Incorporating prior financial domain knowledge into neural networks for implied volatility surface prediction

In this paper we develop a novel neural network model for predicting implied volatility surface. Prior financial domain knowledge is taken into account. A new activation function that incorporates volatility smile is proposed, which is used for the hidden nodes that process the underlying asset price. In addition, financial conditions, such as the absence of arbitrage, the boundaries and the asymptotic slope, are embedded into the loss function. This is one of the very first studies which discuss a methodological framework that incorporates prior financial domain knowledge into neural network architecture design and model training. The proposed model outperforms the benchmarked models with the option data on the S&P 500 index over 20 years. More importantly, the domain knowledge is satisfied empirically, showing the model is consistent with the existing financial theories and conditions related to implied volatility surface.Comment: 8 pages, SIGKDD 202

Option data and modeling BSM implied volatility

This contribution to the Handbook of Computational Finance, Springer-Verlag, gives an overview on modeling implied volatility data. After introducing the concept of Black-Scholes-Merton implied volatility (IV), the empirical stylized facts of IV data are reviewed. We then discuss recent results on IV surface dynamics and the computational aspects of IV. The main focus is on various parametric, semi- and nonparametric modeling strategies for IV data, including ones which respect no-arbitrage bounds.Implied volatility

Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility

We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models

Local Variance Gamma and Explicit Calibration to Option Prices

In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we provide an algorithm for calibrating a pure jump Markov martingale model to match the market prices of European options of multiple strikes and maturities. This algorithm only requires solutions of several one-dimensional root-search problems, as well as application of elementary functions. We show how to construct a time-homogeneous process which meets a single smile, and a piecewise time-homogeneous process which can meet multiple smiles

Option-implied information: What’s the vol surface got to do with it?

We find that option-implied information such as forward-looking variance, skewness and the variance risk premium are sensitive to the way the volatility surface is constructed. For some state-of-the-art volatility surfaces, the differences are economically surprisingly large and lead to systematic biases, especially for out-of-the-money put options. Estimates for risk-neutral variance differ across volatility surfaces by more than 10% on average, leading to variance risk premium estimates that differ by 60% on average. The variations are even larger for risk-neutral skewness. To overcome this problem, we propose a volatility surface that is built with a one-dimensional kernel regression. We assess its statistical accuracy relative to existing state-of-the-art parametric, semi- and non-parametric volatility surfaces by means of leave-one-out cross-validation, including the volatility surface of OptionMetrics. Based on 14 years of end-of-day and intraday S&P 500 and Euro Stoxx 50 option data we conclude that the proposed one-dimensional kernel regression represents option market information more accurately than existing approaches of the literature

APPLICATIONS OF REALIZED VOLATITILY, LOCAL VOLATILITY AND IMPLIED VOLATILITY SURFACE IN ACCURACY ENHANCEMENT OF DERIVATIVE PRICING MODEL

In this research paper, a pricing method on derivatives, here taking European options on Dow Jones index as an example, is put forth with higher level of precision. This method is able to price options with a narrower deviation scope from intrinsic value of options. The finding of this pricing method starts with testing the features of implied volatility surface. Two of three axles in constructed three-dimensional surface are respectively dynamic strike price at a given time point and the decreasing time to maturity within the life duration of one strike-specified option. General features of implied volatility surface are justified by the real trading data. With the dynamic strike price and variable volatility, option price is assumed to reflect the market expectation towards the performance of underlying asset and accompanied uncertainty when approaching the maturity. Therefore, the applications of implied volatility, local volatility and realized volatility are involved in the pricing of derivatives, because of their respective compatibilities of the forward-looking expectation, the stochastic parameter and the tight fit to the real return distribution. In the researching and analysing process, it is found that the realized volatility and the real return distribution are the derivative pricing combination with highest accuracy in the three categories of volatility. The implied volatility fails to fit the derivative price for its emphasis on market expectation and lack of independence from existing model, at the meantime, the local volatility loses its ground in practical application in pricing derivatives, with insufficient small-interval data of transactions