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Abstract:
The "leverage effect" refers to the well-established relationship between stock returns and both implied and realized volatility: volatility increases when the stock price falls. A standard explanation ties the phenomenon to the effect a change in market valuation of a firm's equity has on the degree of leverage in its capital structure, with an increase in leverage producing an increase in stock volatility. We use both returns and directly measured leverage to examine this hypothetical explanation for the "leverage effect" as it applies to the individual stocks in the S&P100 (OEX) index, and to the index itself. We find a strong "leverage effect" associated with falling stock prices, but also numerous anomalies that call into question leverage changes as the explanation. These include the facts that the effect is much weaker or nonexistent when positive stock returns reduce leverage; it is too small with measured leverage for individual firms, but much too large for OEX implied volatilities; the volatility change associated with a given change in leverage seems to die out over a few months; and there is no apparent effect on volatility when leverage changes because of a change in outstanding debt or shares, only when stock prices change. In short, our evidence suggests that the "leverage effect" is really a "down market effect" that may have little direct connection to firm leverage.

Abstract:
Trading in derivatives involves heavy use of quantitative models for valuation and risk management. These models are necessarily imperfect, and when options are involved, the models require a volatility input that must be forecasted, subject to error. This creates "model risk" to which nearly all participants in derivatives markets are exposed. In this paper, we conduct an empirical simulation, with and without hedging, using historical data from 1971-1996 for several important markets. The object is to develop a quantitative assessment of the extent to which the
different sources of model risk can be expected to affect the kind of basic option writing strategy that might be followed by a bank or another financial institution. Specifically, we explore the following problem: If a bank or a similar financial institution writes standard European calls and puts and prices them using the appropriate variant of the Black-Scholes model with a volatility forecast computed optimally from historical data, what are the risk and return characteristics of the trade? More generally, what is the market and model risk exposure faced by a bank that does this transaction repeatedly over time? The results indicate that pricing and hedging errors due to imperfect models and inaccurate volatility forecasts create sizable risk exposure for option writers. We then consider to what extent the bank can limit the damage due to model risk by pricing options using a higher volatility than its best estimate from historical data.

Abstract:
Trading in derivatives involves heavy use of quantitative models for valuation and risk management. These models are necessarily imperfect, and when options are involved, the models require a volatility input that must be forecasted, subject to error. This creates "model risk" to which nearly all participants in derivatives markets are exposed. In this paper, we conduct an empirical simulation, with and without hedging, using historical data from 1976-1996 for several important markets. The object is to develop a quantitative assessment of the extent to which the different sources of model risk can be expected to affect the kind of basic option writing strategy that might be followed by a bank or another financial institution. Specifically, we explore the following problem: If a bank or a similar financial institution writes standard European calls and puts and prices them using the appropriate variant of the Black-Scholes model with a volatility forecast computed optimally from historical data, what are the risk and return characteristics of the trade? More generally, what is the market and model risk exposure faced by a bank that does this transaction repeatedly over time? The results indicate that pricing and hedging errors due to imperfect models and inaccurate volatility forecasts create sizable risk exposure for option writers. We then consider to what extent the bank can limit the damage due to model risk by pricing options using a higher volatility than its best estimate from historical data.

Abstract:
Many exotic derivatives do not have closed-form valuation equations, and must be priced using approximation methods. Where they can be applied, standard lattice techniques based on binomial and trinomial trees will achieve correct valuations asymptotically. They can also generally handle American exercise. But for many problems, including pricing barrier options, convergence may be slow and erratic, producing large errors even with thousands of time steps and millions of node calculations. Options with price barriers that are only monitored at discrete points in time present additional difficulty for lattice models. Standard tree methods increase accuracy by shrinking the time and price step size throughout the lattice, but this increases the number of calculations sharply and much of the additional computation is in regions of the tree where it makes little difference to accuracy. A previous paper, Figlewski and Gao [1999], introduced the Adaptive Mesh Model (AMM), a very flexible approach that greatly increases efficiency in trinomial lattices. Coarse time and price steps are used in most of the tree, but small sections of finer mesh are constructed to improve resolution in specific critical areas. This paper presents an especially effective AMM structure for pricing options with discrete barriers. In a basic example, an AMM with 60 time steps is ten times more accurate than a 5000-step trinomial, but runs more than 1000 times faster.

