Mgr. Jakub Petrásek 1. May 4, 2009

Transcription

1 Dissertation Report - First Steps Petrásek Department of Probability and Mathematical Statistics, Charles University petrasek@karlin.mff.cuni.cz 2 RSJ Invest a.s., Department of Probability and Mathematical Statistics, Charles University, karel.janecek@rsj.cz May 4, 2009

2 Outline

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4 Are continuous type models satisfactory facts of financial time series and how Diffusion models (DM) and s (JM) can capture these facts Sudden movements, Heavy tails DM: extremely large volatility term need to be added JM: generic property DeltaFP Brownian Motion Figure 1.1: Left picture: Changes of Price of Futures contract observed every 6 seconds. In the right one, Brownian Motion with the same mean and variance. 2

5 3 Glance at history (1900) L. Bachelier: probabilistic modelling of financial markets using Brownian Motion. (1st half of 20th cent.) P. : processes introduced. (1963) B. B. Mandelbrot: α-stable distribution to model cotton prices. (1973) Black and Scholes: geometric Brownian motion. (1976) R.C. Merton: (Poisson) Jump-Diffusion model. (1998) O.E. Barndorff-Nielsen: Normal Inverse Gaussian process. ( ) Boom in Jump processes.

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7 5 What are processes Assume a given probability space (Ω, F, (F t ), P), with usual conditions. Definition We say that the process L = (L t, t 0), L 0 = 0 is a process if (i) L has stationary increments: L(L t L s ) = L(L t s ), 0 s < t <, (ii) L has independent increments: L t L s F s, 0 s < t <, (iii) L is continuous in probability: L t P Ls, t s.

8 Examples Poisson process L t Po(λt), λ > 0. Density Characteristic function Brownian motion Remark Characteristic function P(L t = k) = (λt)k e λt, k! ψ Lt (u) = exp ( λt ( e iu 1 )). ψ Lt (u) = exp (µtu 12 σ2 tu 2 ). L is if and only if the distribution of L t is infinitely divisible for all t 0. 6

9 7 Notation We denote a jump size at time t L(t) = L(t) L(t ), 0 t <. For A B(R) bounded below we define N(t, A) = # {0 s t, L(s) A}, 0 t <, which is a Poisson process with intensity ν(a) = E (N(1, A)). We introduce a Poisson integral L t = L s = zn(ds, dz). 0 s t [0,t] R We define a compensated poisson random measure Ñ(t, A) = N(t, A) tν(a).

10 Basic theorem I. Theorem (-Itô Decomposition) If L is a process then there is b R, σ 0 and a Poisson random measure N with a measure ν satisfying (1 z 2 )ν(dz) <, such that L t = bt+σw t + z 1 R zñ(t, dz)+ z >1 zn(t, dz), 0 t <. (2.1) The small jumps part z 1 zñ(t, dz) is an L2 -martingale Large jumps part z >1 zn(t, dz) is of finite variation, but may have no finite moments 8

11 9 Basic theorem II. Theorem (Levy-Khintchine formula) Let L be a process, then u R, t 0 and ψ(u) = ibu 1 2 σ2 u 2 + E e iult = e tψ(u), R\{0} ( e iuz 1 iuzi [ z <1] ) ν(dz). As an immediate result we can see that the law of a process L is uniquely determined by the law of L 1.

12 10 Pathwise properties Essentially driven by jumps, càdlàg paths. As an immediate result of -Itô decomposition we see that for every process L s 2 I [ Ls <1] <, t 0, a.s. 0 s t but we allow 0 s t L s I [ Ls <1] =, in which case L is of infinite variaton. t 0, a.s.

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14 12 Outline of modelling phase 1 Making the series stationary we assume that the nonstationarity is basically caused by variable intensity of trading, overcome by appropriate time change. 2 Selecting a model based on empirical facts (moments, variation, tail behavior). 3 Choosing a fitting procedure and get the parameters if analytical density is known, MLE method is used, otherwise GMM method based on characteristic function can be applied.

15 Variation Remark Let L be a process of the form (2.1), n t = {t 0,..., t n } arbitrary partition of interval [0, t] ( 2 P Lti L ti 1) σ 2 t + ( (L s )) 2, n t 0. n t s [0,t] In other words, our estimator of volatility may be deformed by big jumps. Alternatives BiPower Variation ([4]) L ti L ti 1 L ti 1 L ti 2. n t Truncated Quadratic Variation ([7]) ( ) 2 Lti L I Lti ti 1 Lti 1 <g ti. n t are both consistent estimators of σ 2 t. 13

16 14 Comparison of different estimates of standard deviations :30 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 Figure 3.1: Transformed time: green line = Quadratic variation, red = truncated QV, blue = BiPower variation.

