A data-driven framework for consistent financial valuation and risk measurement

Highlights

• A consistent framework for option pricing and risk-measurment.

• Applicable to case of limited option quotes.

• General approach, requires only an observable underlying time series.

• Fast and accurate computational procedures for practical application.

Abstract

In this paper, we propose a general data-driven framework that unifies the valuation and risk measurement of financial derivatives, which is especially useful in markets with thinly-traded derivatives. We first extract the empirical characteristic function from market-observable time series for the underlying asset prices, and then utilize Fourier techniques to obtain the physical nonparametric density and cumulative distribution function for the log-returns process, based on which we compute risk measures. Then we risk-neutralize the nonparametric density and distribution functions to model-independently valuate a variety of financial derivatives, including path-independent European options and path-dependent exotic contracts. By estimating the state-price density explicitly, and utilizing a convenient basis representation, we are able to greatly simplify the pricing of exotic options all within a consistent model-free framework. Numerical examples, and an empirical example using real market data (Brent crude oil prices) illustrate the accuracy and versatility of the proposed method in handling pricing and risk management of multiple financial contracts based solely on observable time series data.

Introduction

The problem of decision making is central to the field of operations research, where problems with embedded “optionality” play a central role. Examples include capital budgeting of R&D investment projects (Pennings & Lint, 1997), adoption of a new technology (Bobtcheff & Villeneuve, 2010), and the investment project value creation (Borgonovo, Gatti, & Peccati, 2010). To date, the operations research literature studying derivatives markets has focused mainly on parametric options pricing. With the well-accepted belief that “all models are wrong”, there have been investigations on the potential model risk in financial valuation (see Coqueret & Tavin, 2016), as well as a growing interest in the development of nonparametric/model-independent or data-driven techniques. This paper contributes a consistent data-driven framework for the valuation of derivative securities and their risk measurement.

With the rapid development of high frequency trading, the financial market has generated volumes of trades and price quote data suitable for and in need of analysis. A practical question that confronts investors, traders or fund managers is how to fairly determine the price and assess the risk of financial assets and their associated derivatives given the historical times series data of the asset, while maintaining a minimal use of parametric model assumptions. This motivates us to explore the possibility of designing a generic and consistent financial valuation and risk management framework that is fully data-driven, and is well-suited for the case of limited trading in derivatives. The goal of this research is to create a bridge that allows data to be directly translated into useful information that can be utilized for both valuation and risk management purposes. What we propose in this paper is to directly let the data meet the valuation problem without resorting to parametric models, i.e. “let the data directly speak”.

Previously, there have been several successful and popular parametric and nonparametric methods developed and employed for the valuation and risk management of financial derivatives. On the nonparametric side, there are significant developments in nonparametric density estimation through kernel density estimators (see Botev, Grotowski, Kroese, 2010, Figueroa-López, Li, 2020). Although most of the literatures lie in the realm of statistics, the methods proposed can be applied equally well to financial applications. For example, nonparametric density estimators have been applied to estimate transition densities of asset prices (Aït-Sahalia & Park, 2016), the state price densities (Song, Xiu, 2016, Zhang, Brooks, King, 2009), and risk measures such as value-at-risk and expected shortfall (Martins-Filho, Yao, Torero, 2018, Patton, Ziegel, Chen, 2019, Wang, Zhao, 2016). For a review of some of the earlier nonparametric literature pertaining to financial applications, please refer to Fan (2005) and references therein.

Another strand of nonparametric literature focuses on recovering risk-neutral densities from options prices data. Most literature stems from the celebrated (Breeden & Litzenberger, 1978) result, which states that the risk-neutral density is essentially the second derivative of the European call option price with respect to the strike, up to a risk-free discount factor, see for example (Jackwerth & Rubinstein, 1996). It is worthwhile to point out an alternative approach proposed in Rompolis and Tzavalis (2008), which first invokes the Gram-Charlier expansions, and then utilizes the Bakshi and Madan (2000) spanning result. The above referenced work (and related works) assumes that the market already prices a broad spectrum of vanilla options. In contrast, we focus on the other common case when the underlying asset is observable, but the corresponding options market is not available or not mature. An example is the cryptocurrency market, e.g. the Bitcoin market, which still awaits the introduction of options written on Bitcoins. This is also the case when we face with a thinly-traded derivative or an over-the-counter (OTC) derivative whose trading data is obscure or hard to obtain. Examples in this category arise in the diverse commodity markets for energy, agriculture, and metals (see Li & Linetsky, 2014), where the majority of contracts have little or no daily trading volume.¹ Hence we are unable to recover risk-neutral densities from option prices, but are instead given properties of the underlying asset process alone. The research question we aim to address is how to develop a consistent and arbitrage-free option valuation framework given only data of the underlying asset.