Abstract:
Many exotic derivatives do not have closed-form valuation equations, and must be priced using approximation methods. Where they can be applied, standard lattice techniques based on binomial and trinomial trees will achieve correct valuations asymptotically. They can also generally handle American exercise. But for many problems, including pricing barrier options, convergence may be slow and erratic, producing large errors even with thousands of time steps and millions of node calculations. Options with price barriers that are only monitored at discrete points in time present additional difficulty for lattice models. Standard tree methods increase accuracy by shrinking the time and price step size throughout the lattice, but this increases the number of calculations sharply and much of the additional computation is in regions of the tree where it makes little difference to accuracy. A previous paper, Figlewski and Gao [1999], introduced the Adaptive Mesh Model (AMM), a very flexible approach that greatly increases efficiency in trinomial lattices. Coarse time and price steps are used in most of the tree, but small sections of finer mesh are constructed to improve resolution in specific critical areas. This paper presents an especially effective AMM structure for pricing options with discrete barriers. In a basic example, an AMM with 60 time steps is ten times more accurate than a 5000-step trinomial, but runs more than 1000 times faster.

Abstract:
In the reduced-form approach to credit modeling, default frequency has been found to depend on several firm-specific factors, most notably credit rating. But aggregate default rates also vary substantially over time, presumably reflecting changes in general economic conditions. In this paper, we fit Cox intensity models for major credit events, including defaults as well as major upgrades and downgrades in credit rating. The sample covers all corporate issuers in Moody's corporate bond Default Research Database over the period 1981-2002. The models incorporate both firm-specific factors related to a firm's credit rating history and a broad range of macroeconomic variables. Our results show that intensities of occurrence of credit events are significantly influenced by macro factors.

credit risk, default intensity, Cox model

Abstract:
Value at Risk and similar measures of financial risk exposure require predicting the tail of an asset returns distribution. Assuming a specific form, such as the normal, for the distribution, the standard deviation (and possibly other parameters) are estimated from recent historical data and the tail cutoff value is computed. But this standard procedure ignores estimation error, which we find to be substantial even under the best of conditions. In practice, a "tail event" may represent a truly rare occurrence, or it may simply be a not-so-rare occurrence at a time when the predicted volatility underestimates the true volatility, due to sampling error. This problem gets worse the further in the tail one is trying to predict. Using a simulation of 10,000 years of daily returns, we first examine estimation risk when volatility is an unknown constant parameter. We then consider the more realistic, but more problematical, case of volatility that drifts stochastically over time. This substantially increases estimation error, although strong mean reversion in the variance tends to dampen the effect. Non-normal fat-tailed return shocks makes overall risk assessment much worse, especially in the extreme tails, but estimation error per se does not add much beyond the effect of tail fatness. Using an exponentially weighted moving average to downweight older data hurts accuracy if volatility is constant or only slowly changing. But with more volatile variance, an optimal decay rate emerges, with better performance for the most extreme tails being achieved using a relatively greater rate of downweighting.We first simulate non-overlapping independent samples, but in practical risk management, risk exposure is estimated day by day on a rolling basis. This produces strong autocorrelation in the estimation errors, and bunching of apparently extreme events. We find that with stochastic volatility, estimation error can increase the probabilities of multi-day events, like three 1% tail events in a row, by several orders of magnitude. Finally, we report empirical results using 40 years of daily S&P 500 returns which confirm that the issues we have examined in simulations are also present in the real world.

Abstract:
Value at Risk and similar measures of financial risk exposure require predicting the tail of an asset returns distribution. Assuming a specific form, such as the normal, for the
distribution, the standard deviation (and possibly other parameters) are estimated from recent historical data and the tail cutoff value is computed. But this standard procedure
ignores estimation error, which we find to be substantial even under the best of conditions. In practice, a "tail event" may represent a truly rare occurrence, or it may simply be a not-so-rare occurrence at a time when the predicted volatility underestimates the true volatility, due to sampling error. This problem gets worse the further in the tail one is trying to predict.

