Fat Tails Archives - QUANTITATIVE RESEARCH AND TRADING

December 22, 2020

Regime-Switching & Market State Modeling

The Excel workbook referred to in this post can be downloaded here.

Market state models are amongst the most useful analytical techniques that can be helpful in developing alpha-signal generators. That term covers a great deal of ground, with ideas drawn from statistics, econometrics, physics and bioinformatics. The purpose of this short note is to provide an introduction to some of the key ideas and suggest ways in which they might usefully applied in the context of researching and developing trading systems.

Although they come from different origins, the concepts presented here share common foundational principles:

Markets operate in different states that may be characterized by various measures (volatility, correlation, microstructure, etc);
Alpha signals can be generated more effectively by developing models that are adapted to take account of different market regimes;
Alpha signals may be combined together effectively by taking account of the various states that a market may be in.

Market state models have shown great promise is a variety of applications within the field of applied econometrics in finance, not only for price and market direction forecasting, but also basis trading, index arbitrage, statistical arbitrage, portfolio construction, capital allocation and risk management.

REGIME SWITCHING MODELS

These are econometric models which seek to use statistical techniques to characterize market states in terms of different estimates of the parameters of some underlying linear model. This is accompanied by a transition matrix which estimates the probability of moving from one state to another.

To illustrate this approach I have constructed a simple example, given in the accompanying Excel workbook. In this model the market operates as follows:

Where

Y_t is a variable of interest (e.g. the return in an asset over the next period t)

e_t is an error process with constant variance s²

S is the market state, with two regimes (S=1 or S=2)

a₀ is the drift in the asset process

a₁ is an autoregressive term, by which the return in the current period is dependent on the prior period return

b₁ is a moving average term, which smoothes the error process

This is one of the simplest possible structures, which in more general form can include multiple states, and independent regressions X_i as explanatory variables (such as book pressure, order flow, etc):

The form of the error process e_tmay also be dependent on the market state. It may simply be that, as in this example, the standard deviation of the error process changes from state to state. But the changes can also be much more complex: for instance, the error process may be non-Gaussian, or it may follow a formulation from the GARCH framework.

In this example the state parameters are as follows:

	Reg1	Reg 2
s	0.01	0.02
a0	0.005	-0.015
a1	0.40	0.70
b1	0.10	0.20

What this means is that, in the first state the market tends to trend upwards with relatively low volatility. In the second state, not only is market volatility much higher, but also the trend is 3x as large in the negative direction.

I have specified the following state transition matrix:

	Reg1	Reg2
Reg1	0.85	0.15
Reg2	0.90	0.10

This is interpreted as follows: if the market is in State 1, it will tend to remain in that state 85% of the time, transitioning to State 2 15% of the time. Once in State 2, the market tends to revert to State 1 very quickly, with 90% probability. So the system is in State 1 most of the time, trending slowly upwards with low volatility and occasionally flipping into an aggressively downward trending phase with much higher volatility.

The Generate sheet in the Excel workbook shows how observations are generated from this process, from which we select a single instance of 3,000 observations, shown in sheet named Sample.

The sample looks like this:

As anticipated, the market is in State 1 most of the time, occasionally flipping into State 2 for brief periods.

It is well-known that in financial markets we are typically dealing with highly non-Gaussian distributions. Non-Normality can arise for a number of reasons, including changes in regimes, as illustrated here. It is worth noting that, even though in this example the process in either market state follows a Gaussian distribution, the combined process is distinctly non-Gaussian in form, having (extremely) fat tails, as shown by the QQ-plot below.

