Control. Econometric Day Mgr. Jakub Petrásek 1. Supervisor: RSJ Invest a.s.,

Transcription

1 and and Econometric Day 2009 Petrásek Department of Probability and Mathematical Statistics, Charles University, RSJ Invest a.s., petrasek@karlin.mff.cuni.cz 2 Department of Probability and Mathematical Statistics, Charles University, RSJ Invest a.s., karel.janecek@rsj.cz November 26, 2009

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4 and 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 Actual returns Brownian Motion increments Figure: Left picture: Returns observed every 6 seconds. In the right one, Brownian Motion incements with the same mean and variance. 2

5 and 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. 3

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7 and 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. 5

8 and 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 and 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). 7

10 and 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 and 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. 9

12 and 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. 10

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14 and 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. 12

15 and 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 (Barndorff 1998) π L ti L ti 1 L ti 1 L ti 2. 2 n t Truncated Quadratic Variation (Hannig 2009) ( ) 2 Lti L I[ Lti ti 1 Lti 1 <g( ti )]. n t are both consistent estimators of σ 2 t. 13

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

17 and 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. 15

18 and 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). 16

19 and Estimation method Maximum Likelihood method performed 1. NIG model scale ᾱ µ σ θ T T Merton model µ σ γ δ λ T T Table: Comparison of maximum likelihood estimates. 1 Estimation performed in software R. Quasi-Newton optimization method, which allows constraints of parameters, was used. 17

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

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22 and 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 and 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. 21

24 and Personal risk aversion Assume geometric BM model, one needs to consider the maximal proportion of wealth an agent would invest. Example toin coss, winner takes 1.2 of a bet, agent s wealth is

25 and Personal risk aversion Assume geometric BM model, one needs to consider the maximal proportion of wealth an agent would invest. Example toin coss, winner takes 1.2 of a bet, agent s wealth is Gambled money Risk Aversion Coefficient Figure: Maximal (red) and optimal (blue) invested amount of money. 22

26 and Personal risk aversion draw-down probability P p (x) = x 2p 1, It is the probability that the investor s discounted wealth will ever fall below fraction x of the initial wealth. Example Logarithmic utility function: P 1 (x) = x. The probability of losing (1 x) percent of investment is x Power utility function for p = 1/2. An agent loses (1 x) percent of investment with probability 1 for any 0 x 1. 23

27 and Theorem ( Proportion and Consumption) Assume the portfolio (4.2) and the objective (4.3). Let { θp = argmin h(θ p ) = argmin 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 θ p is the optimal proportion, β r(1 p) h(θ p) > 0. (4.4) 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 24

28 and A short comment on the theorem A similar theorem presented for geometric process with ν(dz) <, R which is extremely restrictive, see (Framstad 1998). The authors considered power utility function with 0 < p < 1, which describes an extremely aggressive investor. Assumption (4.4) grants that agent s consumption is positive and that his discounted well-being tends to zero as t. 25

29 and Merton proportion Let us denote Merton proportion Merton consumption c M = A(p) = θ M p = α r pσ 2, β r(1 p) p 1 (α r) 2 1 p 2 σ 2 p. How will the proportion and the consumption be changed after adding jumps into the model? 26

30 and consumption and portfolio - preparation An empirical study was performed. 2 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. 2 Computation performed in software R. Integrals numerically evaluated, adaptive quadrature applied. Nonlinear equation solved by Newton method. 27

31 and consumption and portfolio - results Model Naive Merton NIG Merton Jump p = 4 θp cp p = 10 θp cp p = 40 θp cp p = 70 θp cp Table: Comparison of optimal proportion and consumption for Merton and Jump models. β = 10 %, r = 2 %, α = 7 %, σ =

32 and 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,

33 and 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,

34 and III Jan Hannig. Detecting Jumps from Jump Diffusion Karel Janeček. What is a realistic aversion to risk for real-world individual investors? Robert C. Merton. Option pricing when underlying stock returns are discontinuous. Journal Financial Economics, Bernt Øksendal and Agnès Sulem. Applied stochastic control of jump diffusions. 2nd ed. Universitext. Berlin: Springer.,

35 and IV Nishiyama Y. Sueshi, N. in Mathematical Finance: A Comparative Study. Web page of International Congress on and Simulation,

36 and Thank you for attention 33