HPCFinance research project 8

Transcription

1 HPCFinance research projec 8 Financial models, volailiy risk, and Bayesian algorihms Hanxue Yang Tampere Universiy of Technology March 14, 2016

2 Research projec 8 12/ /2015, Tampere Universiy of Technology, Finland Supervisor: Professor Juho Kanniainen Projec opics: - Realisic financial models o capure he volailiy risk and jump risk. - Opion valuaion under advanced models. - Bayesian esimaion mehods for models wih differen jump srucures.

3 Resuls and secondmens Deliverable 3.5: Opion Pricing Performance of Differen Opion Pricing Models Docoral hesis: Markov Chain Mone Carlo Esimaion of Sochasic Volailiy Models wih Finie and Infinie Aciviy Lévy Jumps: Evidence for Efficien Models and Algorihms Publicaions: - Kanniainen, J., B. Lin, and H. Yang (2014), Esimaing and Using GARCH Models wih VIX Daa for Opion Valuaion, Journal of Banking and Finance, 43, Marino, L., H. Yang, D. Luengo, J. Kanniainen, J. Corander (2015), A Fas Universal Self-uned Sampler wihin Gibbs Sampling, Digial Signal Processing, 47, Yang, H. and J. Kanniainen (2016), Jump and Volailiy Dynamics for he S&P 500: Evidence for Infinie-Aciviy Jumps wih Non-Affine Volailiy Dynamics from Sock and Opion Marke, forhcoming in Review of Finance Secondmens: - Maasrich Universiy, Bgaor,

4 Robus financial models o capure he volailiy risk and jump risk We sudy sochasic volailiy models wih finie/infinie-aciviy Lévy jumps in index reurns, or in boh reurns and spo variance, and wih affine/non-affine variance dynamics and compare heir empirical performance in erms of goodness of fi and opion pricing errors.

5 Models Under he risk-neural measure Q, he log reurns and spo variance of he S&P 500 index follow dy = (r 12 ) v + φ QJ ( i) d + v dw Y, + dj Y,, dv = κ Q (θ Q v )d + αv β (ρdw Y, + ) 1 ρ 2 dw v, + dj v,. Benchmark model: affine Heson model. Finie-aciviy Lévy jumps: compound Poisson process. Infinie-aciviy Lévy jumps: Variance Gamma and Normal Inverse Gaussian processes. Non-affine variance specificaion: β = 1.

6 VIX index pricing formula According o he model assumpions and he generic payoff expansion resul of Carr e al. (2001), he heoreical value of he squared VIX follows: VIX 2 T 10 4 = ( [ 2 S log T F 0 (T ) EQ log S ] ) T S 0 (1) = 2 T r sds 2 ( T r sds + T (M J + φ Q J T 0 T [ T ] ) 0 2 EQ v sds, 0 = 2(M J + φ Q J ( i)) e κq T v 0 T κ Q + θ P κ P + Mv T κ Q 1 e κq T κ Q. T is he mauriy of he VIX index, i.e. 22/252 using rading day coun convenion. M J is he expeced mean size of he reurn jumps under Q. For he Variance Gamma jump, M J = γ Q, for he Normal Inverse Gaussian jump, M J = γ Q /ν Q, and for he compound Poisson jump, M J = λ(µ Q y + ρ J µ v). M v is he expeced mean size of he variance jumps under Q. We assume ha he heoreical value of he VIX is relaed o is marke value as follows: where VIX Marke,T = VIX Model,T eε, ε +1 = ρ εε + σ εɛ ε,+1, ɛ ε,+1 N(0, 1).

7 Mehods Daa: S&P 500 index reurn and 30-day VIX from January 1996 o December Esimaion mehod: Markov Chain Mone Carlo. According o he Bayes rule, he join poserior of laen variables and parameers is p(θ, v, J vix, Y) p(vix, Y, v, J, Θ) where p(θ) is he produc of priors of parameers and = p(vix Y, v, J, Θ)p(Y, v J, Θ)p(J Θ)p(Θ), [ ( ) ( )] vix Marke +1 vix Marke 2 log T 1 1 vix Model ρ ε log +1 vix Model p(vix Y, v, J, Θ) = exp =0 2πσε 2σε 2. T 1 p(y, v J, Θ) =0 1 αv 1/2+β exp 1 ρ 2 { ɛ2 Y,+1 2ρɛ Y,+1ɛ v,+1 + ɛ 2 v,+1 2(1 ρ 2 ) where ɛ Y, = ( Y Y 1 ( r 1 2 v 1 + φ Q J ( i) ηsv 1 ) J Y, ) / v 1 and ɛ v, = ( v v 1 κ P (θ P v 1 ) J v, ) /αv β 1 },

8 Goodness of fi and opion pricing performance Table : Deviance Informaion Crierion (DIC) and opion VRMSEs for affine (i = 0) and non-affine (i = 1) models. Sample A: Opions on Wednesdays from January 1996 o December Sample B: Opions on Thursdays from January 1996 o December Sample C: Opions on Wednesdays and Thursdays from January 2010 o December The bolded and ialic numbers are he bes and wors resuls, respecively, and he models are ranked according o heir performance in Sample A. Model Jump ype β DIC A B C NIG-SQR Normal Inverse Gaussian Heson-SQR None Baes-SQR compound Poisson VG-SQR Variance Gamma Model Jump ype β DIC A B C NIG-linear Normal Inverse Gaussian VG-linear Variance Gamma Heson-linear None Baes-linear compound Poisson

9 Discussion The inclusion of infinie-aciviy reurn jumps is criical for non-affine (linear) variance specificaion. Non-affine variance specificaions improve model robusness.