If we attempt to fit a standard ARMA model to the process, the outcome is very disappointing in terms of the model’s poor explanatory power (R² 0.5%) and lack of fit in the squared-residuals:

ARIMA(1,0,1)

Estimate Std. Err. t Ratio p-Value

Intercept 0.00037 0.00032 1.164 0.244

AR1 0.57261 0.1697 3.374 0.001

MA1 -0.63292 0.16163 -3.916 0

Error Variance^(1/2) 0.02015 0.0004 —— ——

Log Likelihood = 7451.96

Schwarz Criterion = 7435.95

Hannan-Quinn Criterion = 7443.64

Akaike Criterion = 7447.96

Sum of Squares = 1.2172

R-Squared = 0.0054

R-Bar-Squared = 0.0044

Residual SD = 0.0202

Residual Skewness = -2.1345

Residual Kurtosis = 5.7279

Jarque-Bera Test = 3206.15 {0}

Box-Pierce (residuals): Q(48) = 59.9785 {0.115}

Box-Pierce (squared residuals): Q(50) = 78.2253 {0.007}

Durbin Watson Statistic = 2.01392

KPSS test of I(0) = 0.2001 {<1} *

Lo’s RS test of I(0) = 1.2259 {<0.5} *

Nyblom-Hansen Stability Test: NH(4) = 0.5275 {<1}

MA form is 1 + a_1 L +…+ a_q L^q.

Covariance matrix from robust formula.

* KPSS, RS bandwidth = 0.

Parzen HAC kernel with Newey-West plug-in bandwidth.

However, if we keep the same simple form of ARMA(1,1) model, but allow for the possibility of a two-state Markov process, the picture alters dramatically: now the model is able to account for 98% of the variation in the process, as shown below.

Notice that we have succeeded in estimating the correct underlying transition probabilities, and how the ARMA model parameters change from regime to regime much as they should (small positive drift in one regime, large negative drift in the second, etc).

Markov Transition Probabilities

P(.|1) P(.|2)

P(1|.) 0.080265 0.14613

P(2|.) 0.91973 0.85387

Estimate Std. Err. t Ratio p-Value

Logistic, t(1,1) -2.43875 0.1821 —— ——

Logistic, t(1,2) -1.76531 0.0558 —— ——

Non-switching parameters shown as Regime 1.

Regime 1:

Intercept -0.05615 0.00315 -17.826 0

AR1 0.70864 0.16008 4.427 0

MA1 -0.67382 0.16787 -4.014 0

Error Variance^(1/2) 0.00244 0.0001 —— ——

Regime 2:

Intercept 0.00838 2e-005 419.246 0

AR1 0.26716 0.08347 3.201 0.001

MA1 -0.26592 0.08339 -3.189 0.001

Log Likelihood = 12593.3

Schwarz Criterion = 12557.2

Hannan-Quinn Criterion = 12574.5

Akaike Criterion = 12584.3

Sum of Squares = 0.0178

R-Squared = 0.9854

R-Bar-Squared = 0.9854

Residual SD = 0.002

Residual Skewness = -0.0483

Residual Kurtosis = 13.8765

Jarque-Bera Test = 14778.5 {0}

Box-Pierce (residuals): Q(48) = 379.511 {0}

Box-Pierce (squared residuals): Q(50) = 36.8248 {0.917}

Durbin Watson Statistic = 1.50589

KPSS test of I(0) = 0.2332 {<1} *

Lo’s RS test of I(0) = 2.1352 {<0.005} *

Nyblom-Hansen Stability Test: NH(9) = 0.8396 {<1}

MA form is 1 + a_1 L +…+ a_q L^q.

Covariance matrix from robust formula.

* KPSS, RS bandwidth = 0.

Parzen HAC kernel with Newey-West plug-in bandwidth.

There are a variety of types of regime switching mechanisms we can use in state models:

Hamiltonian – the simplest, where the process mean and variance vary from state to state

Markovian – the approach used here, with state transition matrix

Explained Switching – where the process changes state as a result of the influence of some underlying variable (such as interest rate volatility, for example)

Smooth Transition – comparable to explained Markov switching, but without and explicitly probabilistic interpretation.

This example is both rather simplistic and pathological at the same time: the states are well-separated , by design, whereas for real processes they tend to be much harder to distinguish. A difficulty of this methodology is that the models can be very difficult to estimate. The likelihood function tends to be very flat and there are a great many local maxima that give similar fit, but with widely varying model forms and parameter estimates. That said, this is a very rich class of models with a great many potential applications.