Using a simulation of 10,000 years of daily returns, we first examine estimation risk when volatility is an unknown constant parameter. We then consider the more realistic, but more problematical, case of volatility that drifts stochastically over time. This substantially increases estimation error, although strong mean reversion in the variance tends to dampen the effect. Non-normal fat-tailed return shocks makes overall risk assessment much worse, especially in the extreme tails, but estimation error per se does not add much beyond the effect of tail fatness. Using an exponentially weighted moving average to downweight older data hurts accuracy if volatility is constant or only slowly changing. But with more volatile variance, an optimal decay rate emerges, with better performance for the most extreme tails being achieved using a relatively greater rate of
downweighting.

We first simulate non-overlapping independent samples, but in practical risk management, risk exposure is estimated day by day on a rolling basis. This produces strong autocorrelation in the estimation errors, and bunching of apparently extreme events. We find that with stochastic volatility, estimation error can increase the probabilities of multi-day events, like three 1% tail events in a row, by several orders of magnitude. Finally, we report empirical results using 40 years of daily S&P 500 returns which confirm that the issues we have examined in simulations are also present in the real world.

Abstract:
In modern finance, the value of an active investment strategy is measured by comparing its performance against the benchmark of passively holding the market portfolio and the risk less asset. We wish to evaluate the marginal contribution of a theoretical derivatives pricing model in the same way, by comparing its performance against an informationally passive alternative model. All rationally priced options must satisfy a number of conditions to rule out profitable static arbitrage. The Black-Scholes model, and others like it, are obtained by assuming an equilibrium in which there are no profitable dynamic arbitrage opportunities either. The passive model we consider incorporates only the fundamental properties of option prices that must hold to avoid static arbitrage, but has no theoretical content beyond that. We review different measures of model performance and apply them to several versions of the Black-Scholes model and our passive model. As with active portfolio management, it turns out to be not that easy for an active model to do a lot better than a well designed passive alternative. For example, classical Black-Scholes model turns out to be less accurate than the passive benchmark.

Abstract:
In modern finance, the value of an active investment strategy is measured by comparing its performance against the benchmark of passively holding the market portfolio and the riskless asset. We wish to evaluate the marginal contribution of a theoretical derivatives pricing model in the same way, by comparing its performance against an 'informationally passive' alternative model. All rationally priced options must satisfy a number of conditions to rule out profitable static arbitrage. The Black-Scholes model, and others like it, are obtained by assuming an equilibrium in which there are no profitable dynamic arbitrage opportunities either. The passive model we consider incorporates only the fundamental properties of option prices that must hold to avoid static arbitrage, but has no theoretical content beyond that. We review different measures of model performance and apply them to several versions of the Black-Scholes model and our passive model. As with active portfolio management, it turns out to be not that easy for an 'active' model to do a lot better than a well designed passive alternative. For example, the 'classical' Black-Scholes model turns out to be less accurate than the passive benchmark.

informationally passive model, Black-Scholes model, option pricing, dynamic arbitrage

Abstract:
The market's risk neutral probability distribution for the value of an asset on a future date can be extracted from the prices of a set of options that mature on that date, but two key technical problems arise. In order to obtain a full well-behaved density, the option market prices must be smoothed and interpolated, and some way must be found to complete the tails beyond the range spanned by the available options. This paper develops an approach that solves both problems, with a combination of smoothing techniques from the literature modified to take account of the market's bid-ask spread, and a new method of completing the density with tails drawn from a Generalized Extreme Value distribution. We extract twelve years of daily risk neutral densities from S&P 500 index options and find that they are quite different from the lognormal densities assumed in the Black-Scholes framework, and that their shapes change in a regular way as the underlying index moves. Our approach is quite general and has the potential to reveal valuable insights about how information and risk preferences are incorporated into prices in many financial markets.

risk neutral density, implied probabilities, option pricing

Abstract:
Stock index futures and program trading are among the most important financial market innovations of the 1980s. This chapter surveys the literature and provides an overview of the somewhat controversial area of index arbitrage. We begin with a description of how index futures work, how they should be priced in equilibrium according to the â¬Scost of carryâ¬? model, and how index arbitrage works to enforce the theoretical pricing relationship. In theory, index arbitrage is riskless, but we describe how it is affected in practice by transactions costs, execution risk, capital and short sales constraints, and the possibility of unwinding profitable trades before futures expiration. We end with a discussion of the impact of index futures and arbitrage on the volatility of the underlying stock market.

