Narrow Results

In this paper, we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Itô’s formula, we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short-maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the stochastic alpha–beta–rho (SABR), fractional Bergomi and local volatility models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

Synopsis

The research problem

This study investigates the response of stock prices and equity options to negative environmental, social, and governmental (ESG) incidents.

Motivation

Despite the growing significance of ESG news and the prominent role of equity options in portfolio management, empirical research on the relationship between options and ESG news remains notably scarce. Yet, the bulk of active asset managers, hedge funds, and investment banks continuously analyze stock prices and option prices simultaneously, to improve strategy performance and risk management and to search for information signals.

The test hypotheses

We hypothesize that negative ESG news events tend to negatively impact stock prices and trigger an increase in implied volatility (IV). We also postulate that the magnitude of market response is driven by the event characteristics, that is, “severity,” “novelty,” and “reach,” as well as the financial materiality.

Target population

Active asset managers, investment banks, regulators.

Adopted methodology

We employed a two-stage approach to analyze the effect of ESG events on stocks and option IVs. We estimated expected returns and expected change in IV using regression techniques. First, we used these to calculate abnormal returns (AR) and abnormal implied volatility (AIV). We then summed up the ARs and AIVs over several days to obtain cumulative abnormal returns (CAR) and cumulative abnormal changes in volatility (CAIV). Second, we used the CARs and CAIVs as dependent variables for fixed effect panel regressions.

Analyses

We created a large dataset of daily stock and options data on the companies from the S&P 500 index between 2006 and 2021. Most importantly, we use two unique datasets — one on negative ESG news from RepRisk and the other on options prices from OptionMetrics. Out of nearly 3 billion option prices, we extracted option implied volatilities for each stock, applied Bayesian imputation, and obtained daily average IV per stock.

Findings

Results of event studies indicate that, on average, option IV exhibits a minimal day-to-day reaction to ESG incidents. However, more severe incidents and those reported by major business sources result in a more pronounced increase in IV within 10 days following the event. Additionally, our empirical analysis highlights the influence of financial materiality on asset prices, as financially material ESG incidents lead to significant declines in stock prices and an upward surge in IV. Notably, this effect is particularly prominent for incidents related to natural capital. In conclusion, the options market incorporates information regarding ESG events, while stock prices exhibit comparatively lower and less systematic reactivity.

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes.

We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we call small-variation asymptotics, and against Fouque, Papanicolaou, and Sircar's rapid-variation asymptotics.

We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

While the Black–Scholes (BS) model and binomial trees assume that the stock price evolves lognormally with constant volatility, volatility smiles are pronounced in almost all the worlds equity markets. To study the effects of volatility smiles on hedging and arbitrage, the method based on the BS model and implemented with observed market volatility smiles is proposed. The empirical results indicate that the proposed model can reduce risk exposure and increase profits on hedging as compared with the BS model, and hence leading to considerable returns on arbitrage-trading in the Taiwan warrant market. The model is proven to be not only useful for warrant issuers who attempt to reduce vega risk, but also practicable for investors to implement as arbitrage strategies under smile effects.

This paper examines the pricing performance of various discrete-time option models that accept the variation of implied volatilities with respect to the strike price and the time-to-maturity of the option (implied volatility tree models). To this end, data from the S&P 100 options are employed for the first time. The complex implied volatility trees are compared to the standard Cox–Ross–Rubinstein model and the ad-hoc traders model. Various criteria and interpolation methods are used to evaluate the performance of the models. The results have important implications for the pricing accuracy of the models under scrutiny and their implementation.

Crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy which make the investor indifferent to uncertain (down) jumps. This is done in the situation where the investor has a logarithmic utility and where the market coefficients after a possible crash may change. It is scrutinized when and in which sense the crash hedging strategy is optimal. The situation of an investor with incomplete information is considered as well. Finally, introducing the crash horizon, an implied volatility is derived.

