Estimating Jump-Diffusions Using Closed-form Likelihood Expansions - PowerPoint PPT Presentation

Estimating Jump-Diffusions Using Closed-form Likelihood Expansions Chenxu Li Guanghua School of Management Peking University Asymptotic Statistics and Related Topics: Theories and Methodologies September 2-4, 2013 The University of Tokyo, Tokyo, Japan

Motivation ◮ Continuous-time models are widely applied for analyzing financial time series, e.g., for asset pricing, portfolio and asset management, and risk-management. ◮ Examples: diffusion, jump-diffusion, Levy processes, and Levy driven processes, etc. ◮ A key theme in empirical study: statistical inference and econometric assessment based on discretely observed data ◮ Likelihood-based inference (e.g., Maximum-likelihood estimation) is a natural choice among many other methods because of its efficiency. ◮ However, for most sophisticated models, likelihood functions are analytically intractable and thus involve heavy computational load, in particular, in the repetition of valuation for optimization.

For Diffusion Models ◮ Various methods for approximating likelihood functions, e.g., Yoshida (1992), Kessler (1997), Uchida and Yoshida (2012) among many others. ◮ Expansion of (transition densities) likelihood functions: established in A¨ ıt-Sahalia (1999, 2002, 2008) and its extensions and refinements, e.g., Bakshi et al. (2006). ◮ Thanks to the theory of Watanabe-Yoshida (1987, 1992), an alternative widely applicable method has been proposed for approximate maximum-likelihood estimation of any arbitrary multivariate diffusion model; see, Li (2013). ◮ A closed-form small-time asymptotic expansion for transition density (likelihood) was proposed and accompanied by an algorithm for delivering any arbitrary order of the expansion.

Our Goal: How to Deal with Jumps? ◮ Jump-diffusions have been widely used for modeling real-world dynamics of random fluctuations involving both relatively mild diffusive evolutions and discontinuity caused by significant shocks. ◮ Existing expansions: e.g., Schumburg (2001), Yu (2007), and Filipovic (2013). ◮ I propose a closed-form expansion for transition density of jump-diffusion processes, for which any arbitrary order of corrections can be systematically obtained. ◮ As an application, likelihood function is approximated explicitly and thus employed in a new method of approximate maximum-likelihood estimation for jump-diffusion process from discretely sampled data. ◮ Using the theory of Watanabe-Yoshida (1987, 1992) and its generalization to the Levy-driven models in Hayashi and Ishikawa (2012), the convergence related to the density expansion and the approximate estimation method can be theoretically justified under some standard conditions.

A Jump-Diffusion Model dX ( t ) = µ ( X ( t ); θ ) dt + σ ( X ( t ); θ ) dW ( t ) + dJ ( t ; θ ) , X (0) = x 0 (1) where X ( t ) is a d − dimensional random vector; { W ( t ) } is a d − dimensional standard Brownian motion; the unknown parameter θ belonging to a multidimensional open bounded set Θ; J ( t ) is a vector valued jump process modeled by a compounded Poisson process: N ( t ) N ( t ) � � ( Z k , 1 , Z k , 2 , · · · , Z k , d ) ⊤ , J ( t ) ≡ ( J 1 ( t ) , · · · , J d ( t )) T := Z k ≡ k =1 k =1 where { N ( t ) } is a Possion process with an intensity process { λ ( t ) } . Let E ⊂ R d denote the state space of X . We note that various popular jump-diffusion models takes or can be easily transformed into the form of (1), e.g., JD, SVJ, and SVJJ.

The Model and Some Assumptions ◮ Relaxed the condition in the linear drift and diffusion of the affine jump-diffusion model (Duffie et al. (1996)). ◮ As supported by various empirical evidence, the intensity { λ ( t ) } can be choosen as a positive constant λ , which results in the existence and uniqueness of the solution. ◮ For different integers k , Z k = ( Z k , 1 , Z k , 2 , · · · , Z k , d ) ⊤ are i.i.d. multivariate distributions, e.g., normal (double-sided) or (one-sided) exponential. ◮ Without loss of generality, we assume the jump size Z k has a multivariate normal distribution with mean vector α = ( α 1 , α 2 , · · · , α d ) and convariance matrix � � β 2 1 , β 2 2 , · · · , β 2 β =diag ; or Z k has a multivariate d exponential distribution, in which Z k , j ’s are independent and Z k , j has an exponential distribution with intensity γ j .

A Closed-form Expansion of Transition Density ◮ Denote by p (∆ , x | x 0 ; θ ) the conditional density of X ( t + ∆) given X ( t ) = x 0 , i.e. P ( X ( t + ∆) ∈ dx | X ( t ) = x 0 ) = p (∆ , x | x 0 ; θ ) dx . (2) ◮ We will propose a closed-form asymptotic expansion approximation for its transition density (2) in the following form: � 1 � d M � p M (∆ , x | x 0 ; θ ) = √ det D ( x 0 ) Ψ m (∆ , x | x 0 ; θ ) . ∆ m =0 ◮ Here p M denotes an expansion up to the M th order; the functions D ( x 0 ) and Ψ m (∆ , x | x 0 ; θ ) explicitly depending on the drift vector µ , dispersion matrix σ and jump components, will be defined or calculated in what follows. ◮ How to obtain such an expansion and how to pragmatically calculate them symbolically?