Abstract:
Applying modern option valuation theory requires the user to forecast the volatility of the underlying asset over the remaining life of the option, a formidable estimation problem for long maturity instruments. The standard statistical procedures using historical data are based on assumptions of stability, either constant variance, or constant parameters of the variance process, that are unlikely to hold over long periods. This paper examines the empirical performance of different historical variance estimators and of the GARCH (1,1) model for forecasting volatility in important financial markets over horizons up to five years. We find several surprising results: In general, historical volatility computed over many past periods provides the most accurate forecasts for both long and short horizons; root mean squared forecast errors are substantially lower for long term than for short term volatility forecasts; it is typically better to compute volatility around an assumed mean of zero than around the realized mean in the data sample, and the GARCH model tends to be less accurate and much harder to use than the simple historical volatility estimator for this application.

Abstract:
Applying modern option valuation theory requires the user to forecast the volatility of the underlying asset over the remaining life of the option, a formidable estimation problem for long maturity instruments. The standard statistical procedures using historical data are based on assumptions of stability, either constant variance, or constant parameters of the variance process, that are unlikely to hold over long periods. This paper examines the empirical performance of different historical variance estimators and of the GARCH(1,1) model for forecasting volatility in important financial markets over horizons up to five years. We find several surprising results: In general, historical volatility computed over many past periods provides the most accurate forecasts for both long and short horizons; root mean squared forecast errors are substantially lower for long term than for short term volatility forecasts; it is typically better to compute volatility around an assumed mean of zero than around the realized mean in the data sample, and the GARCH model tends to be less accurate and much harder to use than the simple historical volatility estimator for this application.

Abstract:
There has been much discussion of risks tied to trading in derivatives, with some well-informed objective observers arguing that derivatives risks are not significantly greater or different from those associated with traditional financial instruments. Financial risks are often broken down into market risk, credit risk, operational risk and legal risk. We review the standard classification and observe that while derivatives are exposed to these types of risk, they are manifested quite differently in derivatives than in traditional securities. We then consider a 'new' type of risk that is particularly important for derivatives: model risk. Derivatives trading depends heavily on the use of theoretical valuation models, but these are susceptible to error from incorrect assumptions about the underlying asset price process, estimation error on volatility and other inputs that must be forecasted, errors in implementing the theoretical models, and differences between market prices and theoretical values. Empirical evidence drawn from several important asset markets shows that model error can be quite large and can be expected to lead to significant risk in derivatives pricing and risk management.

Abstract:
The market's risk neutral probability distribution for the value of an asset on a future date can be extracted from the prices of a set of options that mature on that date, but two key technical problems arise. In order to obtain a full well-behaved density, the option market prices must be smoothed and interpolated, and some way must be found to complete the tails beyond the range spanned by the available options. This paper develops an approach that solves both problems, with a combination of smoothing techniques from the literature modified to take account of the market's bid-ask spread, and a new method of completing the density with tails drawn from a Generalized Extreme Value distribution. We extract twelve years of daily risk neutral densities from S&P 500 index options and find that they are quite different from the lognormal densities assumed in the Black-Scholes framework, and that their shapes change in a regular way as the underlying index moves. Our approach is quite general and has the potential to reveal valuable insights about how information and risk preferences are incorporated into prices in many financial markets.

Abstract:
The "leverage effect" refers to the well-established relationship between stock returns and both implied and realized volatility: volatility increases when the stock price falls. A standard explanation ties the phenomenon to the effect a change in market valuation of a firm's equity has on the degree of leverage in its capital structure, with an increase in leverage producing an increase in stock volatility. We use both returns and directly measured leverage to examine this hypothetical explanation for the "leverage effect" as it applies to the individual stocks in the S&P100 (OEX) index, and to the index itself. We find a strong "leverage effect" associated with falling stock prices, but also numerous anomalies that call into question leverage changes as the explanation. These include the facts that the effect is much weaker or nonexistent when positive stock returns reduce leverage; it is too small with measured leverage for individual firms, but much too large for OEX implied volatilities; the volatility change associated with a given change in leverage seems to die out over a few months; and there is no apparent effect on volatility when leverage changes because of a change in outstanding debt or shares, only when stock prices change. In short, our evidence suggests that the "leverage effect" is really a "down market effect" that may have little direct connection to firm leverage.

