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1 Financial Modelling in times of crisis: In search of new paradigms By Alex Langnau Allianz Investment Management

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2 Introductory Remarks Note that in , a massive interference of governments around the globe was necessary to avert a global financial melt-down from occuring (at the expense of taking unprecedented debt onto their balance sheets). It must be legitimate to ask the following question(s): Why has the mathematical finance community not collectively been sending out warning signs prior to the crisis similar to scientists warning about Global warming*. Has something gone wrong with our modelling approach? Is the current financial crisis also a crisis of Mathematical Finance? * there are of course individuals like Robert J. Shiller who have been warning of a crisis or several publications foreseeing the possibility of such a event. We refer here more to the lack of collective actions by the community prior to the worst financial crisis in 80 years.

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3 Popular paradigms of mathematical modelling Probably the most common modelling scheme of a stock is based on a random walk hypothesis with average return and volatility and given by Where describes a random (Brownian) shock to the price. In 1905 Einstein used Brownian motion to proof the atomistic interpretation of the world: In a heat-bath larger particles(seen under a microscope) follow a random walk due to kicks by invisible atoms…

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4 Popular paradigms of financial modelling Even when volatility is stochastic, say In this sense Eq.1 also describes the motion of a stock in a heat-bath at constant temperature in equilibrium. Any new information entering the market is instantly thermalised into the price. Note any market-friction or market impact is abstracted away in such a formulation. the picture does not change; it just means that the temperature changes locally in the heat-bath but equilibrium is observed everywhere.

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5 Popular paradigms of financial modelling Adding Levy Jumps can get you out of one equilibrium but instantaneously into another one at different temperature. Standard financial modelling techniques of this type do not explain but only tend to describe! The fact that financial models are largely based on equilibrium assumptions together with their lack of explanatory inside is one of the reasons why the crises went largely undetected by the current mathematical apparatus

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6 Are markets in Equilibrium? Example 1: Cumulative Current Account Balance in Billions over the last 20 year as a measure of net capital flows between countries Net capital flows are far from balanced; an equilibrium is quasi-stable at best.

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7 Are markets in Equilibrium? Example 2: A change of events leading up the crisis could be simplified described as follows: Federal Reserves increases rates in fear of inflation private US household defaults increase Bear Stearns forced to remark their leveraged structured credit book down as credit spreads are widening Creditors force Bear Stearns to unwind part of their book; no one wants to buy and hair- cuts increase all other credits (unrelated to Bear Stearns) need to be re-evaluated across the board with bigger hair-cuts which worsens the credit quality of many portfolios more defaults are happening Standard modelling ansatz of Eq.3 are not even remotely close to capturing this sequence of events !

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8 Are markets in Equilibrium? Summary: Most commonly used modelling techniques that are used in practice are equilibrium models In practice large market moves causes market/human actions which in turn affect the market and so on… Hence the financial world behaves non-linear. However our pricing equations are usually linear equations and hence ignoring feedback effects. Note that events that look extremely remote in a linear world may be far from remote in a non-linear one! Mathematical finance models in most cases completely decouple from capital flows and economics; A mathematical finance person and an economist having lunch may talk about the weather but not much more… (a statement along these lines was presented by Platen during his last visit at the LMU Munich) Requirements for making models more realistic: Correlations between financial assets need to be modelled neither constant or time-dependent but rather state dependent or even path-dependent. In particular in times of crisis market variables become more correlated Market impact and friction as well as finite liquidity creates feedback effects giving rise to non-linear market behaviour. Models should give insights and clues into situations rather than being descriptive only. This could be achieved by making the link between equity and capital flows explicit.