Parameterization ◮ For computational convenience, we start from the following equivalent Stratonovich form: dX ( t ) = b ( X ( t )) dt + σ ( X ( t )) ◦ dW ( t ) + dJ ( t ) , X (0) = x 0 . (3) ◮ We parameterize the dynamics (3) as dX ǫ ( t ) = ǫ [ b ( X ǫ ( t )) dt + σ ( X ǫ ( t )) ◦ dW ( t )+ dJ ( t )] , X ǫ (0) = x 0 . ◮ Therefore, if we obtain an expansion for the transition density p ǫ (∆ , x | x 0 ; θ ) dx = P ( X ǫ (∆) ∈ dx | X ǫ (0) = x 0 ) (4) as a series of ǫ, an approximation for (2) can be directly obtained by plugging in ǫ = 1 .

Pathwise Expansions ◮ Expand X ǫ ( t ) as a power series of ǫ around ǫ = 0 . As X ǫ ( t ) admits M � X m ( t ) ǫ m + O ( ǫ M +1 ) , X ǫ ( t ) = m =0 ◮ It is easy to have X 0 ( t ) ≡ x 0 and X 1 ( t ) = b ( x 0 ) t + σ ( x 0 ) W ( t ) + J ( t ) . ◮ Differentiation of the parameterized SDE on both sides, we obtain an iteration algorithm for obtaining higher-order correction terms: dX m ( t ) = b m − 1 ( t ) dt + σ m − 1 ( t ) ◦ dW ( t ) , for m ≥ 2 , where b m − 1 ( t ) and σ m − 1 ( t ) involves products and summations of X m − 1 ( t ) , X m − 2 ( t ) , ..., X 1 ( t ) , X 0 ( t ) .

Pathwise Expansion ◮ We introduce an iterated Stratonovich integration � t � t 1 � t l − 1 S i , f ( t ) := · · · f l ( t l ) ◦ dW i l ( t l ) · · · f 1 ( t 1 ) ◦ dW i 1 ( t 1 ) , 0 0 0 for an arbitrary index i = ( i 1 , i 2 · · · , i l ) ∈ { 0 , 1 , 2 , · · · , d } l and a stochastic process f = { ( f 1 ( t ) , f 2 ( t ) , · · · , f l ( t )) } ◮ The correction term X n ( t ) can be expressed by iterations and multiplications of Stratonovich integrals. ◮ The integrands involve the step function created by jump arrivals, � � l � ∞ � ( Z i , 1 , Z i , 2 , · · · , Z i , d ) T J ( t ) = 1 [ τ l ,τ l +1 ] ( t ) , l =1 i =1 where τ 1 , τ 2 , · · · , are the jump arrival times.

Expansion for Transition Density ◮ A starting point: p ǫ (∆ , x | x 0 ; θ )= E [ δ ( X ǫ (∆) − x ) | X ǫ (0) = x 0 ] . ◮ To guarantee the convergence, our expansion starts from a standardization of X ǫ (∆) into M � X ǫ (∆) − x 0 Y ǫ (∆) := D ( x 0 ) Y m (∆) ǫ m + O ( ǫ M +1 ) , √ = ǫ ∆ m =0 (5) where D ( x ) is a diagonal matrix depending on σ ( x ) . ◮ As ǫ → 0 , Y ǫ (∆) converges to Y 0 (∆) = D ( x 0 ) √ ( σ ( x 0 ) W (∆) + b ( x 0 )∆ + J (∆)) . (6) ∆ This is nondegerate in the sense of Watanabe-Yoshida (1987, 1992) and Hayashi and Ishikawa (2012).

Expansion of Transition Density: a Road Map ◮ By the scaling property of Dirac Delta function, we have E δ ( X ǫ (∆) − x ) � � d 1 det D ( x 0 ) E [ δ ( Y ǫ (∆) − y )] | y = D ( x 0) = √ � . � x − x 0 ∆ ǫ √ ǫ ∆ ◮ We use the classical rule of differentiation to obtain a Taylor expansion of δ ( Y ǫ (∆) − y ) as M � Φ m ( y ) ǫ m + O ( ǫ M +1 ) , δ ( Y ǫ (∆) − y ) = m =0 ◮ Thus, take expectation to obtain that M � Ψ m ( y ) ǫ m + O ( ǫ M +1 ) , E [ δ ( Y ǫ (∆) − y )] := m =0 where Ψ m ( y ) := E [Φ m ( y )] .

Expansion of Transition Density: a Road Map The M th order expansion of the density p ǫ (∆ , x | x 0 ; θ ): � � d � D ( x 0 ) � x − x 0 �� M � 1 ǫ m . p ǫ √ √ M (∆ , x | x 0 ; θ ) = det D ( x 0 ) Ψ m ǫ ∆ ǫ ∆ m =0 By letting ǫ = 1 , we define a M th order approximation to the transition density p (∆ , x | x 0 ; θ ) as � 1 � d � D ( x 0 ) � � M √ √ p M (∆ , x | x 0 ; θ ) := det D ( x 0 ) Ψ m ( x − x 0 ) . ∆ ∆ m =0

Practical Calculation of the Correction Term Conditioning on the total number of jump arrivals, we have � ∞ Ψ m ( y ) = E [Φ m ( y )] = E [Φ m ( y ) | N (∆) = n ] P ( N (∆) = n ) . n =0 We just need to calculate T m , n ( y ) := E [Φ m ( y ) | N (∆) = n ] . Define N th order approximation of Ψ m ( y ) as � N exp( − λ ∆) λ n ∆ n Ψ m , N ( y ) = T m , n ( y ) . n ! n =0 Thus, the M th order approximation of the transition density is further approximated by the following double summation � 1 � d M N � � exp( − λ ∆) λ n ∆ n √ p M , N (∆ , x | x 0 ; θ ) : = det D ( x 0 ) n ! ∆ m =0 n =0 � D ( x 0 ) � x − x 0 �� √ T m , n . ǫ ∆