Abstract:
The "leverage effect" refers to the well-established relationship between stock returns and both implied and realized volatility: volatility increases when the stock price falls. A standard explanation ties the phenomenon to the effect a change in market valuation of a firm's equity has on the degree of leverage in its capital structure, with an increase in leverage producing an increase in stock volatility. We use both returns and directly measured leverage to examine this hypothetical explanation for the "leverage effect" as itapplies to the individual stocks in the S&P100 (OEX) index, and to the index itself. We find a strong "leverage effect" associated with falling stock prices, but also numerousanomalies that call into question leverage changes as the explanation. These include thefacts that the effect is much weaker or nonexistent when positive stock returns reduce leverage; it is too small with measured leverage for individual firms, but much too large for OEX implied volatilities; the volatility change associated with a given change in leverage seems to die out over a few months; and there is no apparent effect on volatility when leverage changes because of a change in outstanding debt or shares, only when stock prices change. In short, our evidence suggests that the "leverage effect" is really a "down market effect" that may have little direct connection to firm leverage.

Abstract:
This study uncovers trading styles in the transaction records of US Treasury bond futures.It uses transaction-by-transaction data from the Commodity Futures Trading Commissions' (CFTC)Computerized Trade Reconstruction (CTR) records. The data set consists of 30 million transaction - thecomplete US T-bond futures market for 3 years. Each transaction record consists of time (by theminute), price, volume, buy/sell, and an identifier of the specific account.We use statistical clustering techniques to group together trades that are similar. Two sets ofassumptions have to be made: (1) What is a trade? We define a trade to begin when an account opensa position, and to end when its position size returns to zero. We describe each trade by several trade-specificvariables (e.g., length of trade, maximum position size, opening move, long or short) and severalexogenous, market-specific variables (e.g., price, volatility, trading volume). (2) What process generatedthe data? We assume a mixture of Gaussians. An observed trade is interpreted as a noisy realization ofone of the mixture components. This paper assumes identity covariance matrices. Furthermore, eachtrade is fully assigned to a single cluster. We compare this approach to diagonal and to full covariancestructure with probabilistic assignments.Trade profit was held back in the clustering process. It turns out that the clusters differ significantlyin their profit and risk characteristics. Using conditional distributions, we summarize featuresof profitable trading styles and contrast them with losing strategies. We find that profitable styles tendto hold trades longer, trade at higher volatility, and trade earlier in the contracts. We also show howsome clusters uncover "technical" traders. Using the information about the individual accounts, theassignments of accounts to clusters are described by entropy, and the transitions of a given accountthrough clusters is modeled by a first order Markov model.

Abstract:
We examine the intraday behavior of the risk neutral probability density (RND) for the Standard and Poor's 500 Index extracted from a continuous real-time data feed of bid and ask quotes for index options. This allows an exceptionally detailed view of how investors' expectations about returns and attitudes towards risk fluctuated during the financial crisis in the fall of 2008. The increase in risk measures was extraordinary, such as a fivefold increase in minute-to-minute volatility from October 2006 to October 2008. In contrast to moderate positive autocorrelation in the S&P index, the analysis reveals unusually large negative autocorrelation in the mean and standard deviation of the RND. Using quantile regressions, we find a strong pattern in how much different portions of the RND move when the level of the stock index changes, with the middle portion of the RND amplifying the change in the index by a factor of as much as 1.5 or more in some cases. This phenomenon increased in size during the crisis and, surprisingly, was stronger for up moves than for down moves in the market.

risk neutral density, market expectations, 2008 financial crisis, option valuation

Abstract:
In modern finance, the value of an active investment strategy is measured by comparingits performance against the benchmark of passively holding the market portfolio and the riskless asset. We wish to evaluate the marginal contribution of a theoretical derivatives pricing model in the same way, by comparing its performance against an "informationally passive" alternative model. All rationally priced options must satisfy a number of conditions to rule out profitable static arbitrage. The Black-Scholes model, and others like it, are obtained by assuming an equilibrium in which there are no profitable dynamic arbitrage opportunities either. The passive model we consider incorporates only the fundamental properties of option prices that must hold to avoid static arbitrage, but has no theoretical content beyond that. We review different measures of model performance and apply them to several versions of the Black-Scholes model and our passive model. As with active portfolio management, it turns out to be not that easy for an "active" model to do a lot better than a well designed passive alternative. For example, "classical"Black-Scholes model turns out to be less accurate than the passive benchmark.