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9 Part 2; Testing new paradigms: Modelling Dynamic Equity-Bond Correlations The objective of part 2 of this talk is to propose a model that brings together all of the aspects mentioned earlier and applying it to the context of a real world situation This work was done in collaboration with Judith Gampe

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10 Testing new paradigms: Modelling Dynamic Equity-Bond Correlations Some empirical facts on equity bond correlations: * Intuitively one expects correlations between bond (E-B)and equity to be positive: The value of a bond is the sum of its discounted coupons whereas the value of a stock is the sum of its discounted future dividends. Hence a increase in rates should affect both roughly the same way R. Connolly, C. Strivers, L. Sun (Journal of Financial and Quantitative Analysis, Vol. 40) conducted an empirical study and show that E-B correlations vary with the level of the VIX. Probability of correlation < 0 VIX > 40%53.8% VIX > 35%48.8% VIX > 30%46.6% VIX > 25%36.5% VIX < 25%6.08% High values of the VIX increases the probability of negative E-B correlations!

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11 Testing new paradigms: Modelling Dynamic Equity-Bond Correlations Let us formulate the model Eq.4a and 4b describe the dynamics of a Equity index in the presence of stochastic volatility; Eq.4c describes the dynamics of a bond index What makes this model interesting is the occurrence of liquidity parameters that enter explicitly into the dynamics.

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12 Testing new paradigms: Modelling Dynamic Equity-Bond Correlations For the standard Equilibrium finance framework is recovered. describe a situation of finite liquidity as buyer (seller) initiated activity impacts the price. Note that the dynamics of Eq.4 has non-linear feedback as the market activities affect the price and the price in turn impacts the strategies. Eq.4 describes an explicit link between the dynamics of capital market instruments and capital flows The liquidity terms define a non-trivial coupling between the initial individual stochastic differential equations

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13 Testing new paradigms: Modelling Dynamic Equity-Bond Correlations Note that a continuity equation for financial flows must be full-filled,e.g. Intuitively what happens is this: An equity sell-off drives the price down (Eq.4a). Eq.5 implies hence the price goes up. This flight to quality scenario induces in effect negative correlations even though.Think of as the E-B correlations when the markets are in equilibrium. The flight to quality disturbs this equilibrium and induces temporary negative correlations.

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14 Testing new paradigms: Modelling Dynamic Equity-Bond Correlations How should be modelled? Note that the empirical work by R. Connolly and co-workers provides clues. We experimented with several strategies. Our favourite one is a psychology ( risk-aversion ) adjusted volatility-target strategy: Consider a portfolio of equity and bonds,e.g. Where denotes the proportion that is invested in Equities at time t.

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15 Testing new paradigms: Modelling Dynamic Equity-Bond Correlations The strategy is given by Where RVar denotes the variance of returns and For a large value of the psychology parameter the strategy consists of simply keeping the total variance of the portfolio constant (rational limit); for small values of the level of risk gets adjusted down

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16 Modelling Dynamic Equity-Bond Correlations For We find good agreement with the studies of Connolly et.al. Connolly et.alPsychology strategy VIX > 40%53.8%53.99% VIX > 35%48.78%48.54% VIX > 30%46.59%46.43% VIX > 25%36.47%36.31% VIX > 20%6.08%6.42%

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17 Some mechanics of the model During a Monte Carlo simulation Eq.4b simulates a new level of volatility for the next time-step. For the new choice of volatility, strategy Eq.6 defines the new investment decision and hence and also by means of Eq.5 The next time-step is completed by means of Equations 4. Note we calibrated the Heston parameters to the VIX as well as the volatility skew of the Equity index.

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18 Conclusions In this talk we presented a model that allows to deviate from the standard equilibrium formulation of finance where equilibrium is disturbed by means of liquidity terms that explicitly describe the interaction between capital flows (human trading behaviour) and tradeable capital markets instruments that is non-linear in nature and models feedback-effects between the market and investment strategies that has higher explanatory power and provides better clues than models based on Brownian motion and Levy Jumps where feedback-effects explain the non-trivial correlations of stocks and bonds. Similarly also credit-spreads and the gold-price is possible that allows completely new ways of looking at risk

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