September 29, 2020

Robustness in Quantitative Research and Trading

What is Strategy Robustness? What is its relevance to Quantitative Research and Trading?

One of the most highly desired properties of any financial model or investment strategy, by investors and managers alike, is robustness. I would define robustness as the ability of the strategy to deliver a consistent results across a wide range of market conditions. It, of course, by no means the only desirable property – investing in Treasury bills is also a pretty robust strategy, although the returns are unlikely to set an investor’s pulse racing – but it does ensure that the investor, or manager, is unlikely to be on the receiving end of an ugly surprise when market conditions adjust.

Robustness is not the same thing as low volatility, which also tends to be a characteristic highly prized by many investors. A strategy may operate consistently, with low volatility in certain market conditions, but behave very differently in other. For instance, a delta-hedged short-volatility book containing exotic derivative positions. The point is that empirical researchers do not know the true data-generating process for the markets they are modeling. When specifying an empirical model they need to make arbitrary assumptions. An example is the common assumption that assets returns follow a Gaussian distribution. In fact, the empirical distribution of the great majority of asset process exhibit the characteristic of “fat tails”, which can result from the interplay between multiple market states with random transitions. See this post for details:

http://jonathankinlay.com/2014/05/a-quantitative-analysis-of-stationarity-and-fat-tails/

In statistical arbitrage, for example, quantitative researchers often make use of cointegration models to build pairs trading strategies. However the testing procedures used in current practice are not sufficient powerful to distinguish between cointegrated processes and those whose evolution just happens to correlate temporarily, resulting in the frequent breakdown in cointegrating relationships. For instance, see this post:

http://jonathankinlay.com/2017/06/statistical-arbitrage-breaks/

Modeling Assumptions are Often Wrong – and We Know It

We are, of course, not the first to suggest that empirical models are misspecified:

“All models are wrong, but some are useful” (Box 1976, Box and Draper 1987).

Martin Feldstein (1982: 829): “In practice all econometric specifications are necessarily false models.”

Luke Keele (2008: 1): “Statistical models are always simplifications, and even the most complicated model will be a pale imitation of reality.”

Peter Kennedy (2008: 71): “It is now generally acknowledged that econometric models are false and there is no hope, or pretense, that through them truth will be found.”

During the crash of 2008 quantitative Analysts and risk managers found out the hard way that the assumptions underpinning the copula models used to price and hedge credit derivative products were highly sensitive to market conditions. In other words, they were not robust. See this post for more on the application of copula theory in risk management:

http://jonathankinlay.com/2017/01/copulas-risk-management/

Robustness Testing in Quantitative Research and Trading

We interpret model misspecification as model uncertainty. Robustness tests analyze model uncertainty by comparing a baseline model to plausible alternative model specifications. Rather than trying to specify models correctly (an impossible task given causal complexity), researchers should test whether the results obtained by their baseline model, which is their best attempt of optimizing the specification of their empirical model, hold when they systematically replace the baseline model specification with plausible alternatives. This is the practice of robustness testing.

Robustness testing analyzes the uncertainty of models and tests whether estimated effects of interest are sensitive to changes in model specifications. The uncertainty about the baseline model’s estimated effect size shrinks if the robustness test model finds the same or similar point estimate with smaller standard errors, though with multiple robustness tests the uncertainty likely increases. The uncertainty about the baseline model’s estimated effect size increases of the robustness test model obtains different point estimates and/or gets larger standard errors. Either way, robustness tests can increase the validity of inferences.

Robustness testing replaces the scientific crowd by a systematic evaluation of model alternatives.

Robustness in Quantitative Research

In the literature, robustness has been defined in different ways:

as same sign and significance (Leamer)
as weighted average effect (Bayesian and Frequentist Model Averaging)
as effect stability We define robustness as effect stability.