We propose a quasi-Monte Carlo (qMC) algorithm to simulate variates from the normal inverse Gaussian (NIG) distribution. The algorithm is based on a Monte Carlo technique found in Rydberg [13], and is based on sampling three independent uniform variables. We apply the algorithm to three problems appearing in finance. First, we consider the valuation of plain vanilla call options and Asian options. The next application considers the problem of deriving implied parameters for the underlying asset dynamics based on observed option prices. We employ our proposed algorithm together with the Newton Method, and show how we can find the scale parameter of the NIG-distribution of the logreturns in case of a call or an Asian option. We also provide an extensive error analysis for this method. Finally we study the calculation of Value-at-Risk for a portfolio of nonlinear products where the returns are modeled by NIG random variables.

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility.

An arbitrage-free CEV economy driven by Brownian motion in independent, continuous random time is introduced. European options are priced by the no-arbitrage principle as conditional averages of their classical CEV values over the CEV-modified random time to maturity. A novel representation of the classical CEV price is used to investigate the asymptotics of the average implied volatility. It is shown that the average implied volatility of the at-the-money call option is lower and of deep out-of-the-money call options, under appropriate sufficient conditions, greater than the implied CEV volatilities. Unlike in the classical CEV model, the shape of the out-of-the-money tail can be both downward and upward sloping depending on the tails of random time. The model is implemented in limit lognormal time. Its multiscaling law is shown to imply a term structure of implied volatility that is qualitatively more sensitive to changes in the time to maturity than is the classical CEV model.

In this paper, we address the problem of recovering the local volatility surface from option prices consistent with observed market data. We revisit the implied volatility problem and derive an explicit formula for the implied volatility together with bounds for the call price and its derivative with respect to the strike price. The analysis of the implied volatility problem leads to the development of an ansatz approach, which is employed to obtain a semi-explicit solution of Dupire's forward equation. This solution, in turn, gives rise to a new expression for the volatility surface in terms of the price of a European call or put. We provide numerical simulations to demonstrate the robustness of our technique and its capability of accurately reproducing the volatility function.

We discuss how implied volatilities for OTC traded Asian options can be computed by combining Monte Carlo techniques with the Newton method in order to solve nonlinear equations. The method relies on accurate and fast computation of the corresponding vegas of the option. In order to achieve this we propose the use of logarithmic derivatives instead of the classical approach. Our simulations document that the proposed method shows far better results than the classical approach. Furthermore we demonstrate how numerical results can be improved by localization.

We examine the asymptotic behaviour of the call price surface and the associated Black-Scholes implied volatility surface in the small time to expiry limit under the condition of no arbitrage. In the final section, we examine a related question of existence of a market model with non-convergent implied volatility. We show that there exist arbitrage free markets in which implied volatility may fail to converge to any value, finite or infinite.

Using a complete sample of US equity options, we analyze patterns of implied volatility in the cross-section of equity options with respect to stock characteristics. We find that high-beta stocks, small stocks, stocks with a low-market-to-book ratio, and non-momentum stocks trade at higher implied volatilities after controlling for historical volatility. We find evidence that implied volatility overestimates realized volatility for low-beta stocks, small caps, low-market-to-book stocks, and stocks with no momentum and vice versa. However, we cannot reject the null hypothesis that implied volatility is an unbiased predictor of realized volatility in the cross section.

Motivated by the desire to integrate repeated calibration procedures into a single dynamic market model, we introduce the notion of a "tangent model" in an abstract set up, and we show that this new mathematical paradigm accommodates all the recent attempts to study consistency and absence of arbitrage in market models. For the sake of illustration, we concentrate on the case when market quotes provide the prices of European call options for a specific set of strikes and maturities. While reviewing our recent results on dynamic local volatility and tangent Lévy models, we present a theory of tangent models unifying these two approaches and construct a new class of tangent Lévy models, which allows the underlying to have both continuous and pure jump components.

