Lecture on advanced volatility models Erik Lindstrm FMS161/MASM18 - PowerPoint PPT Presentation

Lecture on advanced volatility models Erik Lindström FMS161/MASM18 Financial Statistics Erik Lindström Lecture on advanced volatility models

Stochastic Volatility (SV) Let r t be a stochastic process. ◮ The log returns (observed) are given by r t = exp ( V t / 2 ) z t . ◮ The volatility V t is a hidden AR process V t = α + β V t − 1 + e t . ◮ Or more general A ( · ) V t = e t . ◮ More flexible than e.g. EGARCH models! ◮ Multivariate extensions. Erik Lindström Lecture on advanced volatility models

A simulation of Taylor (1982) 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 0 50 100 150 200 250 300 350 400 450 500 Erik Lindström Lecture on advanced volatility models

Long Memory Stochastic Volatility (LMSV) The autocorr. of volatility decays slower than exp. rate ◮ The returns (observed) are given by ( ?? ) r t = exp ( V t / 2 ) z t . ◮ The volatility V t is a hidden, fractionally integrated AR process A ( · )( 1 − q − 1 ) b V t = e t , where b ∈ ( 0 , 0 . 5 ) . ◮ This gives long memory! Erik Lindström Lecture on advanced volatility models

Long Memory Stochastic Volatility (LMSV) ◮ The long memory model can be approximated by a large AR process, cf. ( ? , p 520). ◮ It can be shown that ∞ ( 1 − q − 1 ) b = π j q − j , ∑ j = 0 where Γ( j − b ) π j = Γ( j + 1 )Γ( − b ) . Erik Lindström Lecture on advanced volatility models

Stochastic Volatility in continuous time A popular application of stoch. volatility models is option valuation. ◮ Several parameterizations. ◮ The Heston model is the most used model, mainly due to computational properties V t S t d W ( S ) � d S t = µ S t d t + t V t d W ( V ) � d V t = κ ( θ − V t ) d t + σ t d W ( S ) d W ( V ) = ρ d t t t ◮ Note that the drift and squared diffusion have affine form. ◮ This reduces the task of computing prices to inversion of a Fourier integral. Erik Lindström Lecture on advanced volatility models

Continuous time volatility ◮ We can compute the volatility in a continuous time model. ◮ Advantage: A continuous time model can use data from any time scale, and does not assume that data is equidistantly sampled. ◮ Can derive a limit theory when data is sampled at high frequency. ◮ This is based on the general theory on quadratic variation. Erik Lindström Lecture on advanced volatility models

Quadratic variation ◮ Let { S } be a general semimartingale. ◮ Let π N = { 0 = τ 0 < τ 1 < ... < τ N = T } be a partition of [ 0 , T ] , and denote ∆ = τ n − τ n − 1 , where ∆ = T / N . ◮ Define N ( S ( τ n ) − S ( τ n − 1 )) 2 . ∑ Q N = n = 1 ◮ What are the properties of Q N ? ◮ Q N converges to the quadratic variation . Erik Lindström Lecture on advanced volatility models

Quadratic variation, cont Let S t = σ W t . ◮ Then N ( S ( τ n ) − S ( τ n − 1 )) 2 . ∑ Q N = n = 1 ◮ Note that ( S ( τ n ) − S ( τ n − 1 )) 2 ∼ σ 2 ∆ χ 2 ( 1 ) . ◮ Remember E [ χ 2 ( p )] = p , V [ χ 2 ( p )] = 2 p . ◮ What are the properties of Q N ? ◮ E [ Q N ] = σ 2 ∆ E [ χ 2 ( N )] = σ 2 ∆ N = σ 2 T . � 2 V [ χ 2 ( N )] = � σ 4 T 2 � σ 2 ∆ ◮ V [ Q N ] = � 2 N → 0 N 2 p ◮ Chebyshev’s inequality then states that Q N → σ 2 T . Erik Lindström Lecture on advanced volatility models

Quadratic variation of daily log returns for the Black-Scholes model 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 50 100 150 200 250 300 350 400 450 500 Erik Lindström Lecture on advanced volatility models

Quadratic variation, cont ◮ For a diffusion process d X t = µ ( t , X t ) d t + σ ( t , X t ) d W t , � σ 2 ( s , X s ) d s . the quadratic variation converge to Q N → ◮ For a jump diffusion d X t = µ ( t , X t ) d t + σ ( t , X t ) d W t + d Z t , where { Z } is a Poisson process N t with random jumps of size J i the quadratic variation yields N t � σ 2 ( s , X s ) d s + J 2 ∑ Q N → i . i = 0 Erik Lindström Lecture on advanced volatility models

Realized variation ◮ The quadratic (realized) variation is estimated as N ( S ( τ n ) − S ( τ n − 1 )) 2 . ∑ QV N = n = 1 ◮ The Bipower variation ( ? ) is estimated as N BPV N = π ∑ | S ( τ n + 1 ) − S ( τ n ) || S ( τ n ) − S ( τ n − 1 ) | . 2 n = 1 ◮ It can be shown that the Bipower variation converge to � σ 2 ( s , X s ) d s , for a jump diffusion process (and BPV N → even for a general semimartingale). ◮ The difference between the realized variation and Bipower variation is used to estimate the size of the jump component. Erik Lindström Lecture on advanced volatility models

Example: Realised variation for daily log return of Black-Scholes QV BPV 0.15 0.1 0.05 100 200 300 400 500 600 700 800 900 −3 QV−BPV (jumps ?) x 10 2 1 0 −1 −2 −3 100 200 300 400 500 600 700 800 900 Erik Lindström Lecture on advanced volatility models

Example: Realised variation for daily log return of OMXS30 QV 1 BPV 0.8 0.6 0.4 0.2 1995 2000 2005 2010 QV−BPV (jumps ?) 0.04 0.03 0.02 0.01 0 1995 2000 2005 2010 Erik Lindström Lecture on advanced volatility models