Naturally there are pros and cons associated with various parametric and nonparametric methods. Parametric models are widely used in industry for option pricing, since they are convenient and possess desirable statistical properties when the underlying model’s assumptions hold. In the literature on parametric models for options pricing, popular stochastic models include stochastic volatility models (see Bernard, Cui, & McLeish, 2017) and exponential Lévy models (see Lian, Zhu, Elliott, & Cui, 2017). Various financial contracts with path-dependent payoffs are considered, such as the arithmetic Asian options (see Cui, Lee, & Liu, 2018b). However, misspecification of the true underlying model results in inconsistent estimators. Thus it is preferable to develop an alternative approach that minimizes the impacts of choices of models. There is a strand of literature that incorporates model uncertainty in the valuation, however, it is still not fully nonparametric, or data-driven. This research is in line with recent efforts in a data-driven approach to simulation-based model calibration (Peng, Fu, Heidergott, & Lam, 2019), which aim to reduce or eliminate model-misspecification errors. Instead of carrying out inference of input parameters, our focus here is on directly extracting information from the output-level data (i.e. historical time series of asset prices) to estimate prices of financial contracts and compute risk measures within a unified framework.

Our approach is nonparametric in nature, but it inherits some nice properties akin to parametric methods in that no-arbitrage constraints are satisfied by design. By utilizing the continuous nonparametric density estimator designed in this paper, and pricing through integration with respect to this density, the smoothness of the pricing function is guaranteed similar to that with a parametric approach. Thus our approach is fully data-driven while at the same time it also preserves the smoothness of the pricing function. This distinguishes our method from other nonparametric methods. There exists literature that combines parametric and nonparametric features². A paper closely related is Barone-Adesi et al. (2008), in which they propose a parametric asymmetric GARCH model, while they estimate the innovation distribution in a nonparametric way. There are at least two differences from our approach. First, their basis is a GARCH model assumption, while we assume an underlying Lévy model. Second, they use Monte Carlo simulation to price derivatives, while our method allows for analytical Fourier inversion valuations. A detailed comparison between the two approaches is left to future research.

Besides valuation and pricing, accurate and prompt risk measurements are also important in everyday financial practice. For example, as required by the Dodd-Frank act³ or Basel III⁴, financial institutions have to report their financial positions and control the risk exposures to illiquid assets. Similarly the solvency requirements (e.g. Solvency II Directive⁵) in Europe imposes that insurance and reinsurance companies have to hold certain amount of capital (aka solvency capital requirements (SCR)) for risk planning. The risk measures serve this reporting purpose, and two of the most popular ones are the value at risk (VaR), and the expected shortfall (ES), which is also named as the conditional value at risk (CVaR). There have been some recent literatures on the nonparametric estimation of risk measures (see Chen, 2007) and also portfolio optimization problems with nonparametric risk measures as constraints (see Cui, Sun, Zhu, Jiang, & Li, 2018a). In a recent paper (Cai & Wang, 2008), they consider the application of the double kernel local linear estimator and the weighted Nadaraya-Watson estimator, and construct a distribution estimator that is continuous and differentiable. In contrast, the method proposed in this paper allows us to obtain risk estimates in a consistent manner within the same framework as valuation.

The approach taken in this paper centers around the interplay between observable financial time series under the physical measure, and the need for risk management under the physical measure and valuation of financial derivatives under the risk-neutral measure. Instead of formulating a parametric model such as a diffusion model, or adopting the risk-neutralization of a well-known times series model (such as the Heston & Nandi, 2000 GARCH model), we propose a new alternative approach, which is model-independent, and fully data-driven. The novelty of our method lies in that we carry out the risk-neutralization directly at the empirical distribution function level. Thus we are able to utilize time series to valuate different types of options contracts under the risk-neutral measure, including both path-independent and path-dependent options.

The contributions of the paper are three-fold:

• We propose a fully data-driven approach for financial valuation and risk management, which is based on the empirical characteristic function and the novel use of Fourier techniques. The input is historical time series of the financial asset, while the output are risk measures, and option prices for various types of payoffs, ranging from vanilla European call options to path-dependent Bermudan and Asian options.

• Our proposed approach is accurate and efficient, when certain conditions are met for the data generating process (DGP). We test the method using both known parametric models and real market data DGP, and demonstrate that our method yields consistently accurate computation of risk measures and valuation of a variety of financial contracts. We also carry out backtests of our method, which yield highly-accurate results.

• We propose a general data-driven framework which enables consistent no arbitrage option pricing and risk calculation without sacrificing accuracy. In general, market practitioners use the risk-neutral measure to price, and calculate VaR and ES under the physical measure. Our proposed approach uses a consistent estimator for both, and can be treated as a unified calibration method for financial valuation and risk management, which is especially useful for opaque OTC markets or markets with thinly-traded derivatives.

The rest of the paper is organized as follows: Section 2 describes the main methodology behind the nonparametric option pricing, where we propose in Section 2.1.1 the density estimator based on B-splines. Section 3 discusses the model-free option pricing method base on the Fourier method, i.e. the projection (PROJ) method. in particular, two risk-neutralization methods are discussed and compared with each other. Section 4 considers the application to risk measurement. More specifically, we consider the Value at Risk (VaR) in Section 4.1 and the Expected Shortfall (ES) in Section 4.2. Section 5 presents the numerical experiments and an empirical example, which illustrate the accuracy and efficiency of our proposed data-driven approach. We also carry out backtests in this section. Section 6 concludes the paper with discussions of potential future research directions.