Forward Volatility Research Articles

Natural Volatility and Option Pricing

In this paper we recover the Black-Scholes and local volatility pricing engines in the presence of an unspecified, fully stochastic volatility. The input volatility functions are allowed to fluctuate randomly and to depend on time to expiration in a systematic way, bringing the underlying theory in line with industry experience and practice. More generally we show that to price a European-exercise path-(in)dependent option, it is enough to model the evolution of the variance of instantaneous returns over the natural filtration of the underlying security. We call the square root of this new process natural volatility. We develop the associated concept of path-conditional forward volatility, via which the natural volatility can be directly specified in an economically meaningful way.

Hedging Forward Volatility

Hedging Forward Volatility

Hedging Forward Volatility

Hedging Forward Volatility

Predicting S&P 500 volatility for intermediate time horizons using implied forward volatility

This paper finds that the implied forward volatility of S&P 500 futures options contains significant explanatory power regarding sunsequent realized volatility during intermediate future time intervals. It provides rational, unbiased, and informationally efficient predictions and dominates all alternative volatility forecasts considered.

The Term Structure of Implied Forward Volatility: Recovery and Informational Content in the Corn Options Market

Using a flexible method, we develop the term structure of volatility implied by corn futures options with differing maturities, and evaluate its ability to predict subsequent realized price volatility. The implied forward volatilities anticipate realized volatility well. For the nearby interval, the implied forward volatilities provide unbiased forecasts, and are superior to forecasts based on historical volatilities. For more distant intervals, early‐year options predict the direction and magnitude of future volatility changes about as well as a three‐year moving average and better than a naïve forecast. However, later‐year options display less forecast power in part due to reduced trading activity.

Markov Functional Modeling of Equity, Commodity and Other Assets

In this short note we show how to setup a one dimensional single asset model, e.g. equity model, which calibrates to a full (two dimensional) implied volatility surface. We show that the efficient calibration procedure used in LIBOR Markov functional models may be applied here too. In a addition to the calibration to a full volatility surface the model allows the calibration of the joint asset-interest rate movement (i.e. local interest rates) and forward volatility. The latter allows the calibration of compound or Bermudan options. The Markov functional modeling approach consists of a Markovian driver process x and a mapping functional representing the asset states S(t) as a function of x(t). It was originally developed in the context of interest rate models, see [Hunt Kennedy Pelsser 2000]. Our approach however is similar to the setup of the hybrid Markov functional model in spot measure, as considered in [Fries Rott 2004]. For equity models it is common to use a deterministic Numeraire, e.g. the bank account with deterministic interest rates. In our approach we will choose the asset itself as Numeraire. This is a subtle, but crucial difference to other approaches considering Markov functional modeling. Choosing the asset itself as Numeraire will allow for a very efficient numerically calibration procedure. As a consequence interest rates have to be allowed to be stochastic, namely as a functional of x too. The Black-Scholes model with deterministic interest rates is a special case of such a Markov functional model. The most general form of this modeling approach will allow for a simultaneous calibration to a full two dimensional volatility smile, a prescribed joint movement of interest rates and a given forward volatility structure.

Path-Conditional Forward Volatility

In derivatives modelling, it has often been necessary to make assumptions about the volatility of the underlying variable over the life of the contract. This can involve specifying an exact trajectory, as in the Black and Scholes (1973), Merton (1973) or Black (1976) models; one that depends on the level of the underlying variable as in the local volatility models of Dupire (1994), Derman and Kani (1994) and Rubinstein (1994); or fixing the parameters of a more general stochastic volatility process as in Hull and White (1987) or Heston (1993). These forward-looking assumptions are by their very nature destined to be disproved, and what is more are at odds with the frequent model recalibration that (rightly) takes place in practice. In Carey (2005), the Black-Scholes analytical framework is extended, via the definition of higher-order volatilities and the derivation of moment formulae for the case where they are deterministic. In this paper, we show that the same formulae can be obtained under markedly weaker assumptions, which leave the future volatilities unspecified. Instead, we impose constraints on new, related quantities, which we term path-conditional forward volatilities. Under this scheme, the model inputs are no longer the future spot volatilities, but rather their forward counterparts. One consequence, we show, is that contrary to conventional wisdom, the Black-Scholes formula can in principle be used without any reference to future volatility.

