Lecture on advanced volatility models

Transcription

1 FMS161/MASM18 Financial Statistics

2 Stochastic Volatility (SV) Let r t be a stochastic process. The log returns (observed) are given by (Taylor, 1982) r t = exp(v t /2)z t. The volatility V t is a hidden AR process V t = α + βv t 1 + e t. Or more general A( )V t = e t. More flexible than e.g. EGARCH models! Multivariate extensions.

3 A simulation of Taylor (1982)

4 Long Memory Stochastic Volatility (LMSV) The autocorr. of volatility decays slower than exp. rate The returns (observed) are given by (Breidt and Crato and de Lima, 1998; Harvey, 1998) 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!

5 Long Memory Stochastic Volatility (LMSV) The long memory model can be approximated by a large AR process, cf. (Brockwell and Davis, 1991, p 520). It can be shown that (1 q 1 ) b = j=0 π j q j, where π j = Γ(j b) Γ(j + 1)Γ( b).

6 Stochastic Volatility in continuous time A popular application of stoch. volatility models is option valuation. Several parameterizations. The Heston model (Heston, 1993) is the most used model, mainly due to computational properties ds t = µs t dt + V t S t dw (S) t dv t = κ(θ V t )dt + σ V t dw (V ) t dw (S) t dw (V ) t = ρdt Note that the drift and squared diffusion have affine form. This reduces the task of computing prices to inversion of a Fourier integral.

7 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.

8 Quadratic variation References 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 Q N = N n=1 What are the properties of Q N? (S(τ n ) S(τ n 1 )) 2. Q N converges to the quadratic variation.

9 Quadratic variation, cont Let S t = σw t. Then Q N = N n=1 (S(τ n ) S(τ n 1 )) 2. 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)] = 2p. What are the properties of Q N? E[Q N ] = σ 2 E[χ 2 (N)] = σ 2 N = σ 2 T. V[Q N ] = ( σ 2 ) ( ) 2 V[χ 2 (N)] = σ 4 T 2 2N 0 N 2 Chebyshev s inequality then states that Q N p σ 2 T.

10 Quadratic variation of daily log returns for the Black-Scholes model

11 Quadratic variation, cont For a diffusion process dx t = µ(t,x t )dt + σ(t,x t )dw t, the quadratic variation converge to Q N σ 2 (s,x s )ds. For a jump diffusion dx t = µ(t,x t )dt + σ(t,x t )dw t + dz t, where {Z } is a Poisson process N t with random jumps of size J i the quadratic variation yields Q N σ 2 (s,x s )ds + N t i=0 J 2 i.

12 Realized variation References The quadratic (realized) variation is estimated as QV N = N n=1 (S(τ n ) S(τ n 1 )) 2. The Bipower variation (Barndorff-Nielsen and Shephard, 2004) is estimated as BPV N = π 2 N S(τ n+1 ) S(τ n ) S(τ n ) S(τ n 1 ). n=1 It can be shown that the Bipower variation converge to BPV N σ 2 (s,x s )ds, for a jump diffusion process (and even for a general semimartingale). The difference between the realized variation and Bipower variation is used to estimate the size of the jump component.

13 Example: Realised variation for daily log return of Black-Scholes 0.15 QV BPV x 10 3 QV BPV (jumps?)

14 Example: Realised variation for daily log return of OMXS QV BPV QV BPV (jumps?)

15 References Barndorff-Nielsen, Ole E. and Shephard, Neil. (2004). Power and Bipower Variation with Stochastic Volatility and Jumps, Journal of Financial Econometrics, 2, Breidt, F. J., Crato, N., and de Lima, P. (1998). The detection and estimation of long memory in stochastic volatility. Journal of Econometrics, 83, Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods, Springer-Verlag,Second Edition. Harvey, A. C. (1998). Long memory in stochastic volatility. In: Knight, J., Satchell, S. (Eds.), Forecasting volatility in financial markets. Butterworth-Heinemann, London. Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies, 6, Taylor, S.J. (1982), Financial Returns Modelled by the Product of Two Stochastic Processes, a Study of Daily Sugar Prices , in O.D. Anderson (eds.), Time Series Analysis: Theory and Practice, North-Holland, Amsterdam,