Lecture on Parameter Estimation for Stochastic Differential Equations Erik Lindström
Recap (1) hedging 9.2.2.1 ( Advanced hedging (Greeks 9.2.2 and quadratic Risk management Model validation Why? 0 0 Stochastic Integral Equations )) ◮ We are interested in the parameters θ in the ∫ t ∫ t X ( t ) = X ( 0 )+ µ θ ( s , X ( s )) d s + σ θ ( s , X ( s )) d W ( s )
Recap (1) hedging 9.2.2.1 ( Advanced hedging (Greeks 9.2.2 and quadratic Risk management Why? 0 Stochastic Integral Equations 0 )) ◮ We are interested in the parameters θ in the ∫ t ∫ t X ( t ) = X ( 0 )+ µ θ ( s , X ( s )) d s + σ θ ( s , X ( s )) d W ( s ) ◮ Model validation
Recap (1) hedging 9.2.2.1 ( Advanced hedging (Greeks 9.2.2 and quadratic Why? 0 0 Stochastic Integral Equations )) ◮ We are interested in the parameters θ in the ∫ t ∫ t X ( t ) = X ( 0 )+ µ θ ( s , X ( s )) d s + σ θ ( s , X ( s )) d W ( s ) ◮ Model validation ◮ Risk management
Recap Stochastic Integral Equations 0 0 (1) Why? ◮ We are interested in the parameters θ in the ∫ t ∫ t X ( t ) = X ( 0 )+ µ θ ( s , X ( s )) d s + σ θ ( s , X ( s )) d W ( s ) ◮ Model validation ◮ Risk management ◮ Advanced hedging (Greeks 9.2.2 and quadratic hedging 9.2.2.1 ( P / Q ))
2 N Some asymptotics n t 0 (4) The MLE for the diffusion ( ) parameter is given by 2 1 N 1 N 1 0 W t 0 X t n 1 X t n 2 d 2 1 N 1 t N W t N Consider the arithmetic Brownian motion 1 (2) The drift is estimated by computing the mean, and compensating for the sampling t n 1 t n 1 N N n t 0 0 X t n 1 X t n (3) Expanding this expression reveals that the MLE is given by X t N X t 0 t N (5) d X ( t ) = µ d t + σ d W ( t )
2 N Some asymptotics 0 The MLE for the diffusion ( ) parameter is given by 2 1 N 1 N 1 n X t n t 0 1 X t n 2 d 2 1 N 1 (4) t N Consider the arithmetic Brownian motion Expanding this expression reveals that the MLE is (2) The drift is estimated by computing the mean, and W t 0 (3) (5) given by X t N X t 0 t N t 0 W t N d X ( t ) = µ d t + σ d W ( t ) compensating for the sampling δ = t n + 1 − t n N − 1 ∑ µ = 1 ˆ X ( t n + 1 ) − X ( t n ) . δ N n = 0
2 N Some asymptotics 0 2 1 N 1 N 1 n 1 X t n (4) X t n 2 d 2 1 N 1 The MLE for the diffusion ( ) parameter is given by (5) Consider the arithmetic Brownian motion (2) The drift is estimated by computing the mean, and given by Expanding this expression reveals that the MLE is (3) d X ( t ) = µ d t + σ d W ( t ) compensating for the sampling δ = t n + 1 − t n N − 1 ∑ µ = 1 ˆ X ( t n + 1 ) − X ( t n ) . δ N n = 0 µ = X ( t N ) − X ( t 0 ) = µ + σ W ( t N ) − W ( t 0 ) ˆ . t N − t 0 t N − t 0
Some asymptotics (3) d 1 (4) Consider the arithmetic Brownian motion given by Expanding this expression reveals that the MLE is (5) (2) The drift is estimated by computing the mean, and d X ( t ) = µ d t + σ d W ( t ) compensating for the sampling δ = t n + 1 − t n N − 1 ∑ µ = 1 ˆ X ( t n + 1 ) − X ( t n ) . δ N n = 0 µ = X ( t N ) − X ( t 0 ) = µ + σ W ( t N ) − W ( t 0 ) ˆ . t N − t 0 t N − t 0 The MLE for the diffusion ( σ ) parameter is given by N − 1 → σ 2 χ 2 ( N − 1 ) σ 2 = ∑ ˆ ( X ( t n + 1 ) − X ( t n ) − ˆ µδ ) 2 δ ( N − 1 ) N − 1 n = 0
t n X t n t n X t n A simple method log T t X t t X t t X t covariance P and Normal distribution with argument x , mean m and x m P is the density for a multivariate where (7) X t n 1 X t n 1 Many data sets are sampled at high frequency, t X t d W t making the bias due to discretization of the SDEs some of the schemes in Chapter 12 acceptable. The simplest discretization, the Explicit Euler method, would for the stochastic differential equation d X t t X t d t (6) n correspond the Discretized Maximum Likelihood (DML) estimator given by DML arg max N 1 (8)
A simple method correspond the Discretized Maximum Likelihood covariance P and Normal distribution with argument x , mean m and (7) Many data sets are sampled at high frequency, (DML) estimator given by (8) (6) would for the stochastic differential equation The simplest discretization, the Explicit Euler method, some of the schemes in Chapter 12 acceptable. making the bias due to discretization of the SDEs d X ( t ) = µ ( t , X ( t )) d t + σ ( t , X ( t )) d W ( t ) N − 1 ˆ ∑ θ DML = arg max log φ ( X ( t n + 1 ) , X ( t n ) + µ ( t n , X ( t n ))∆ , Σ( t n , X ( t n θ ∈ Θ n = 1 where φ ( x , m , P ) is the density for a multivariate Σ( t , X ( t )) = σ ( t , X ( t )) σ ( t , X ( t )) T .