Parameter Stability and Properties of Robustness

Robustness is the share of the probability density distribution of the baseline model that falls within the 95-percent confidence interval of the baseline model. In formulaeic terms:

Robustness is left-–right symmetric: identical positive and negative deviations of the robustness test compared to the baseline model give the same degree of robustness.
If the standard error of the robustness test is smaller than the one from the baseline model, ρ converges to 1 as long as the difference in point estimates is negligible.
For any given standard error of the robustness test, ρ is always and unambiguously smaller the larger the difference in point estimates.
Differences in point estimates have a strong influence on ρ if the standard error of the robustness test is small but a small influence if the standard errors are large.

Robustness Testing in Four Steps

Define the subjectively optimal specification for the data-generating process at hand. Call this model the baseline model.
Identify assumptions made in the specification of the baseline model which are potentially arbitrary and that could be replaced with alternative plausible assumptions.
Develop models that change one of the baseline model’s assumptions at a time. These alternatives are called robustness test models.
Compare the estimated effects of each robustness test model to the baseline model and compute the estimated degree of robustness.

Model Variation Tests

Model variation tests change one or sometimes more model specification assumptions and replace with an alternative assumption, such as:

change in set of regressors
change in functional form
change in operationalization
change in sample (adding or subtracting cases)

Example: Functional Form Test

The functional form test examines the baseline model’s functional form assumption against a higher-order polynomial model. The two models should be nested to allow identical functional forms. As an example, we analyze the ‘environmental Kuznets curve’ prediction, which suggests the existence of an inverse u-shaped relation between per capita income and emissions.

Note: grey-shaded area represents confidence interval of baseline model

Another example of functional form testing is given in this review of Yield Curve Models:

http://jonathankinlay.com/2018/08/modeling-the-yield-curve/

Random Permutation Tests

Random permutation tests change specification assumptions repeatedly. Usually, researchers specify a model space and randomly and repeatedly select model from this model space. Examples:

sensitivity tests (Leamer 1978)
artificial measurement error (Plümper and Neumayer 2009)
sample split – attribute aggregation (Traunmüller and Plümper 2017)
multiple imputation (King et al. 2001)

We use Monte Carlo simulation to test the sensitivity of the performance of our Quantitative Equity strategy to changes in the price generation process and also in model parameters:

http://jonathankinlay.com/2017/04/new-longshort-equity/

Structured Permutation Tests

Structured permutation tests change a model assumption within a model space in a systematic way. Changes in the assumption are based on a rule, rather than random. Possibilities here include:

sensitivity tests (Levine and Renelt)
jackknife test
partial demeaning test

Example: Jackknife Robustness Test

The jackknife robustness test is a structured permutation test that systematically excludes one or more observations from the estimation at a time until all observations have been excluded once. With a ‘group-wise jackknife’ robustness test, researchers systematically drop a set of cases that group together by satisfying a certain criterion – for example, countries within a certain per capita income range or all countries on a certain continent. In the example, we analyse the effect of earthquake propensity on quake mortality for countries with democratic governments, excluding one country at a time. We display the results using per capita income as information on the x-axes.

Upper and lower bound mark the confidence interval of the baseline model.

Robustness Limit Tests

Robustness limit tests provide a way of analyzing structured permutation tests. These tests ask how much a model specification has to change to render the effect of interest non-robust. Some examples of robustness limit testing approaches:

unobserved omitted variables (Rosenbaum 1991)
measurement error
under- and overrepresentation
omitted variable correlation

For an example of limit testing, see this post on a review of the Lognormal Mixture Model:

http://jonathankinlay.com/2018/08/the-lognormal-mixture-variance-model/

Summary on Robustness Testing

Robustness tests have become an integral part of research methodology. Robustness tests allow to study the influence of arbitrary specification assumptions on estimates. They can identify uncertainties that otherwise slip the attention of empirical researchers. Robustness tests offer the currently most promising answer to model uncertainty.

December 10, 2019

Modeling Asset Processes

Introduction

Over the last twenty five years significant advances have been made in the theory of asset processes and there now exist a variety of mathematical models, many of them computationally tractable, that provide a reasonable representation of their defining characteristics.

