ESTIMATING RISK USING STOCHASTIC VOLATILITY MODELS AND PARTICLE STOCHASTIC APPROXIMATION EXPECTATION MAXIMIZATION HENRIK KRAGH Master’s thesis CENTRUM SCIENTIARUM MATHEMATICARUM 2020:E62 Faculty of Science Centre for Mathematical Sciences Mathematical Statistics Abstract In this thesis several stochastic volatility models are presented and used to estimate the risk of a collection of Swedish stocks, as well as of a portfolio consisting of said stocks. Model parameters are estimated using the PSAEM algorithm. It is concluded that these model are adequate at estimating the one day ahead five percent Value at Risk of the data in terms of conditional coverage. Contents 1 Introduction 1 1.1 Stochastic Volatility . .1 1.2 Thesis Objective . .2 2 Theory and Concepts 3 2.1 Log Returns . .3 2.2 Value at Risk . .3 2.2.1 Backtesting Value at Risk . .3 2.3 Hidden Markov Model . .4 2.4 Maximum Likelihood . .5 2.5 Exponential Family Distributions . .5 3 Sequential Monte Carlo Methods 7 3.1 Expectation Maximization . .7 3.2 Monte Carlo EM . .7 3.3 Sequential Monte Carlo . .8 3.4 Particle Filter . .9 3.5 Particle Gibbs . .9 3.6 Particle Gibbs with Ancestor Sampling . 10 3.7 Particle Stochastic Approximation EM . 11 4 Method 15 4.1 Data . 15 4.2 Simple Univariate Model . 16 4.3 Correlated Univarite Model . 17 4.4 Multivariate Model . 18 4.5 Cholesky Decomposition . 18 4.6 Spherical Parameterization . 19 4.7 Portfolio . 19 5 Results 21 5.1 Simple Univariate Model . 21 5.1.1 Simulation Study . 21 5.1.2 Parameter Estimates . 22 5.1.3 Model Evaluation . 23 5.2 Correlated Univariate Model . 25 5.2.1 Parameter Estimates . 25 5.2.2 Model Evaluation . 26 5.3 Multivariate Model . 27 5.3.1 Parameter Estimates . 28 5.3.2 Model Evaluation . 29 6 Discussion and Conclusion 31 6.1 PSAEM . 31 6.2 Simulation Study . 31 6.3 Simple Univariate Model . 31 6.4 Correlated Univariate Model . 32 6.5 Multivariate Model . 32 6.6 Conclusion . 33 6.7 Future Studies . 33 A Appendix 35 A.1 Exponential Form of the Multivariate Model . 35 A.2 Figures . 36 A.3 Full Parameter Estimates of the Multivariate Model . 39 7 References 41 1 Introduction Financial risk management has become a big part of the global economy, allowing companies as well as private investors to maximize potential earnings while controlling the amount of risk they are exposed to. A big part of risk management within finance is the construction of portfolios consisting of several assets in such a way that the risk of the portfolio matches the risk preferences of its holder. To manage this risk it is essential to be able to estimate and predict it, which requires sufficient understanding of the behaviour of the returns of the assets. Thus finding ways of modeling these returns hold great relevance. These models often have complex likelihoods, as well as require optimization over a very large parameter space (such as the hidden state of a hidden Markov model). For this reason models for the returns of financial assets are commonly applied using Monte Carlo methods, which will be done in this thesis. 1.1 Stochastic Volatility Volatility is a measure of the unforeseeable dispersion that occurs within the value of an asset over time. Thus any method of approximating risk must take into consideration the volatility of the asset. Figure 1: Daily log returns of Holmen B from 2006-01-02 to 2020-01-23. The returns of an asset often varies between periods with low volatility, and periods with high volatility. This behaviour is known as volatility clustering [5]. An example of this is presented in Figure 1, where a cluster of high volatility can be seen at the end of 2008 due to the subprime mortgage crisis. Modern financial models capture this behaviour by allowing volatility to change over time. This often result in models where the volatility is 1 driven by randomness, which are known as stochastic volatility models. 1.2 Thesis Objective The aim of this thesis is twofold. Firstly, to evaluate the performance of the Particle Stochastic Approximation Expectation Maximisation (PSAEM) algorithm when applied to financial models. Secondly, and primarily, to present several new stochastic volatility models for estimating the risk of daily asset returns. These models are evaluated via backtesting of the one day Value at Risk of individual Swedish stocks, as well as of a portfolio consisting of several stocks. 