17 15 Normal Inverse Gaussian Process can be expressed as L t = B(T t ), where T t = inf {s > 0; W s + αs = δt}, and B t is a Brownian motion with drift θ and volatility σ. Pure jump model with infinite variation. Exponential tail decay. Probability density in a closed (analytical) form (Bessel function), i.e. MLE possible.

18 16 Merton Jump-Diffusion Process can be expressed as N t L t = αt + σw t + Y i, t 0, i=1 i.e. Brownian motion with big gaussian jumps. Tails a little heavier than gaussian. Probability density function can be expressed in a series expansion. We use first order approximation f L t (x) = (1 λ t)f W t (x) + λ t (f W t +Y 1 ) (x).

19 17 Estimation method Maximum Likelihood method performed (used software R with quasi-newton optimization method, which allows constraints of parameters.) NIG model scale ᾱ µ σ θ T T Merton model µ σ γ δ λ T T Table 1: Comparison of maximum likelihood estimates.

20 Graphical inference Figure 3.2: Estimated probability density function: green (solid) line = NIG, red (dashed) = Merton Jump, blue (dotted) = Gaussian. 18

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22 Model set-up I. Consider an investor placing his money into two assets riskfree, paying interest rate r risky asset with dynamics An investor controls df t = αdt + σdw t + the number of F t, t 0 in his portfolio by t, consumption C t 0. i.e. the dynamics of his portfolio is of the form dx t = t (αdt + σdw t + with X (0) = x, t F t (predictable), C t F t. zñ(dt, dz). (4.1) ) zñ(dt, dz) + rx t dt C t (4.2) dt. 20

23 21 Model Set-up II. The objective of an investor is v(x) = sup ( t,c t) A(x) 0 e βt E U(C t )dt, (4.3) where A(x) is the set of admissible strategies, β a discount factor and U denotes a power utility function of the form Notation U(x) = x 1 p 1 p, p > 1. θ p (t) = t X t is the number of assets in the portfolio per one money unit at time t and let c t = Ct X t denotes the proportional consumption.

24 22 Theorem ( Proportion and Consumption) Assume the portfolio (4.2) and the objective (4.3). Let { θp = argmax h(θ p ) = argmax αθ p (1 p) 1 2 σ2 θpp(1 2 p) ( + (1 + θp z) 1 p 1 θ p z(1 p) ) } ν(dz). Assume also that Then β (r c )(1 p) h(θ p) > 0. (4.4) θ p is the optimal proportion, c = (K(1 p)) 1/p is the optimal consumption, v(z) = Kz 1 p is the value function, where K = 1 1 p ( β r(1 p) h(θ ) p p. ) p

25 23 A short comment on the theorem A similar theorem presented for geometric process with ν(dz) <, which is extremely restrictive, see [6]. R Assumption (4.4) grants that agent s consumption is positive and that his discounted well-being tends to zero as t.

26 consumption and portfolio - preparation It is known that 1 θ J p θ C p, 2 c J c C for p > 1, c J c C for p < 1 but how significant is the difference? An empirical study was performed. 1 Futures is a martingale with respect to the risk neutral measure. To compare optimal portfolios based on different models we: standardized the data, so that σ 30%, α is set as 7%. Assume that our (Futures) returns behave like stock log-returns but with different volatility and drift. 1 Computation performed in software R. Integrals numerically evaluated, adaptive quadrature applied. Nonlinear equation solved by Newton method. 24

27 25 consumption and portfolio - results Model Naive Merton NIG Merton Jump p = 2 θp cp p = 4 θp cp p = 6 θp cp Table 2: Comparison of optimal proportion and consumption for Merton and Jump models. β = 10 %, r = 2 %, α = 7 %, σ = 0.3.

28 26 Future work More extensive empirical study. Study of different models (to get better fit of the tail behavior...). Iterated time changing algorithm.

29 27 I Mandelbrot Benoìt B. The variation of certain speculative prices. Journal of Business, XXXVI, Ole E. Barndorff-Nielsen. of Normal Inverse Gaussian Type. Finance and Stochastic, Ole E. Barndorff-Nielsen and Neil Shephard. Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics, 2004.

30 28 II Ole E. Barndorff-Nielsen, Neil Shephard, and Matthias Winkel. Limit theorems for multipower variation in the presence of jumps. Stochastic and Their Applications, Rama Cont and Peter Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series., Nils Chr. Framstad, Bernt Øksendal, and Agnès Sulem. consumption and portfolio in a jump diffusion market. In A. Shyriaev et al (eds): Workshop on Mathematical Finance, 1998.

31 29 III Jan Hannig. Detecting Jumps from Jump Diffusion Bernt Øksendal and Agnès Sulem. Applied stochastic control of jump diffusions. 2nd ed. Universitext. Berlin: Springer., Nishiyama Y. Sueshi, N. in Mathematical Finance: A Comparative Study. Web page of International Congress on and Simulation, 2005.

32 30 Thank you for attention