Estimating Volatility For Risk Management

Introduction

The financial world is full of unknowns. In making decisions we either gain or lose depending on how we read the trends and make accurate forecasts. We cannot do away with risk but we can manage it to our advantage. Risk is simply defined as ‘the possibility of loss, injury, disadvantage or destruction’ (Simmons). Risk is not constant, it can be volatile. Volatility means rapid or unexpected changes (Mikhailov).

Accurate measurement of volatility is of key importance in the field of financial risk management for two reasons (Mikhailov):  1) the number of companies practicing risk management and  2)  huge amount of derivatives in traded in the exchange markets. These factors except volatility itself is observable from the market or can be specified by option contract.  It is very difficult to estimate volatility but it is very crucial for risk evaluation and options pricing.

The many notions of volatility include (Mikhailov):  Actual volatility (AV);  Historical volatility (HV); Future volatility (FV); Implied volatility (IV);  non-constant volatility; local volatility and stochastic volatility.

Actual volatility, although unobservable,  is a measure of randomness or uncertainty or irregularity of price behaviour at present. It is approximated by historical volatility.  Historical volatility is based on past price levels usually the past 20 or 60 days period. It can be used for stationary time series (Mikhailov).

Future volatility (FV) is the actual volatility over period until the expiry of  the option. It is a measure of uncertainty about future price movements. This defines the option price in the Black-Scholes (BS) model.

Implied volatility is the volatility implied by the market price of a derivative based on a theoretical pricing model (Wikipedia, http://en.wikipedia.org/wiki/Implied_volatility). It estimates FV based on the Black-Scholes model assuming that option is a traded asset. The BS is used for instruments with log-normal prices (Wikipedia).

Non-constant volatility  assumes a continuum of strikes (as in the strike dependency of implied volatility of vanilla market) and maturities (Mikhailov).  A time-dependent volatility (Merton’s model, 1990) gives a correct price of an option for only one strike and all maturities. This points to the need to developed volatitility models in a more general setting. Three types of continuous models have been developed: local volatility models, jump-diffusion models and stochastic volatility models(Mikhailov).

Stochastic volatility as opposed to  deterministic volatilities, is probabilitic and not dependent only on one factor like market price. Volatility is treated as a stochastic variable. The need for stochastic models arose from the trouble of result financial engineers to recalibrate model parameters everyday to the new market data.

            This paper deals with volatility estimation and models as they relate to: (a) Portfolio insurance strategies  and (b) The accuracy of Value at Risk models.

(a) Portfolio insurance strategies

Definitions:

A portfolio is a collection of investments (stocks, bonds) owned by a same individual or organization (Portfolio – http://www.investorwords.com/3741/portfolio.html). A portfolio insurance is a strategy of hedging a stock portfolio against market risk by selling stock index future short or buying stock index put options. (Portfolio Insurance (http://www.investorwords.com/3742/portfolio_insurance.html)). Strategy is a an action plan carried over a long period of time for achieving a goal such as making a profit and avoid losses  (Strategy http://www.investorwords.com/4775/strategy.html).

It would be nice to participate in the market when it goes up and be protected from losses when market falls. (Dybvig, 1997). In terms of market Mt, the portfolio insurance offers a payoff:

Portfolio insurance strategy was developed by Rubinstein and Leland using replicating strategy from optimum pricing theory. It made possible the creation of desired terminal payoff through asset allocation between stocks and bonds without trading options.

A European put option gives the owner the right (not obligation) to sell one’s share at a strike price of f.  This payoff is the same as investing kW ₀ in the market and a put option on that investment with and exercise price of f.

The Black-Scholes model gives the call option price as

          S N (_{x 1}) – B N(_{x 2})

Where S is the value of the underlying stock and B is the value of a discount bond with face equal the strike price  and maturing with the option, N( .) is the cumulative normal distribution where:

The first term gives the amount to be invested in sock while the second term gives the amount to invest in riskfree bond. Once the variance parameter is estimated, the portfolio insurance strategy can be computed (Dybvig, 1997).

This assumed a constant riskless rate. However it happened that the market crashed. Many with portfolio insurance were not able to execute trades needed to adjust hedges as the market fell. It is probably prudent for a portfolio insurer to maintain futures options to reduce need for trading in general and automatically adjust risk exposure during such crash. Another strategy is buying-out-of-the money puts on the market which is more expensive (Dybvig, 1977).

Dybvig (1977) suggest to use futures contracts to manage required day-to-day changes in risk exposure which is less expensive than trading the underlying stocks all the time.

There are sources of tracking errors: hedge is not exactly the theoretical one; volatility is different from the expected volatility on the average;  and high volatility especially repeated large up and down moves in a net flat market. This known as “whip-saw”.  Tracking errors create minor problem in continuation, since the original strategy is supposed to be infeasible. The solution is to keep the promise f fixed but vary k to be consistent with current wealth (Dybvik)

Gomes (2000) listed the choices for optimal portfolio for a retirement saving system: life-cycle portfolio allocation; portfolio choice with return predictability; and trading with loss-averse investors. In life-cycle portfolio allocation a finitely-lived investor faces mortality risk, borrowing and short-sale constraints and receiving labor income. The optimal fortfolio and savings decisions are solved numerically to find that labour income significantly increases the demand for equities and that optimal portfolio share invested in stocks   decreases with age. The model evaluates the welfare loss of alternative investment strategies and extend the model to study the welfare implications of  alternative retirement savings systems.