Pricing Prepayment Option in C&I Loans at Origination

This paper presents a pricing model of commercial and industrial (C&I) loan prepayment option. Modeling of prepayment is essential in pricing mortgage contracts since prepayment truncates the timing and amount of expected cash flows. Lenders normally charge a penalty for prepayment, for example interest on unused principal. We first propose that prepayment penalties should be viewed as fees in arrears reflecting the value of a call option at origination. We then rationalize business prepayment behavior with both refinancing and non-refinancing incentives. Business borrowers make rational decisions to refinance when interest rates fall or their credit standing improves. Business borrowers also prepay with internal equity to optimize capital structure and reduce debt. Prepayment value in C&I loans reflects the option value depending on the market conditions, borrower's creditworthiness and the need to reduce debt. In the rest of the paper, we propose a contingent claims approach to the pricing C&I loan prepayment contract, given loans being defaultable, and transaction costs and fees being proportional to the loan amount. An implementation example is provided with prepayment option value being determined via a double non-recombining tree. The two underlying state variables are forward credit spread and forward interest rate specified in the HJM framework and their volatilities are exponentially dampened and proportional to the spot rates. We find that prepayment option value concentrates on the first few periods, and locking out these periods would reduce the option value signicantly. We also find that prepayment option value on a variablerate loan is insensitive to the forward interest rate volatility but very sensitive to the forward credit spread volatility.

Consistent Pricing and Hedging of an FX Options Book

In the foreign exchange (FX) options market away-from-the-money options are quiteactively traded, and quotes for the same type of instruments are available everydaywith very narrow spreads (at least for the main currencies). This makes it possibleto devise a procedure for extrapolating the implied volatilities of non-quoted options,providing us with reliable data to which to calibrate our favorite model.In this article, we test the goodness of the Brigo, Mercurio and Rapisarda (2004) modelas far as some fundamental practical implications are concerned. This model, which isbased on a geometric Brownian motion with time-dependent coefficients that are notknown initially and whose value is randomly drawn at an infinitesimal future time, canaccommodate very general volatility surfaces and, in case of the FX options market,can lead to a perfect fit to the main volatility quotes.We first show the fitting capability of the model with an example from real marketdata. We then support the goodness of our calibration by providing a diagnostic on theforward volatilities implied by the model. We also compare the model prices of someexotic options with the corresponding ones given by a market practice. Finally, weshow how to derive bucketed sensitivities to volatility and how to hedge accordingly atypical options book.Keywords: foreign exchange options market, uncertain Black-Scholes parameters,calibration, forward volatilities, bucketed sensitivities, options bookhedging.JEL Classification Numbers: C14, C15, C61, F31, G13, G24

Intermediate Volatility Forecasts Using Implied Forward Volatility: The Performance of Selected Agricultural Commodity Options

Options with different maturities can be used to generate an implied forward volatility, a volatility forecast for non-overlapping future time intervals. Using five commodities with varying characteristics, we find that the implied forward volatility dominates forecasts based on historical volatility information, but that the predictive accuracy is affected by the commodity's characteristics. Unbiased and efficient corn and soybeans market forecasts are attributable to the well-established volatility during crucial growing periods. For soybean meal, wheat, and hogs volatility is less predictable, and investors appear to demand a risk premium for bearing volatility risk.

Extended Libor market models with stochastic volatility

This paper introduces stochastic volatility to the Libor market model of interest rate dynamics. As in Andersen and Andreasen (2000a) we allow for non-parametric volatility structures with freely specifiable level dependence (such as, but not limited to, the CEV formulation), but now also include a multiplicative perturbation of the forward volatility surface by a general mean-reverting stochastic volatility process. The resulting model dynamics allow for modeling of non-monotonic volatility smiles while explicitly allowing for control of the stationarity properties of the resulting model dynamics. Using asymptotic expansion techniques, we provide closed-form pricing formulas for caps and swaptions that are robust, accurate, and well-suited for both model calibration and general mark-to-market of plain-vanilla instruments. Monte Carlo schemes for the proposed model are proposed and examined.

