Cliquet Options Research Articles

Recent Developments in Financial Risk and the Real Economy

In this article, we review recent developments in macroeconomics and finance on the relationship between financial risk and the real economy. We focus on three specific topics: (a) the term structure of uncertainty, (b) time variation—specifically, the long-term decline—in the variance risk premium, and (c) time variation in conditional skewness. We also introduce two new data series: implied volatility from one-day options on grains for the period 1906–1936 and prices of cliquet options, which provide insurance against single-day crashes on the S&P 500. Both series give some context to the recent rise in trade in extremely short-dated options. Finally, we discuss new avenues for future research.

Neural Networks Based Dynamic Implied Volatility Surface

A pricing model is tied to its ability of capturing the dynamics of the spot price process. Its misspecification will lead to pricing and hedging errors. Parametric pricing formula depends on the particular form of the dynamics of the underlying asset. For tractability reasons, some assumptions are made which are not consistent with the multifractal properties of market returns. On the other hand, non-parametric models such as neural networks use market data to estimate the implicit stochastic process driving the spot price and its relationship with contingent claims. We propose to use neural networks for learning the dynamics of the implied volatility surface. Rather than learning the dynamics of the whole volatility surface, we perform a dimensionality reduction and choose to learn that of the main volatility risk factors from the data. These risk factors form a vector basis for the implied volatility surface. They can be the first three eigenmodes of a PCA decomposition, the parameters of a polynomial model, or any other model parameters from a stochastic volatility model. We assume that these factors are driven by explanatory variables, such as spot price, volume, estimated volatility, and financial indicators like the VIX and options from the main indices. We use neural networks to capture the relation between the volatility risk factors and their explanatory variables. We model the dynamics of the IV surfaces by letting the parameters of a polynomial model, or a stochastic volatility model with an explicit expression for the smile such as the SVI and the SABR model, be statistically evolved, by considering one neural network per parameter. The smile is reconstructed by using the polynomial model, or the parametric smile representation, where each parameter is replaced by a trained network. That is, not only do we learn the model parameters at a fixed time, but we use the fact that the networks are universal interpolators to learn about the dynamics of their term-structures. We can then reconstruct the volatility surface and use it for pricing and hedging options as well as for risk analysis, forecasting, and volatility trading. We also describe how to use the model for pricing forward starting contracts such as cliquet options.

CVA for Cliquet options under Heston model

Credit value adjustment (CVA) is an important pricing component in the counterparty credit risk (CCR) management. Cliquet options are a popular volatility product with protection against downside risk as well as significant upside potential. This paper aims to study the CVA for Cliquet options under stochastic volatility models. A partial differential equation (PDE) is first derived to price Cliquet options under the Heston model. Numerical schemes are then provided to solve the PDE and calculate exposure and CVA. Numerical tests are also carried out to examine the scheme accuracy and impacts of wrong way risk to CVA. Test results show that the numerical schemes are accurate. Wrong way risk plays an important role in pricing CVA for Cliquet options and the impact is crucial from the perspective of CCR management.

Variance Reduction of Ordinary Monte-Carlo Estimates with the Brownian-Bridge Path Construction

It is well known that the Brownian-bridge path construction is usually very effective at reducing the variance of quasi Monte-Carlo estimates, with some exceptions – cliquet options in particular. It will however not change the variance of the value of a financial derivative contract obtained by a standard Monte-Carlo simulation, where pseudo-random numbers are used instead of quasi-random numbers (or low discrepancy sequences such as the Sobol sequence). We will see here that the Brownian-bridge path construction can still significantly reduce the variance of the Theta estimate, that is the sensitivity towards time of a financial derivative contract, even without relying on quasi-random numbers.

Cliquet option pricing with Meixner processes

We investigate the pricing of cliquet options in a geometric Meixner model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a pure-jump Meixner--L\'{e}vy process yielding Meixner distributed log-returns. In this setting, we infer semi-analytic expressions for the cliquet option price by using the probability distribution function of the driving Meixner--L\'{e}vy process and by an application of Fourier transform techniques. In an introductory section, we compile various facts on the Meixner distribution and the related class of Meixner--L\'{e}vy processes. We also propose a customized measure change preserving the Meixner distribution of any Meixner process.

Cliquet Option Pricing in a Jump-Diffusion LLvy Model

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Levy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Levy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Levy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several greeks while putting emphasis on the vega.

Calibration Design of Implied Volatility Surfaces

The calibration of option pricing models leads to the minimization of an error functional. We show that its usual specification as a root mean squared error implies fluctuating exotics prices and possibly wrong prices. We propose a simple and natural method to overcome these problems, illustrate drawbacks of the usual approach and show advantages of our method. To this end, we calibrate the Heston model to a time series of DAX implied volatility surfaces and then price cliquet options.