Consistency Approximate ML estimators (13.5) are, provided enough computational resources are allocated Simulation based estimators Fokker-Planck based estimators Series expansions. GMM-type estimators (13.6) are consistent if the moments are correctly specified (which is a non-trivial problem!) ◮ The DMLE is generally NOT consistent.
Consistency enough computational resources are allocated GMM-type estimators (13.6) are consistent if the moments are correctly specified (which is a non-trivial problem!) ◮ The DMLE is generally NOT consistent. ◮ Approximate ML estimators (13.5) are, provided ◮ Simulation based estimators ◮ Fokker-Planck based estimators ◮ Series expansions.
Consistency enough computational resources are allocated moments are correctly specified (which is a non-trivial problem!) ◮ The DMLE is generally NOT consistent. ◮ Approximate ML estimators (13.5) are, provided ◮ Simulation based estimators ◮ Fokker-Planck based estimators ◮ Series expansions. ◮ GMM-type estimators (13.6) are consistent if the
x t x s x t x Simultion based estimators s Works very well for Multivariate models! (2012) and Whitaker et al (2016) etc. Improved by Durham-Gallant (2002), Lindström This is the Pedersen algorithm. (9) s t p E p Then it follows that and is easily (...) extended to Levy driven SDEs. ◮ Discretely observed SDEs are Markov processes
Simultion based estimators (9) This is the Pedersen algorithm. Improved by Durham-Gallant (2002), Lindström (2012) and Whitaker et al (2016) etc. Works very well for Multivariate models! and is easily (...) extended to Levy driven SDEs. ◮ Discretely observed SDEs are Markov processes ◮ Then it follows that p θ ( x t | x s ) = E θ [ p θ ( x t | x τ ) |F ( s )] , t > τ > s
Simultion based estimators (9) This is the Pedersen algorithm. (2012) and Whitaker et al (2016) etc. Works very well for Multivariate models! and is easily (...) extended to Levy driven SDEs. ◮ Discretely observed SDEs are Markov processes ◮ Then it follows that p θ ( x t | x s ) = E θ [ p θ ( x t | x τ ) |F ( s )] , t > τ > s ◮ Improved by Durham-Gallant (2002), Lindström
Simultion based estimators (9) This is the Pedersen algorithm. (2012) and Whitaker et al (2016) etc. ◮ Discretely observed SDEs are Markov processes ◮ Then it follows that p θ ( x t | x s ) = E θ [ p θ ( x t | x τ ) |F ( s )] , t > τ > s ◮ Improved by Durham-Gallant (2002), Lindström ◮ Works very well for Multivariate models! ◮ and is easily (...) extended to Levy driven SDEs.
Some key points estimate - use CRNs or importance sampling control variates) process, as it reduces variance AND improves the asymptotics. albeit somewhat restrictive in terms of the class of feasible models. ◮ Naive implementation only provides a point wise ◮ Variance reduction helps (antithetic variates, ◮ The near optimal importance sampler is a Bridge ◮ There is a version that is completely bias free,
t x t x 0 x 2 t p x t x 0 p x t 2 2 2 1 p x t x t p x t where (12) Fokker-Planck Consider the expectation p the Ito formula. Equating these yields Two possible ways to compute this, direct and using (11) and then (10) (13) ∫ E [ h ( X ( t )) |F ( 0 )] = h ( x ( t )) p ( x ( t ) | x ( 0 )) d x ( t ) ∂ ∂ t E [ h ( X ( t )) |F ( 0 )]
t x t x 0 x 2 t Fokker-Planck p x t 2 2 2 1 p x t x t p x t where (12) p x t x 0 p Consider the expectation Equating these yields o formula. Two possible ways to compute this, direct and using (11) and then (10) (13) ∫ E [ h ( X ( t )) |F ( 0 )] = h ( x ( t )) p ( x ( t ) | x ( 0 )) d x ( t ) ∂ ∂ t E [ h ( X ( t )) |F ( 0 )] the It ¯
Fokker-Planck (11) 2 where (12) Consider the expectation o formula. Equating these yields Two possible ways to compute this, direct and using (13) and then (10) ∫ E [ h ( X ( t )) |F ( 0 )] = h ( x ( t )) p ( x ( t ) | x ( 0 )) d x ( t ) ∂ ∂ t E [ h ( X ( t )) |F ( 0 )] the It ¯ ∂ p ∂ t ( x ( t ) | x ( 0 )) = A ⋆ p ( x ( t ) | x ( 0 )) ∂ ∂ 2 ( ) A ⋆ p ( x ( t )) = − ∂ x ( t ) ( µ ( · ) p ( x ( t )))+ 1 σ 2 ( · ) p ( x ( t )) . ∂ x 2 ( t )
Example of the Fokker-Planck equation From (Lindström, 2007) Figure: Fokker-Planck equation computed for the CKLS process 400 300 p(s,x s ;t,x t ) 200 0.08 100 0.07 0 0.06 0.08 0.075 0.05 0.07 0.04 0.065 0.06 0.03 0.055 Time 0.05 0.02 Grid
Comments on the PDE approach Generally better than the Monte Carlo method in Figure: Comparing Monte Carlo, 2nd order and 4th order numerical approximations of the Fokker-Planck equation low dimensional problems. Durham−Gallant Poulsen −1 10 Order2−Pade(1,1) Order4−Pade(2,2) −2 10 −3 MAE 10 −4 10 −5 10 2 3 10 10 time
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