10 Convenional MCMC mehods: Gibbs sampling and Meropolis-Hasings algorihms The general idea of MCMC algorihms is o generae a Markov chain by Mone Carlo echniques, whose equilibrium disribuion is a given arge disribuion, ha is, he join poserior p(θ, X Y), where Θ represens he se of model parameers, X represens he sae variable, and Y represens he se of observaions. Suppose we would like o esimae Θ = {κ} and v = {v } in he Heson model from index reurns Y = {Y }: Gibbs sampling: M is he pre-specified number of MCMC runs, Sep 1 Iniialisaion: Assume he iniial sae: (κ (0), v (0) ), Sep 2 Recursion: For i = 1,..., M, 1. Sample v (i) from p(v (i) Y, v (i 1) +1, v(i) 1, κ(i 1) ), for = 1,..., T, 2. Sample κ (i) from p(κ (i) v (i), Y) = N(a, b), Meropolis-Hasings (MH): for = 1,..., T, 1. Sample v (i) 2. Accep v (i) α(v (i) from he proposal densiy q(v (i) wih probabiliy (, v (i 1) ) = min where π( ) is he poserior. q(v (i 1) q(v (i) v (i 1) ) = N(v (i 1), σ 2 ), v (i) )π(v (i) ) ( ) v (i 1) )π(v (i 1), 1 = min ) π(v (i) ) π(v (i 1), 1 ) The success of MCMC algorihms lies in an appropriae choice of he proposal densiy. ),

11 Advanced MCMC mehods Adapive Meropolis (AM), proposed by Haario e al. (1999), updaes he variance of a Gaussian proposal based on pas samples generaed in he MCMC chain: σ i = s cov(x (1), x (2),..., x (i 1) ) + s d ɛ. Paricle MCMC mehods: Paricle Marginal Meropolis-Hasings (PMMH), proposed by Andrieu e al. (2010), and Paricle Gibbs Ancesor Sampling (PGAS) by Lindsen e al. (2014) approximae he performance of he marginal MH or Gibbs sampling algorihm by generaing a paricle se for sae variables o consruc he proposal disribuion used in he MH algorihm.

12 Fas Universal Self-uned Sampler (FUSS) Proposed by Marino, L., H. Yang, D. Luengo, J. Kanniainen, J. Corander (2015), A Fas Universal Self-uned Sampler wihin Gibbs Sampling, Digial Signal Processing, 47, Iniialisaion: Choose a se of suppor poins, S M = {s 1,..., s M }, such ha s 1 < s 2 <... < s M. 2. Pruning: Remove suppor poins according o a pre-specified crierion, aaining a final se S m, wih m < M. 3. Consrucion: Build a proposal funcion p(x S m) given S m, using some appropriae pre-defined mechanism. 4. MCMC algorihm: Perform K seps of he MCMC mehod using p(x S m) as a proposal pdf, hus yielding a se of samples {x 1,..., x K }. V (x) W (x) s5,v(s5) (x) p(x) (x) p(x) x x (a) Example of he proposal consrucion wih 9 suppor poins, in he log-domain. (b) The corresponding unnormalised densiies p(x) = e W (x) and π(x) = e V (x). (c) The corresponding normalised densiies p(x) p(x) and π(x) π(x).

13 Mehods Simulaion sudies: we esimae he Heson, Baes, and Normal Inverse Gaussian (NIG) models using simulaed one-year log asse prices. Only he physical model dynamics is esimaed. Empirical sudies: we esimae he non-affine NIG model using he join informaion of he S&P 500 index reurns and he 30-day VIX index in

14 Seleced resuls of simulaion sudies Table: Esimaion resuls of he Heson, Baes, and NIG models wih differen algorihms. This able shows he mean squared error of spo variance v (MSE v), he log-likelihood of he parameers and laen sae variables, and he compuaion ime. The compuaion ime is he running ime for he whole algorihm, normalised wih respec o he ime required by he MH algorihm wih a chain of 100,000 runs. Heson model Algorihms Chain lengh Number of paricles MSE v Log-likelihood Time MH E AM E FUSS-RC-P E PMMH-PF E PGAS E Baes model Algorihms Chain lengh Number of paricles MSE v Log-likelihood Time MH E AM E FUSS-RC-P E PMMH-PF E PGAS E NIG model Algorihms Chain lengh Number of paricles MSE v Log-likelihood Time MH E AM E FUSS-RC-P E PMMH-PF E PGAS E

15 v 300 v 600 v 300 v 600 Seleced resuls of empirical sudies: Auocorrelaion funcion (ACF) v 400 v v 400 v v 700 v 800 v v 700 v 800 v (a) MH: ACFs of variance a seleced ime poins. (b) AM: ACFs of variance a seleced ime poins. 1 v v v 400 v 500 v 600 v v 400 v 500 v 800 v v 600 v 700 v 800 v (c) FUSS-RC-P4: ACFs of variance a seleced ime poins. (d) PGAS: ACFs of variance a seleced ime poins.

16 Discussion AM algorihm: able o significanly increase he accepance rae of he MH algorihm; less efficien when models become complex. FUSS algorihms: able o generae virually independen samples wih he fases mixing speed, sable regardless of model specificaions; compuaionally expensive. PMCMC mehods: compuaionally cheaper, efficien in mixing; vulnerable o large independen jumps.