), ( )) x x t t u u u � These two state variables translate into and for X V the characteristic function: = ( , ) X X V 11

Application to Heston´s model � Joint characteristic function for Heston process: ( ) + ν + ϕ = C ( X , V ) D ( X , V ) ( t ) D ( X , V ) x ( t ) , 2 , 1 , ( ), ( ) uv uv u uv u X V x t v t e uv u u ~ ⎛ − τ ⎞ κ − ρσ + − d ( ) ( ) Xi d g g e ⎜ ⎟ = = , D X , V D uv X V iX ⎜ ⎟ ~ 2 − τ , 2 σ − , 1 uv d ⎝ 1 ⎠ g e ⎛ ~ ⎞ ⎛ − τ ⎞ κθ − d ( ) 1 g ( ) ⎟ e ⎜ ⎟ ⎜ ⎟ = μ τ + − + κ − ρσ − τ , 2 ln C X V i X Xi d ⎜ ⎟ ⎜ ~ σ − uv 2 ⎝ 1 ⎠ g ⎝ ⎠ 2 κ − ρσ − − σ κ − ρσ − Xi d iV Xi d ~ = = g g 2 κ − ρσ + − σ κ − ρσ + Xi d iV Xi d ( ) ( ) 2 = κ − ρσ + σ + 2 d Xi X i X 12

Application to Heston´s model � Characteristic function with time-dependent t parameters at maturity : v ( ) + ν + ϕ = C ( X , V ) D ( X , V ) ( t ) D ( X , V ) x ( t ) 0 0 , 2 0 , 1 0 , ( ), ( ) v v uv X V x t v t e 0 0 0 v ( ) ( ) ( ) ⎧ − = + 1 , , , ( , ) C X V C X V C X i D X V 0 0 , 2 v uv u uv ⎪ ( ) ⎪ ( ) − = 1 ⎨ , , ( , ) D X V D X i D X V 0 , 2 0 , 2 , 2 v u uv ⎪ = ⎪ ( , ) D X V iX ⎩ 0 , 1 v a ( ) ( ) ϕ ϕ X x X x 0 u 0 uv u τ = τ = t − t t 0 u u uv v u t t 0 u v 13

Application to Heston´s model � Valuation of vanilla options: ⎛ ⎞ ( ) ( ) ( ) ⎜ ⎟ + = − = − x ( ) C DF E S K DF ⎜ E e 1 K E 1 ⎟ T { } { } > > ln ln T T T x K x K 1 4 2 4 4 3 4 1 4 2 4 3 ⎜ ⎟ T T ⎝ ⎠ Asset or nothing Cash or nothing � Characteristic function for cash or nothing option: ∫ ⋅ ϕ = ϕ = CN iX x ( ) ( , 0 ) ( , ) X x X x e f x v x dx dv T 0 0 0 0 0 0 T T T T T T T R � Inversion formula: cummulative density in terms of characteristic function. ∞ ϕ ϕ − ⎛ ⎞ 1 1 1 ( ) ( ) X X ∫ > = + − ⎜ ⎟ ( ) P x a dX − π ⎝ iXa iXa ⎠ 2 2 iX e e 0 14

Application to Heston´s model � Characteristic function for asset or nothing option: ( ) ( ) − ϕ − ( ) i X i x iXx x ( , 0 ) x E E X i e e e T T T ϕ = = = AN 0 0 T ( ) x X ϕ − 0 0 T x ( , 0 ) ( ) ( ) i x E e E S T 0 0 T T ( ) AN , , f x v x v 6 4 4 4 4 7 4 4 4 4 8 0 T T T 0 0 x ( ) e T ∫ ∫ ϕ = AN iXx ( , ) , , X x v e dx f x v x v dv T 0 0 0 0 0 0 T T T T T T ( ) E S T R R � Final expression of vanillas: ( ) ( ) = > − > AN CN ( ln ) ( ln ) C DF E S P x K KP x K T T T T 15