Adaptive mixture for a controlled smile: the LT model

We build here an arbitrage-free model for an equity-type asset that allows the forward volatility surface to be uncertain today and change profiles over time with some stationarity. Spot and forward distributions are both expressed in the same format through convenient lognormal mixtures ensuring consistent recombinations. The underlying spot process is obtained as the product of a lognormal and a jump variable that can be easily simulated. Closed-form solutions are derived for standard and forward-start European options. The model addresses the drawbacks of local volatilities and is very tractable with respect to jump-diffusion and stochastic vol models. It allows to calibrate, as needed, on both cliquet and vanilla options and offers a convenient framework to observe and control volatility surface evolution.

Extended Libor Market Models with Stochastic Volatility

This paper introduces stochastic volatility to the Libor market model of interest rate dynamics. As in Andersen and Andreasen (2000a) we allow for non-parametric volatility structures with freely specifiable level dependence (such as, but not limited to, the CEV formulation), but now also include a multiplicative perturbation of the forward volatility surface by a general mean-reverting stochastic volatility process. The resulting model dynamics allow for modeling of non-monotonic volatility smiles while explicitly allowing for control of the stationarity properties of the resulting model dynamics. Using asymptotic expansion techniques, we provide closed-form pricing formulas for caps and swaptions that are robust, accurate, and well-suited for both model calibration and general mark-to-market of plain-vanilla instruments. Monte Carlo schemes for the proposed model are proposed and examined.

The Term Structure of Interest Rates as a Random Field

Forward rate dynamics are modeled as a random field. In contrast to multifactor models, random field models offer a parsimonious description of term structure dynamics, while eliminating the self-inconsistent practice of recalibration. The form of the drift of the instantaneous forward rate process necessary to preclude arbitrage under the risk-neutral measure is obtained. Forward risk-adjusted measures are identified and used to price a bond option when the forward volatility structure depends on the square root of the current spot rate. Several classes of tractable random field models are presented. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

The reduction of forward rate dependent volatility HJM models to Markovian form: pricing European bond options

We consider a single factor Heath-Jarrow-Morton model with a forward rate volatility function depending upon a function of time to maturity, the instantaneous spot rate of interest and a forward rate to a fixed maturity. With this specification the stochastic dynamics determining the prices of interest rate derivatives may be reduced to Markovian form. Furthermore, the evolution of the forward rate curve is completely determined by the two rates specified in the volatility function and it is thus possible to obtain a closed form expression for bond prices. The prices of bond options are determined by a partial differential equation involving two spatial variables. We discuss the evaluation of European bond options in this framework by use of the ADI method.

Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options

Extensions to the Black-Scholes model have been suggested recently that permit one to calculate worst-case prices for a portfolio of vanilla options or for exotic options when no a priori distribution for the forward volatility is known. The Uncertain Volatility Model (UVM) by Avellaneda and Parás finds a one-sided worstcase volatility scenario for the buy resp. sell side within a specified volatility range. A key feature of this approach is the possibility of hedging with options: risk cancellation leads to super resp. sub-additive portfolio values. This nonlinear behaviour causes the combinatorial complexity of the pricing problem to increase significantly in the case of barrier options. In the paper, it is shown that for a portfolio P of n barrier options and any number of vanilla options, the number of PDEs that have to be solved in a hierarchical manner in order to solve the UVM problem for P is bounded by O (n 2). A numerically stable implementation is described and numerical results are given.

The Term Structure of Interest Rates as a Random Field

Note: This abstract was revised by the author since June 1997. Forward rate dynamics are modeled as a random field. In contrast to multi-factor models, random field models offer a parsimonious description of term structure dynamics, while eliminating the self-inconsistent practice of recalibration. The form of the drift of the instantaneous forward rate process necessary to preclude arbitrage under the risk neutral measure is obtained. Forward measures are characterized, and used to price a bond option when the forward volatility structure depends upon the square root of the current spot rate. In addition, it is demonstrated that random field models offer a parsimonious method to account for parameter uncertainty, inherently predicting that the best hedging instrument for a given asset is one of similar maturity. Finally, a random field is shown to be supported within a general equilibrium framework, allowing the risk-neutral measure and risk premia to be identified.