The thermodynamics of risk

Doctoral thesis English OPEN
Mascie-Taylor, Jonathan Hugo
  • Subject: HG

It is now routine to consider the full probability distribution of downturns in many sectors. In the financial services sector regulators (both internal and external) require corporations not only to measure their risk, but also to hold a sufficient amount of capital to cover potential losses given that risk. Another example is in emergency service vehicle routing, where one needs to be able to reliably get to a destination within a fixed limit of time, rather than taking a route which may have a shorter expected travel time but could, under certain travel conditions, take significantly longer [Samaranayake et al., 2012]. Further examples can be found in food hygiene [Pouillot et al., 2007] and technology infrastructure [Buyya et al., 2009].\ud \ud In the first part of the thesis we consider the implications of risk in portfolio optimisation. We construct an algorithm which allows for the efficient optimisation of a portfolio at various risk points. During this work we assume that the value at risk can only be estimated via sampling; this is because it would be near impossible to analytically capture the probability distribution of a large portfolio. We focus initially on optimising a single risk point but later expand the work to the optimisation of multiple risk points. We study the ensemble defined by the algorithm, and also various approximations of it are then used to both improve the algorithm but also to question exactly what we should be optimising when we wish to minimise risk. The key challenge in constructing such an algorithm is to consider how much the optimisation method biases the samples used to estimate the value at risk. We wish to select genuinely better solutions; not just solutions which were somehow lucky, and hence treated more favourably, during the optimisation process.\ud \ud In the second part of the thesis we switch our focus to considering how we can understand when large losses will occur. In the financial services sector this translates to asking the question: under what market conditions will I make a (very) significant loss, or even go bankrupt? We consider various methods of answering this question. The initial algorithm relies heavily on an understanding of how our portfolio is modelled but we work to extend this algorithm so that no prior knowledge of the system is required.\ud \ud In the final chapter we discuss some further implications and possible future directions of this work.
  • References (14)
    14 references, page 1 of 2

    Chapter 1 Introduction 1 1.1 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Portfolio Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Reverse Stress Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    Chapter 2 Background 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 The Gamma and Beta Functions . . . . . . . . . . . . . . . . 10 2.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . 12 2.3.3 Generalised Hyperbolic Distributions . . . . . . . . . . . . . . 12 2.4 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.1 Monte Carlo Sampling . . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . 17 2.5.3 Rare Event Sampling . . . . . . . . . . . . . . . . . . . . . . 18 2.5.4 Thermal Integration . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.5 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . 19

    Chapter 3 Optimising Quantile Risk 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Problem De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 The Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Calculating the Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Numerical Convergence of Estimators . . . . . . . . . . . . . . . . . 41 3.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7.1 Example Speci c Comments . . . . . . . . . . . . . . . . . . . 56 3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    Chapter 4 Optimising Quantile Risk: Interpreting the Physics 59 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Problem De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 New Risk Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Bias to Zero Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.5.1 The Approximation . . . . . . . . . . . . . . . . . . . . . . . 64 4.5.2 An Example VD(R) . . . . . . . . . . . . . . . . . . . . . . . 66 4.5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.4 Algorithmic Implications . . . . . . . . . . . . . . . . . . . . 75 4.5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Adapting k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6.1 The Approximation . . . . . . . . . . . . . . . . . . . . . . 84 4.6.2 An Example VD(R) . . . . . . . . . . . . . . . . . . . . . . . 85 4.6.3 Algorithmic Implications . . . . . . . . . . . . . . . . . . . . 86 4.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.7 System Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    Chapter 6 Estimating Extreme Risk 108 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2.1 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . 110 6.2.2 Estimating Extreme Risk in the NIG Distribution . . . . . . 112 6.2.3 Extending to d Dimensions . . . . . . . . . . . . . . . . . . . 113 6.3 Simple Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4 Large Single Stock Loss . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.5 Large Portfolio Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    Chapter 7 Estimating Extreme Risk: Thermal Integration 123 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.2.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2.2 Expected Performance . . . . . . . . . . . . . . . . . . . . . . 126 7.2.3 Sampler: Log Normal Distribution . . . . . . . . . . . . . . . 127 7.2.4 Sampler: Chi Adjustment . . . . . . . . . . . . . . . . . . . . 128 7.2.5 Algorithm Parameters . . . . . . . . . . . . . . . . . . . . . . 129 7.3 Example 1: 1D NIG Distribution . . . . . . . . . . . . . . . . . . . . 129 7.4 Example 2: Financial Stocks (one dimension) . . . . . . . . . . . . . 132 7.5 Example 3: Financial Portfolio (n dimensions) . . . . . . . . . . . . 133 7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

    h 0 0 0 0 0 0 0 in

    C 1 1 1 1 1 1 1 t

    E. Anderson and M. Ferris. A direct search algorithm for optimization with noisy function evaluations. SIAM J. Optim., 11 (3):837{857, 2001.

    Domenico Cuocoa and Hong Liu. An analysis of var-based capital requirements. Journal of Financial Intermediation, 15:362{394, 2006.

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