While the Geometric Brownian Motion model remains a staple of stochastic calculus theory, it is no longer the only game in town. Other models, many more sophisticated, have been developed to address the shortcomings in the original. There now exist models that provide a good explanation of some of the key characteristics of asset processes that lie beyond the scope of models couched in a simple Gaussian framework. Features such as mean reversion, long memory, stochastic volatility, jumps and heavy tails are now readily handled by these more advanced tools.

In this post I review a critical selection of asset process models that belong in every financial engineer’s toolbox, point out their key features and limitations and give examples of some of their applications.

Modeling Asset Processes

December 3, 2019

Conditional Value at Risk Models

One of the most widely used risk measures is the Value-at-Risk, defined as the expected loss on a portfolio at a specified confidence level. In other words, VaR is a percentile of a loss distribution.
But despite its popularity VaR suffers from well-known limitations: its tendency to underestimate the risk in the (left) tail of the loss distribution and its failure to capture the dynamics of correlation between portfolio components or nonlinearities in the risk characteristics of the underlying assets.

One method of seeking to address these shortcomings is discussed in a previous post Copulas in Risk Management. Another approach known as Conditional Value at Risk (CVaR), which seeks to focus on tail risk, is the subject of this post. We look at how to estimate Conditional Value at Risk in both Gaussian and non-Gaussian frameworks, incorporating loss distributions with heavy tails and show how to apply the concept in the context of nonlinear time series models such as GARCH.

Var, CVaR and Heavy Tails

November 26, 2019

Copulas in Risk Management

Copulas in Risk Managment

September 27, 2018

Crash-Protecting Your Portfolio With CrashMetrics

In a post on LinkedIn I referred to the concept of CrashMetrics and how it can be used for portfolio protection. It’s a simple approach to the management of extreme risk that works rather well. It can be summarized as “CAPM for crashes”. Here’s how it works.

Let’s take Proctor and Gamble as our example stock. We’ll use daily data from 1970-2014 for the stock and for the S&P 500 Index, as follows:

DateListPlot[PG=TimeSeries[FinancialData[“PG”,{{1970,1,2},{2014,12,31}}]],Filling->Axis]

DateListPlot[SP500=TimeSeries[FinancialData[“^GSPC”,{{1970,1,2},{2014,12,31}}]],Filling->Axis]

We are also going to need an estimate of the risk free rate of return. We’ll use a 30-day T-Bill rate:

DateListPlot[TBill=TimeSeries[FinancialData[“^IRX”,{{1970,1,2},{2014,12,31}}]],Filling->Axis]

CAPM Beta Estimation

Next we convert the annual Bill yields into estimates of the continuously compounded daily return and subtract these from the gross returns for PG and the S&P 500 Index, to create series of excess returns for the stock and the index.

SP500Dates=SP500[“Times”]
PGReturns=Log[PG[Drop[SP500Dates,1]]]-Log[PG[Drop[SP500Dates,-1]]];
SP500Returns=Log[Drop[SP500[“Values”],1]]-Log[Drop[SP500[“Values”],-1]];
TBillDailyRate=Log[TBill[Drop[SP500Dates,1]]]/250 /. Indeterminate->0;
Histogram[PGXReturns=PGReturns-TBillDailyRate,{-0.05,0.05,0.001}]

Excess Returns PG 1970-2014

Excess Returns S&P 500 Index 1970-2014

We are now ready to estimate the stock beta for PG, using a simple linear regression model of the excess returns in the stock vs. the excess returns in the S&P 500 Index:

dataset=Partition[Riffle[SP500XReturns,PGXReturns],2
CAPM = LinearModelFit[dataset,x,x]

From which we estimate the beta for PG to be around 0.78 (the slope of the regression line in the scatterplot below). That seems plausible for a large, diversified consumer goods manufacturer, which is likely to be less volatile than the broad index during normal market conditions.