2 2 Theory and Concepts 2.1 Log Returns The log returns (y1, . ,yT ) = y1:T of a process S0:T is given by the logarithm of the current price divided by the price one time unit back. That is, ! St yt = log . (1) St−1 Log returns are typically used to describe changes in value of derivative assets, as they have the nice property that they are time additive. That is, the log return over several time steps is the sum of all individual log returns in the interval. 2.2 Value at Risk The Value at Risk at level α, referred to as VaR(α), of a stochastic return Y is defined as −1 VaR(α) = − max{v : P (Y ≤ v) < α} = −FY (α). That is, the negative α quantile of Y [7]. In this thesis VaR will be used to calculate the one day risk of different assets, based on parametric estimates of the distribution of the conditional log returns yt+1|y1:t,x0:t. At time t we wish to estimate VaR (α) = −F −1 (α). t+1 yt+1|y1:t,x0:t,θ 2.2.1 Backtesting Value at Risk For a given return process y1:T with a corresponding α level Value at Risk process VaR1:T (α), consider the indication process I1:T (α) that is defined by 1 if yt ≤ − VaRt(α) It(α) = (2) 0 if yt > − VaRt(α). For a VaR process which correctly predicts the α quantiles this sequence will have the properties • P (It(α) = 1) = α, for t = 1, . ,T. • Any pair {Ii(α),Ij(α)}, i 6= j are independent. The first property implies that the VaR sequence must correctly predict the α quantile of the return at all time points, and is known as the unconditional coverage property. The second property means that the occurrence of a VaR exceedance must not give any information about future exceedances, known as the independence property. Showing that these two properties holds true turns out to be equivalent to showing that 3 i.i.d. It(α) ∼ B(α) for t = 1, . ,T where B(α) is a Bernoulli distribution with parameter α [4]. This is known as the condi- tional coverage property. To test that the unconditional coverage property of the indicator sequence holds, i.e. that P (It(α) = 1) = α, one can perform the likelihood ratio test for a model where I ∼ B(α) compared to a Bernoulli distribution with a free parameter. This will give the likelihood ratio ! (1 − α)T −nαn LR = −2 log (3) 1 (1 − αˆ)T −nαˆn PT n where n = 1 It(α) and αˆ = T . If the unconditional coverage property holds, LR1 will 2 asymptotically follow a χ1 distribution. Therefore it can be rejected with confidence level 0 −1 0 α if LR1 > F 2 (α ). χ1 The independence property of the indicator sequence can be tested with a likelihood ratio test on whether modeling I1:T as i.i.d. B(ˆα) is sufficient, or if the sequence would be bet- ter modeled as a binary Markov chain with transition probabilities P (It = 0|It−1 = 0) = αˆ00,P (It = 1|It−1 = 1) =α ˆ1,1. This is done using the likelihood ratio ! (1 − αˆ)T −nαˆn LR2 = −2 log (4) n00 n01 n10 n11 (ˆα00) (1 − αˆ00) (1 − αˆ11) αˆ11 nij where n is the number of transitions in I : from state i to state j, and αˆ = . i,j 0 T ij ni0+ni1 2 LR2 will under the second property of I0:T (α) converge to a χ1 distribution. As the three models used in the previous likelihood ratio tests are nested, a test can be constructed which simultaneously test both properties of the indicator sequence. This means that ! (1 − α)T −nαn LRVaR = −2 log (5) α n00 n01 n10 n11 (ˆα00) (1 − αˆ00) (1 − αˆ11) αˆ11 2 will be asymptotically χ2 distributed. Thus the hypothesis that conditional coverage of the 2 indicator sequence holds true can be tested by comparing LRVaRα to the quantiles of a χ2 distribution [4]. 2.3 Hidden Markov Model A pair of stochastic processes {x0:T , y1:T } is called a hidden Markov model (HMM) if x is a Markov process that is not observed. 1:T (6) P (yt|y1, . ,yT ,x1, . ,xT ) = P (yt|xt) for 1 < t < T. This means that at each time step, a new hidden state xt is generated from a distribution that only depends on xt−1, and then a observation yt is generated from xt. 4 2.4 Maximum Likelihood The perhaps most frequently used estimate model parameters based on observations is the maximum likelihood estimator, the parameter values for which the observations are as likely as possible. For an observation y, it is defined as ˆ θML = arg max pθ(y). θ In the case of a stochastic process y1:T with a hidden state x0:T , the likelihood of the process is equal to their joint likelihood integrated over the distribution of x0:T . That is, ˆ θML = arg max pθ(y1:T ) = arg max pθ(y1:T , x0:T )dx0:T .