Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.     

A Note on the Term Structure of Implied Volatilities for the Yen/U.S. Dollar Currency Option

This paper tests the relationship between short dated and long dated implied volatilities obtained from Tokyo market currency option prices by employing three different volatility models: a mean reverting model, a GARCH model, and an EGARCH model. We document evidence that long dated average expected volatility is higher than that predicted by the term structure relationship during the dramatic appreciation of yen/dollar exchange in the early 1990's. This revised version was published online in August 2006 with corrections to the Cover Date.     

Overreactions in the Options Market

This paper examines the “term structure” of options' implied volatilities, using data on S&P 100 index options. Because implied volatility is strongly mean reverting, the implied volatility on a longer maturity option should move by less than one percent in response to a one percent move in the implied volatility of a shorter maturity option. Empirically, this elasticity turns out to be larger than suggested by rational expectations theory—long-maturity options tend to “overreact” to changes in the implied volatility of short-maturity options.     

From implied to spot volatilities

This paper is concerned with the relation between spot and implied volatilities. The main result is the derivation of a new equation which gives the dynamics of the spot volatility in terms of the shape and the dynamics of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We first observe that the spot volatility can be recovered from the limit, as the expiry tends to zero, of at-the-money implied volatilities. Then, we derive the semimartingale decomposition of implied volatilities at any expiry and strike from the no-arbitrage condition. Finally the spot volatility dynamics is found by performing an asymptotic analysis of these dynamics as the expiry tends to zero. As a consequence of this equation, we give general formulas to compute the shape of the implied volatility surface around the at-the-money strike and for short expiries in general spot volatility models.     

Asymptotic analysis for stochastic volatility: martingale expansion

A general class of stochastic volatility models with jumps is considered and an asymptotic expansion for European option prices around the Black–Scholes prices is validated in the light of Yoshida’s martingale expansion theory. Several known formulas of regular and singular perturbation expansions are obtained as corollaries. An expansion formula for the Black–Scholes implied volatility is given which explains the volatility skew and term structure. The leading term of the expansion is always an affine function of log moneyness, while the term structure of the coefficients depends on the details of the underlying stochastic volatility model. Several specific models which represent various types of term structure are studied.     

Analyzing interest rate risk: Stochastic volatility in the term structure of government bond yields

We propose a Nelson–Siegel type interest rate term structure model where the underlying yield factors follow autoregressive processes with stochastic volatility. The factor volatilities parsimoniously capture risk inherent to the term structure and are associated with the time-varying uncertainty of the yield curve’s level, slope and curvature. Estimating the model based on US government bond yields applying Markov chain Monte Carlo techniques we find that the factor volatilities follow highly persistent processes. We show that yield factors and factor volatilities are closely related to macroeconomic state variables as well as the conditional variances thereof.     

An Empirical Comparison of Two Stochastic Volatility Models using Indian Market Data

We conduct an empirical comparison of hedging strategies for two different stochastic volatility models proposed in the literature. One is an asymptotic expansion approach and the other is the risk-minimizing approach applied to a Markov-switched geometric Brownian motion. We also compare these with the Black–Scholes delta hedging strategies using historical and implied volatilities. The derivatives we consider are European call options on the NIFTY index of the Indian National Stock Exchange. We compare a few cases with profit and loss data from a trading desk. We find that for the cases that we analyzed, by far the better results are obtained for the Markov-switched geometric Brownian motion.     

Markov models for commodity futures: theory and practice

The objective of this paper is to develop a generic, yet practical, framework for the construction of Markov models for commodity derivatives. We aim for sufficient richness to permit applications to a broad variety of commodity markets, including those that are characterized by seasonality and by spikes in the spot process. In the first, largely theoretical, part of the paper we derive a series of useful results concerning the low-dimensional Markov representation of the dynamics of an entire term structure of futures prices. Extending previous results in the literature, we cover jump-diffusive models with stochastic volatility as well as several classes of regime-switching models. To demonstrate the process of building models for a specific commodity market, the second part of the paper applies a selection of our theoretical results to the exercise of constructing and calibrating derivatives trading models for USD natural gas. Special attention is paid to the incorporation of empirical seasonality effects in futures prices, in implied volatilities and their ‘smile’, and in correlations between futures contracts of different maturities. European option pricing in our proposed gas model is closed form and of the same complexity as the Black–Scholes formula.     

Implied Volatility in Options Markets and Conditional Heteroscedasticity in Stock Markets

This study examines whether or not the volatility of stock index returns forecasted by a GARCH-M specification is consistent with the implied volatility observed in options markets. Recent data for the New York Stock Exchange Composite Index and Standard & Poor's 500 Index and their options are employed. The patterns of the term structure of implied volatility are compared with those of volatility estimates obtained from the GARCH process. The results indicate that the GARCH process appears to partially explain the variation of implied volatilities and the term structure of implied volatilities.     

Options markets, self-fulfilling prophecies, and implied volatilities

This paper answers the following often asked question in option pricing theory: if the underlying asset's price does not satisfy a lognormal distribution, can market prices satisfy the Black-Scholes formula just because market participants believe it should? In complete markets, if the underlying asset's objective distribution is not lognormal, then the answer is no. But, in an incomplete market, if the underlying asset's objective distribution is not lognormal and all traders believe it is, then the answer is yes! The Black-Scholes formula can be a self-fulfilling prophecy. The proof of this second assertion consists of generating an economy where self-confirming beliefs sustain the Black-Scholes formula as an equilibrium. An asymmetric information model is provided, where the underlying asset's price has stochastic volatility and drift. This model is distinct from the existing pricing models in the literature, and it provides new empirical implications concerning Black-Scholes implied volatilities and the bid/ask spread. Similar to stochastic volatility models, this model is consistent with the implied volatility “smile” pattern in strike prices. In addition, it is consistent with implied volatilities being biased predictors of future volatilities.     

Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment

We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed 'bursty' or persistent nature of stock price volatility. Empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by- tick fluctuations of the index value, but it is fast mean- reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to 'fit the skew' from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log- moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk are not needed for the asymptotic pricing formulas for European derivatives, and we derive the formula for a knock-out barrier option as an example. The extension to American and path-dependent contingent claims is the subject of future work. This revised version was published online in August 2006 with corrections to the Cover Date.     

Unspanned Stochastic Volatility: Evidence from Hedging Interest Rate Derivatives

Most existing dynamic term structure models assume that interest rate derivatives are redundant securities and can be perfectly hedged using solely bonds. We find that the quadratic term structure models have serious difficulties in hedging caps and cap straddles, even though they capture bond yields well. Furthermore, at‐the‐money straddle hedging errors are highly correlated with cap‐implied volatilities and can explain a large fraction of hedging errors of all caps and straddles across moneyness and maturities. Our results strongly suggest the existence of systematic unspanned factors related to stochastic volatility in interest rate derivatives markets.     

Asymmetric Mean Reversion in European Interest Rates: A Two-factor Model

Abstract

This paper tests for asymmetric mean reversion in European short-term interest rates using a combination of the interest rate models introduced by Longstaff and Schwartz (Longstaff, F.A., Schwarts, E.S. (1992) Interest rate volatility and the ferm structure: A two factor general equilibrium model, Journal of Finance, 48, pp. 1259–1282.) and Bali (Bali, T. (2000) Testing the empirical performance of stochastic volatility models of the short-term interest rates, Journal of Financial and Quantitative Analysis, 35, pp. 191–215.). Using weekly rates for France, Germany and the United Kingdom, it is found that short-term rates follow in all instances asymmetric mean reverting processes. Specifically, interest rates exhibit non-stationary behavior following rate increases, but they are strongly mean reverting following rate decreases. The mean reverting component is statistically and economically stronger thus offsetting non-stationarity. Volatility depends on past innovations past volatility and the level of interest rates. With respect to past innovations volatility is asymmetric rising more in response to positive innovations. This is exactly opposite to the asymmetry found in stock returns.     

General approximation schemes for option prices in stochastic volatility models

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.     

Coupling smiles

The present paper addresses the problem of computing implied volatilities of options written on a domestic asset based on implied volatilities of options on the same asset expressed in a foreign currency and the exchange rate. It proposes an original method together with explicit formulae to compute the at-the-money implied volatility, the smile's skew, convexity, and term structure for short maturities. The method is completely free of any model specification or Markov assumption; it only assumes that jumps are not present. We also investigate how the method performs on the particular example of the currency triplet dollar, euro, yen. We find a very satisfactory agreement between our formulae and the market at one week and one month maturities.     

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.     

Neural network volatility forecasts

We analyse whether the use of neural networks can improve ‘traditional’ volatility forecasts from time-series models, as well as implied volatilities obtained from options on futures on the Spanish stock market index, the IBEX-35. One of our main contributions is to explore the predictive ability of neural networks that incorporate both implied volatility information and historical time-series information. Our results show that the general regression neural network forecasts improve the information content of implied volatilities and enhance the predictive ability of the models. Our analysis is also consistent with the results from prior research studies showing that implied volatility is an unbiased forecast of future volatility and that time-series models have lower explanatory power than implied volatility. Copyright © 2008 John Wiley & Sons, Ltd.     

The volatility term structure in a lognormal process for the short rate

One-factor Markov models are widely used by practitioners for pricing financial options. Their simplicity facilitates their calibration to the intial conditions and permits fast computer Implementations. Nevertheless, the danger remains that such models behave unrealistically, if the calibration of the volatility is not properly done. Here, we study a lognormal process and investigate how to specify the volatility constraints in such a way that the term structure of volatility at future times, as implied by the short rate process, has a realistic and stable shape. However, the drifting down of the volatility term structure is unavoidable. As a result, there is a tendency to underestimate option prices.     

On the stability of stablecoins

This paper investigates the volatility processes of stablecoins and their potential stochastic interdependencies with Bitcoin volatility. We employ a novel approach to choose the optimal combination for the power law exponent and the minimum value for the volatilities bending the power law. Our results indicate that Bitcoin volatility is well-behaved in a statistical sense with a finite theoretical variance. Surprisingly, the volatilities of stablecoins are statistically unstable and contemporaneously respond to Bitcoin volatility. Also, whereas the volatilities of stablecoins are not Granger-causal for Bitcoin volatility, lagged Bitcoin volatility exhibits Granger-causal effects on the volatilities of stablecoins. We conclude that Bitcoin volatility is a fundamental factor that drives the volatilities of stablecoins.     

“Life Expectancy in the Future: A Summary of a Discussion among Experts”, Robert B. Friedland,October 1998

Abstract

In this paper we present an econometric model of implied volatilities of S&;P500 index options. First, we model the dynamics the CBOE VIX index as a proxy for the general level of implied volatilities. We then describe a parametric model of the implied volatility surface for options with a term of up to two years. We show that almost all of the variation in the implied volatility surface can be explained by the VIX index and one or two other uncorrelated factors. Finally, we present a model of the dynamics of these factors.