Show[ListPlot[dataset],Plot[CAPM[x],{x,-0.05,0.05},PlotStyle->Red]]

The CAPM regression shows that around 40% of the variation in excess returns in PG is explained by movements in the broad market (the remainder is due to stock-specific risk factors):

CAPM[“AdjustedRSquared”] = 0.40

That’s a typically scenario with the CAPM model, which is based on some fairly simple, but rather heroic assumptions that we need not delve too deeply into here.

CrashMetrics Approach

In CrashMetrics we focus exclusively in the left tail of the distribution. For the S&P 500 index the average excess return is very close to zero, while the daily standard deviation of returns is just over 1.5%. So let’s focus on down-moves that are, say, at least 3xSD, or larger. We create a reduced data set comprising days on which the index declined by at least 4.5%, and repeat the regression procedure using just those 54 days:

Dimensions[reducedDataset=Select[dataset,#[[1]] < -0.045&]{54,2}

CrashM = LinearModelFit[reducedDataset,x,x]

Show[ListPlot[reducedDataset],Plot[CrashM[x],{x,0, -0.25},PlotStyle->Red]]

Two points are especially noteworthy.

The first is that the beta for PG during major market down-moves is a lot higher than during normal markets (around 1.34 vs 0.78) and being greater than 1, indicates that during adverse conditions PG tends to exacerbate the down-turn in the broad market.

The second is that the regression R-squared is much higher (0.68) for the CrashMetrics regression model, reflecting the tendency of stocks to correlate more closely with the market index during major sell-offs. In that sense, the “crash-beta” estimate is a more reliable estimate than the regular CAPM beta.

How to Use CrashMetrics

How is this technique helpful to the portfolio manager?

To begin, you might want to estimate crash-betas for all of the stocks in your portfolio and for the portfolio as a whole, to give you a handle on how the portfolio is likely to behave under extreme stress.

You could then choose to make adjustments to the portfolio composition to reduce its crash exposure. This can be done by reducing the allocations to high crash-beta stocks in favor of low crash-beta stocks. Alternatively, you can buy tail protection using out-of-the-money put options in high-crash beta stocks. What’s interesting about this technique is that you might end up paying less for crash-protection than you might think.

Taking our PG test case as an example, this is typically seen as a less risky stock and its options are priced accordingly. Consequently, the Gamma in the options looks cheap when considering how the stock behaves during market crashes. Conversely, options in very volatile stocks (AAPL springs to mind, for example), are likely to be relatively highly priced, but may offer less protection during a crash scenario, depending on the behavior of the stock during major market declines.

September 6, 2018

Quantitative Analysis of Fat Tails – JonathanKinlay.com

In this quantitative analysis I explore how, starting from the assumption of a stable, Gaussian distribution in a returns process, we evolve to a system that displays all the characteristics of empirical market data, notably time-dependent moments, high levels of kurtosis and fat tails. As it turns out, the only additional assumption one needs to make is that the market is periodically disturbed by the random arrival of news.

NOTE: if you are unable to see the Mathematica models below, you can download the free Wolfram CDF player and you may also need this plug-in.

You can also download the complete Mathematica CDF file here.

Stationarity

A stationary process is one that evolves over time, but whose probability distribution does not vary with time. As the word implies, such a process is stable. More formally, the moments of the distribution are independent of time.

Let’s assume we are dealing with such a process that have constant mean μ and constant volatility (standard deviation) σ.

Φ=NormalDistribution[μ,σ]

Here are some examples of Normal probability distributions, with constant mean μ = 0 and standard deviation σ ranging from 0.75 to 2

Plot[Evaluate@Table[PDF[Φ,x],{σ,{.75,1,2}}]/.μ→0,{x,-6,6},Filling→Axis]

The moments of Φ are given by:

Through[{Mean, StandardDeviation, Skewness, Kurtosis}[Φ]]

{μ, σ, 0, 3}

They, too, are time – independent.