Abstract:
Trading strategies and contingent claims with path-dependent returns are difficult to model analytically. Monte Carlo simulation, the standard solution technique, is computationally expensive and provides a solution only for the specific parameter values used in the simulation. We present an alternate "quasi-analytic" procedure that combines the power of the simulation approach with the computational efficiency of an analytical solution. Our method uses simulation results to construct an analytic function that maps the input parameters to the returns distribution function. This analytic function can then be used to directly estimate the returns distribution for other parameter values without further simulation. We illustrate the approach by analyzing the performance of a path-dependent long term protective put strategy that requires rolling over a series of short term options. The returns to the strategy depend on the investor's choice of put strike and rollover policy. We use our method to examine a risk averse investor's optimal trading strategy, a problem that is exceedingly time-consuming using standard Monte Carlo simulation. For example, the simulation approach takes more than 45 minutes to solve for just one particular volatility scenario. Our method provides the answer in a matter of seconds.

Abstract:
Trading strategies and contingent claims with path-dependent returns are difficult to model analytically. Monte Carlo simulation, the standard solution technique, is computationally expensive and provides a solution only for the specific parameter values used in the simulation. We present an alternative "quasi-analytic" procedure that combines the power and flexibility of the simulation approach with the computational efficiency of an analytical solution. Our method uses simulation results to construct an analytic function that provides an approximate mapping from the input parameters to the returns distribution function. This analytic function can then be used to estimate the returns distribution for other parameter values directly without further simulation.We illustrate the approach by analyzing the performance of a path-dependent long-term protective put strategy that requires rolling over a series of short-term options. The returns to the strategy depend on the investor's choice of put strike and rollover policy. We use our method to examine a risk-averse investor's optimal trading strategy, a problem that is time-consuming using standard Monte Carlo simulation. In one example, the simulation approach takes more than forty-five minutes to solve for just one particular volatility scenario, while our method provides the answer in a matter of seconds.

Abstract:
This monograph puts together results from several lines of research on the general topic of volatility forecasting for option pricing applications. I offer empirical evidence and a variety of observations about volatility prediction and the ways in which it is customarily approached. Along with describing the theory and the implementation of the standard techniques, I point out several areas in which common procedures and ways of thinking about volatility forecasting turn out to involve assumptions or ideas that do not stand up under close examination.We consider both general classes of prediction methods: those based on estimating volatility statistically using historical price data and those based on deriving implied volatilities from observed option prices. The major theme we explore is the connection, or perhaps more correctly, the possibility of a disconnection between theory and practice in dealing with volatility prediction and its role in option valuation.We first discuss the statistical theory used in fitting models of price behavior in financial markets. Chapter I brings out the distinction between physical processes and economic processes in terms of the stability of their internal structure and the prospects for making accurate predictions about them. We argue that routinely applying the classical estimation methodology appropriate for physical processes to the economic process of price behavior in a financial market can lead one to build models that are too complex and to hold inappropriately high expectations about the potential accuracy of volatility forecasts from them. Evidence is presented that simpler more robust estimation techniques tend to be more successful at forecasting than those that may be theoretically more elegant, but fragile.Conflict between theory and practice also arises in the use of implied volatility from option market prices, because of the significant disparity between the trading strategies arbitrage-based derivatives valuation models assume investors follow and what options market participants actually do. In theory, the implied volatility is the options market's well-informed prediction of the underlying asset's future volatility. Academic researchers typically treat it as such. In practice, however, the arbitrage trading that is supposed to force option prices into conformance with the market's volatility expectations may not be done very actively at all. In many markets the trade is very hard to execute, and it also will normally be less profitable and riskier than a simple market making strategy that reacts to the market, maximizes order flow and earns profits from the bid-ask spread. The latter, however, may do little to enforce theoretical pricing against the noisy forces of supply and demand in the market. Thus the implied volatility derived from market option prices need not be a good proxy for the market's best forecast of future volatility of the underlying asset. And in fact, it almost never passes the standard unbiasedness test for rational forecasting.In both cases, I try to point out important implications for volatility estimation that tend to be overlooked by those following traditional lines of thought, and hope that in the end, the reader will acquire a broader perspective on what is involved in obtaining the volatility input to a derivatives valuation model, and what questions need to be asked of any proposed technique.