We derive a closed-form expression for the stock price density under the modified SABR model [see section 2.4 in Islah (2009)] with zero correlation, for β = 1 and β < 1, using the known density for the Brownian exponential functional for μ = 0 given in Matsumoto and Yor (2005), and then reversing the order of integration using Fubini's theorem. We then derive a large-time asymptotic expansion for the Brownian exponential functional for μ = 0, and we use this to characterize the large-time behaviour of the stock price distribution for the modified SABR model; the asymptotic stock price "density" is just the transition density p(t, S₀, S) for the CEV process, integrated over the large-time asymptotic "density" associated with the Brownian exponential functional (re-scaled), as we might expect. We also compute the large-time asymptotic behaviour for the price of a call option, and we show precisely how the implied volatility tends to zero as the maturity tends to infinity, for β = 1 and β < 1. These results are shown to be consistent with the general large-time asymptotic estimate for implied variance given in Tehranchi (2009). The modified SABR model is significantly more tractable than the standard SABR model. Moreover, the integrated variance for the modified model is infinite a.s. as t → ∞, in contrast to the standard SABR model, so in this sense the modified model is also more realistic.

We introduce a class of randomly time-changed fast mean-reverting stochastic volatility (TC-FMR-SV) models. Using spectral theory and singular perturbation techniques, we derive an approximation for the price of any European option in the TC-FMR-SV setting. Three examples of random time-changes are provided and are shown to induce distinct implied volatility surfaces. The key features of the TC-FMR-SV framework are that (i) it is able to incorporate jumps into the price process of the underlying asset (ii) it allows for the leverage effect and (iii) it can accommodate multiple factors of volatility, which operate on different time-scales.

In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

We model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the price process is driven by Brownian motion, an associated "master equation" for the dynamics of the conditional probability density is derived and expressed in integral form. By a "model" for the conditional density process we mean a solution to the master equation along with the specification of (a) the initial density, and (b) the volatility structure of the density. The volatility structure is assumed at any time and for each value of the argument of the density to be a functional of the history of the density up to that time. In practice one specifies the functional modulo sufficient parametric freedom to allow for the input of additional option data apart from that implicit in the initial density. The scheme is sufficiently flexible to allow for the input of various types of data depending on the nature of the options market and the class of valuation problem being undertaken. Various examples are studied in detail, with exact solutions provided in some cases.

In this paper, we study the asymptotic behavior of the implied volatility in stochastic asset price models. We provide necessary and sufficient conditions for the validity of asymptotic equivalence in Lee's moment formulas, and obtain new asymptotic formulas for the implied volatility in asset price models without moment explosions. As an application, we prove a modified version of Piterbarg's conjecture. The asymptotic formula suggested by Piterbarg may be considered as a substitute for Lee's moment formula for the implied volatility at large strikes in the case of models without moment explosions. We also characterize the asymptotic behavior of the implied volatility in several special asset price models, e.g., the CEV model, the finite moment log-stable model of Carr and Wu, the Heston model perturbed by a compound Poisson process with double exponential law for jump sizes, and SV1 and SV2 models of Rogers and Veraart.

We study the term structure of the implied volatility in the presence of a symmetric smile. Exploiting the result by Tehranchi (2009) that a symmetric smile generated by a continuous martingale necessarily comes from a mixture of normal distributions, we derive representation formulae for the at-the-money (ATM) implied volatility level and curvature in a general symmetric model. As a result, the ATM curve is directly related to the Laplace transform of the realized variance. The representation formulae for the implied volatility and its curvature take semi-closed form as soon as this Laplace transform is known explicitly. To deal with the rest of the volatility surface, we build a time dependent SVI-type (Gatheral, 2004) model which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, the SVI approximation displays considerable performances in a wide range of maturities and strikes. All these results can be applied to skewed smiles by considering a displaced model. Finally, a noteworthy fact is that all along the paper we avoid dealing with any complex-valued function.