We can simulate some observations from such a process, with, say, mean μ = 0 and standard deviation σ = 1:

ListPlot[sampleData=RandomVariate[Φ /.{μ→0, σ→1},10^4]]

Histogram[sampleData]

If we assume for the moment that such a process is an adequate description of an asset returns process, we can simulate the evolution of a price process as follows :

ListPlot[prices=Accumulate[sampleData]]

An Empirical Distribution

Lets take a look at a real price series, comprising 1 – minute bar data in the June ‘ 14 E – Mini futures contract.

As with our simulated price process, it is clear that the real price process for Emini futures is also non – stationary.

What about the returns process?

ListPlot[returnsES]

Notice the banding effect in returns, which results from having a fixed, minimum price move of $12 .50, rather than a continuous scale.

Histogram[returnsES]

Through[{Min,Max,Mean,Median,StandardDeviation,Skewness,Kurtosis}[returnsES]]

{-0.00867214, 0.0112353, 2.75501×10^{-6, 0., 0.000780895, 0.35467, 26.2376}}

The empirical returns distribution doesn’ t appear to be Gaussian – the distribution is much more peaked than a standard Normal distribution with the same mean and standard deviation. And the higher moments don’t fit the Normal model either – the empirical distribution has positive skew and a kurtosis that is almost 9x greater than a Gaussian distribution. The latter signifies what is often referred to as “fat tails”: the distribution has much greater weight in the tails than a standard Normal distribution, indicating a much greater likelihood of an extreme value than a Normal distribution would predict.

A Quantitative Analysis of Non-Stationarity: Two States

Non – stationarity arises when one or more of the moments of a distribution vary over time. Let’s take a look at how that can arise, and its effects.Suppose we have a Gaussian returns process for which the mean, or drift, or trend, fluctuates over time.

Let’s consider a simple example where the process drift is μ1 and volatility σ1 for most of the time and then for some proportion of time k, we get addition drift μ2 and volatility σ2. In other words we have:

Φ1=NormalDistribution[μ1,σ1]

Through[{Mean,StandardDeviation,Skewness,Kurtosis}[Φ1]]

{μ1, σ1, 0, 3}

Φ2=NormalDistribution[μ2,σ2]

Through[{Mean,StandardDeviation,Skewness,Kurtosis}[Φ2]]

{μ2, σ2, 0, 3}

This simple model fits a scenario in which we suppose that the returns process spends most of its time in State 1, in which is Normally distributed with drift is μ1 and volatility σ1, and suffers from the occasional “shock” which propels the systems into a second State 2, in which its distribution is a combination of its original distribution and a new Gaussian distribution with different mean and volatility.

Let’ s suppose that we sample the combined process y = Φ1 + k Φ2. What distribution would it have? We can represent this is follows :

y=TransformedDistribution[(x1+k x2),{x1~Φ1,x2~Φ2}]

Through[{Mean,StandardDeviation,Skewness,Kurtosis}[y]]

Plot[PDF[y,x]/.{μ1→0,μ2→0,σ1 →1,σ2 →2, k→0.5},{x,-6,6},Filling→Axis]

The result is just another Normal distribution. Depending on the incidence k, y will follow a Gaussian distribution whose mean and variance depend on the mean and variance of the two Normal distributions being mixed. The resulting distribution in State 2 may have higher or lower drift and volatility, but it is still Gaussian, with constant kurtosis of 3.

In other words, the system y will be non-stationary, because the first and second moments change over time, depending on what state it is in. But the form of the distribution is unchanged – it is still Gaussian. There are no fat-tails.

Non – Stationarity : Random States

In the above example the system moved between states in a known, predictable way. The “shocks” to the system were not really shocks, but transitions. But that’s not how financial markets behave: markets move from one state to another in an unpredictable way, with the arrival of news.

We can simulate this situation as follows. Using the former model as a starting point, lets now relax the assumption that the incidence of the second state, k, is a constant. Instead, let’ s assume that k is itself a random variable. In other words we are going to now assume that our system changes state in a random way. How does this alter the distribution?