Return predictability can be low-frequency predictability, this time the infinitely-lived investor faces a time-varying equity premium. The portfolio rule (solved numerically) is approximately linear in the state variable while the log consumption-wealth ratio is approximately quadratic (Gomes, 2000).

Gomes (2000) also studied portfolio rebalancing motives due to (high-frequency) time variation in the equity premium and (high-frequency) time variation in the volatility of stock returns. Theoretical utility gains are large with bigger share resulting from timing the mean equity premium. These market timing rules yield positive risk-adjusted rates of return with small gain.

In trading with loss-averse investors, the demand function becomes discontinuous. Investors follow a generalized portfolio insurance as wealth rises beyond a threshold. Risk-averse investors will not hold stocks unless expected equity premium is quite high.  Dynamic model with symmetric information generates ARCH and excess kurtosis in stock returns. It yields a realistic volume series that is positively autocorrelated and positively correlated with stock return volatility (Gomes, 2000).

The market crash of 1987 has been known as Black Monday. Its causes and effects are enumerated in (Black Monday: Causes and Effects, http://www.ncs.pvt.k12.va.us/ryerbury/pasc/pasc.htm).  It mentioned poor choices of portfolio insurance as one of its causes(Arbel and Kaff, 1989). The portfolio insurance professionals relied on their very risky intuition instead of reliable information. Massive selling when stocks were high price, caused the value of decrease below their true value and because of the low value the process would be repeated (Sobel, 1987).

1. b) The accuracy of Value at Risk models:

Volatility is important in risk management if its fluctuations can be forecast (Diebold, 1997). Volatility forecastability, depends on time horizon, the further into the future, the less accurate is the forecast. Assessment methods are plagued by the joint assessments of volatility forecastability and an assumed model although there are now several models (as discussed later) developed for many unique situations.

Andersen et al. (2001) criticized the parametric multivariate ARCH or stochastic volatility models as performing poorly at intraday frequencies. They used traditional time series procedures for modelling and forecasting. They linked conditional covariance matrix with realized volatility built on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation. Their model used simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities. They stated that their practical model performed as good as ARCH models. (Andersen, et al. 2001).

The random-walk process or  “Brownian motion” is considered the most important building block of  modern finance (Fuentes). The simplest random-walk model described by Bachelier (1900) stated that successive difference of price of stocks were independent random variables with a normal distribution with variance proportional to the time interval of the differences. This model was later refined to “Geometric Brownian motion” in which the differences of the prices were distributed as a log-normal. Although these models were simple they did not account for some of the characteristics of the real financial market data, and hence were not accurate. This was followed by the Black and Scholes (1973) model for option pricing.

The Black-Sholes is a model of varying price over time of financial instruments particularly stocks. It is a mathematical formula for the theoretical value of European put and call stock options. It states that call option is implicitly priced if the stock is traded. (Wikipedia, http://en.wikipedia.org/wiki/Black-Scholes). This model assumes that price of the underlying instrument follows a geometric Brownian movement with constant drift (expected gain) and volatility σ. The model is extended to European options on instruments paying dividends.

Many assumptions of the Black-Scholes models are not practiced. There are some “holes” (smile effect). Experiments established that the distribution of the assets returns have nonzero third and fourth moments. (Mikhailov). Volatility smiles reflects non-lognormal distributions of the returns. The mean and volatility parameterise lognormal distribution. Black-Scholes can give only one correct price of an option for only one strike and only one maturity. (Mikhailov).

The shortcomings of these models were demonstrated in the crash of 1987 which showed that market volatility is not constant across time (Fuentes). Before this crash the Black-Scholes model was good description of the Vanilla market: the strike dependency of the implied volatility was negligible.  This scenario was described earlier by a mathematician Benoit Mandelbrot (1963) in a phenomenon called ‘volatility clustering’ which stated that large price changes are not isolated in between periods of slow change.

ARCH and GARCH Models

Two basic models that measure volatility are: ARCH (Autoregressive conditional heteroskedasticity) and GARCH (Generalized ARCH or Generalised Autoregressive Conditional Heteroskedastic). These models use recent past information to give one-period forecast variance. For the simple first-order autoregression:

y_t =α y _t-1 + u_t,

Where u_t is white noise with variance V(u_t) = σ ² with unconditional mean y_t is zero and α y _t-1 is the conditional mean. Engle proposed for heteroskedasticity the following model:

y_t =Є _t σ _t ,

σ _t² = α ₀ + α₁y _t-1²

Where Є _t is independently distributed with mean 0 and variance 1. σ _t is a positive, time-varying function at time t-1. The variance can be extended to include fu)rther back in time:

                        σ _t² = h(y _t-1, y _t-2, …., y _t-p, α),

Where p is called the order of the ARCH process, and α is a vector of unknown parameters (Fuentes).

These two model types have spawned many derivatives and models. Bera and Higgins (1993) remarked that the ARCH literature showed that the apparent changes in volatility of economic time series may be predictable. Such changes result from specific type of nonlinear dependence rather than exogenous structural changes in variables (Bera and Higgins, 1983)

The assumption of constant volatility over time is both logically inconsistent and statistically inefficient  when the resulting series moves through time (Campbell, Lo, and MacKinlay. 1997). As an example they gave the case of financial data where large and small errors tend to occur in clusters: more large returns follow large returns and more small returns follow small returns.

Such returns are serially correlated. They made distinction between nonlinearities and linear time series where shocks are assumed t be uncorrelated but not necessarily identically independently distributed (iid). With nonlinear time series shocks are assumed iid but there is a nonlinear function relating observed time series and the underlying shocks.

The following are interpretations of process found in the history of ARCH literature: Heteroskedasticity may be caused by a time dependence in the rate of information arrival to the market (Lamoureux and Lastrapes, 1990) for example   daily trading volume of stock market. Contemporaneous errors in expectations are linked with past errors in the  same expectations (Mizrach, 1990). This is related to “adaptable expectations hypothesis” in macroeconomics.

Another interpretation is that any economic variable evolves on ‘operational’ time scale while in practice it is measured on a ‘calendar’ time scale (Stock, 1998). This may lead to volatility clustering since variable may evolve more quickly or slowly relative to calendar time (Bera and Higgins, 1990; Diebold, 1986)

History of ARCH model

Robert Engle (1982) introduced his ARCH model (AutoRegressive Conditional Heteroscedasticity) which modelled the time variation of second- and higher-order moments. The  ARCH family of model incorporates recent information to make better forecasts. The recent past gives one-period forecast variance. The ARCH processes are serially uncorrelated with mean zero, constant unconditional variance and non-constant condition on the past. Engle (1982) proposed a phenomenon known as ‘heteroskedasticity’ which meant that variance is not constant over time. Extensions of the ARCH model (Fuentes)  included: GARCH, IGARCH (integrated GARCH (Engle and Bollerslev, 1986), ARCH-M (ARCH-mean model, Engle et al., 1987) and EGARCH (Exponential GARCH, Nelson, 1991)

Bollerslev (1986) made a more flexible lag structure which was known as ‘Generalized ARCH or GARCH (p,q), where p is the autorregresive term and a polynomial (L) of order q, the moving average term. The disturbances in the ARCH-M model are also heteroskedastic but includes additional information since the standard deviation at each time t affects the mean of  the same observation. EGARCH  solved the problem of asymmetry in stock prices (Nelson, 1991).

ARCH-type models will become better at understanding time series analysis and predicting future pricing. Such models that treats stochastic volatility may however become complex that we are put in a situation where we have to compromise between efficiency and accuracy of models. (Fuentes)

Stochastic volatility models:

Heston’s stochastic volatility model( http://www.tsresearch.com/public/volatility/heston/) takes into account the leverage effect (negative correlation between price and volatility) and describes the mean-reverting property of volatility. This model is considered as the successor to the Black and Scholes model. However, it has some disadvantages and open questions:  some option prices were negative or at least lying below the usual arbitrage bounds which makes BS volatility inversion impossible. It did not eliminate all biases and did not perform consistently well across various maturities. The model implicitly considered systematic volatility risk through a linear specification for the volatility risk premium.

Estimating and testing ARCH Models (Perrelli):

How do we know when to use any of the ARCH models and how do we estimate the parameters?  Johnston and DiNardo (1997) suggest a very simple test for the presence of

ARCH problems. The basic menu (step-by-step) is: 1) Regress y (scalar random variable) on x (observed time series data) by obtain the residuals (Alexander), underlying shocks. 2) Compute OLS regression ε_t² = σ₀ + σ₁ ε_t-1²+ …+ σ_p ε_t-p²  + error   3) test joint significance of  σ₁…,  σ_p.   4) If any σ is significant, do a straight-forward method of estimation(correction) provided by Greene (1997).

Conclusions:

Esimating future volatilities is really a risky business with so many foreseen and unforeseen factors coming into the picture.  As the saying goes history repeats itself. An investor should study historical volatilities to find the signs of impending disasters such as what happened to the crash of  1987. Study what went wrong and what could be done in case it happens again.

Integrate the past, the present and the estimate of future volatilities.

Since accurate estimations of volatility is very important, it is equally important to the appropriate model to estimate it based on prevailing and expected conditions. Minimize risk or maximize profit using appropriate formulas and make decisions based on these.

Bonham (2005) explained how a portfolio of projects can be aligned with everchanging marketplace. The strategy as it was applied to information technology (IT)- based portfolio consists of a central strategy; maximized overall return on investment; balanced risks across the organization.

            With all the models and computer softwares (e.g., Yip, 2005) available the final decisions still lie on the person. There remains a need to integrate analytical (mathematical reasoning, logic, etc.) and intuitive (abstract reasoning, emotion, creativity) aspects of ones personality into a properly balanced and powerful trading system. (McMaster, 1999).

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