An appropriate model for λ might be a Poisson process, which is often used as a model for unpredictable, discrete events, ranging from bus arrivals to earthquakes. PDFs of Poisson distributions with means λ=5, 10 and 20 are shown in the chart below. These represent probability distributions for processes that have mean arrivals of 5, 10 or 20 events.

DiscretePlot[Evaluate@Table[PDF[PoissonDistribution[λ],k],{λ,{5,10,20}}],{k,0,30},PlotRange→All,PlotMarkers→Automatic]

Our new model now looks like this :

y=TransformedDistribution[{x1+k*x2},{x1⎡Φ1,x2⎡Φ2,k⎡PoissonDistribution[λ]}]

The first two moments of the distribution are as follows :

Through[{Mean,StandardDeviation}[y]]

As before, the mean and standard deviation of the distribution are going to vary, depending on the state of the system, and the mean arrival rate of shocks, . But what about kurtosis? Is it still constant?

Kurtosis[y]

Emphatically not! The fourth moment of the distribution is now dependent on the drift in the second state, the volatilities of both states and the mean arrival rate of shocks, λ.

Let’ s look at a specific example. Assume that in State 1 the process has volatility of 7.5 %, with zero drift, and that the shock distribution also has zero drift with volatility of 65 %. If the mean incidence rate of shocks λ = 10 %, the distribution kurtosis is close to that seen in the empirical distribution for the E-Mini.

Kurtosis[y] /.{σ1→0.075,μ2→0,σ2→0.65,λ→0.1}

{35.3551}

More generally :

ListLinePlot[Flatten[Kurtosis[y]/.Table[{σ1→0.075,μ2→0,σ2→0.65,λ→i/20},{i,1,20}]],PlotLabel→Style[“Kurtosis vs Mean Shock Arrival Rate”, FontSize→18],AxesLabel->{“Incidence Rate (%)”, “Kurtosis”},Filling→Axis, ImageSize→Large]

Thus we can see how, even if the underlying returns distribution is Gaussian in form, the random arrival of news “shocks” to the system can induce non – stationarity in overall drift and volatility. It can also result in fat tails. More specifically, if the arrival of news is stochastic in nature, rather than deterministic, the process may exhibit far higher levels of kurtosis than in its original Gaussian state, in which the fourth moment was a constant level of 3.

Quantitative Analysis of a Jump Diffusion Process

Nobel – prize winning economist Robert Merton extended this basic concept to the realm of stochastic calculus.

In Merton’s jump diffusion model, the stock price follows the random process

∂S_t / S_t =μdt + σdW_t+(J-1)dN_t

The first two terms are familiar from the Black–Scholes model : drift rate μ, volatility σ, and random walk W_t (Wiener process).The last term represents the jumps :J is the jump size as a multiple of stock price, while N_t is the number of jump events that have occurred up to time t.is assumed to follow the Poisson process.

PDF[PoissonDistribution[λt]]

where λ is the average frequency with which jumps occur.

The jump size J follows a log – normal distribution

PDF[LogNormalDistribution[m, ν], s]

where m is the average jump size and v is the volatility of the jump size.

In the jump diffusion model, the stock price S_t follows the random process dS_t/S_t=μ dt+σ dW_t+(J-1) dN(t), which comprises, in order, drift, diffusive, and jump components. The jumps occur according to a Poisson distribution and their size follows a log-normal distribution. The model is characterized by the diffusive volatility σ, the average jump size J (expressed as a fraction of S_t), the frequency of jumps λ, and the volatility of jump size ν.

The Volatility Smile

The “implied volatility” corresponding to an option price is the value of the volatility parameter for which the Black-Scholes model gives the same price. A well-known phenomenon in market option prices is the “volatility smile”, in which the implied volatility increases for strike values away from the spot price. The jump diffusion model is a generalization of Black–Scholes in which the stock price has randomly occurring jumps in addition to the random walk behavior. One of the interesting properties of this model is that it displays the volatility smile effect. In this Demonstration, we explore the Black–Scholes implied volatility of option prices (equal for both put and call options) in the jump diffusion model. The implied volatility is modeled as a function of the ratio of option strike price to spot price.