Cliquet Option Pricing with Meixner Processes

We investigate the pricing of cliquet options in a geometric Meixner model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a pure-jump Meixner-Levy process yielding Meixner distributed log-returns. In this setting, we infer semi-analytic expressions for the cliquet option price by using the probability distribution function of the driving Meixner-Levy process and by an application of Fourier transform techniques. In an introductory section, we compile various facts on the Meixner distribution and the related class of Meixner-Levy processes. We also propose a customized measure change preserving the Meixner distribution of any Meixner process.

Minimum Relative Entropy and Cliquet Hedging

The goal of this paper is to show a very practical technique for hedging and risk measures of the exotic equity derivative named a cliquet. This methodology was developed and successfully applied by the author in his day-to-day work as an equity derivatives senior quant. The technique is based on the following pillars: Econometric GARCH family for time-series modeling. Minimum relative entropy methodology, as presented by Avellaneda et al. (2001); also known as weighted Monte Carlo. Mean/variance hedging loss function equivalences (Schal, 1994). Based on these we develop a very useful pricing and hedging methodology that has been applied very successfully during the first decade of the 21st century. Concretely, it has been used to hedge a several-hundred-million EUR cliquet option book.

Numerical experiments on hedging cliquet options

In this paper, we conduct pricing and hedging experiments in order to check whether simple stochastic volatility models are capable of capturing forward volatility and forward skew risks correctly. As a reference, we use the Bergomi model, which treats these risks accurately. The results of our experiments show that the cost of poor volatility modeling in the Heston model, the Barndorff-Nielsen–Shephard model and a variance-gamma model with stochastic arrival is too high when pricing and hedging cliquet options.

Efficient Valuation of Equity-Indexed Annuities Under LLvy Processes Using Fourier-Cosine Series

Equity-Indexed Annuities (EIAs) are deferred annuities which accumulate value over time according to crediting formulas and realized equity index returns. We propose an efficient algorithm to value two popular crediting formulas found in EIAs - Annual Point-to-Point (APP) and Monthly Point-to-Point (MPP) - under general Levy-process based index returns. APP contracts observe returns of referenced indexes annually and credit EIA accounts, subject to minimum and maximum returns. MPP contracts incorporate both local/monthly caps and global/annual floors on index credits. MPP contracts have payoffs of a cliquet option. Our algorithm, based on the COS method (Fang and Oosterlee, 2008), expands the present value of an EIA contract using Fourier-cosine series, expresses the value of the EIA contract as a series of terms involving simple characteristic function evaluations. We present several examples with different Levy processes, including the Black-Scholes model and the CGMY model. These examples illustrate the efficiency of our algorithm as well as its versatility in computing annuity market sensitivities, which could facilitate the hedging and pricing of annuity contracts.

Pricing Forward Skew Dependent Derivatives. Multifactor Versus Single‐Factor Stochastic Volatility Models

AbstractEmpirical evidence shows that, in equity options markets, the slope of the skew is largely independent of the volatility level. Single‐factor stochastic volatility models are not flexible enough to account for the stochastic behavior of the skew. On the other hand, multifactor stochastic volatility models are able to account for the existence of stochastic skew. This study studies the effects of introducing stochastic skew in the valuation of forward skew dependent exotic options. In particular, I consider cliquet, as well as reverse cliquet structures. The study also derives a semi‐closed‐form solution for the price of forward‐start options under the multifactor stochastic specification. The empirical results indicate that the consideration of additional volatility factors in the context of stochastic volatility models allows us to generate more flexible smile patterns. This additional flexibility has a relevant impact on the valuation of forward skew dependent derivatives. In this sense, this study shows that similar calibrations of single factor and multifactor stochastic volatility models to the current market prices of plain vanilla options can lead to important discrepancies in the pricing of exotic forward skew dependent derivatives such as regular cliquet structures and reverse cliquet options. © 2013 Wiley Periodicals, Inc. Jrl Fut Mark 34:124–144, 2014

Pricing and Hedging of Cliquet Options and Locally Capped Contracts

This paper provides a new approach for pricing and hedging popular highly path-dependent equity-linked contracts. We illustrate our technique with two examples: the locally capped contracts (a popular design on the exchange-listed retail investment contracts on the American Stock Exchange) and the cliquet option (extensively sold by insurance companies). Wilmott [Wilmott Magazine, 2002, pp. 78--83] describes these types of contracts as the “height of fashion in the world of equity derivatives.” Existing literature proposes methods based on partial differential equations, Monte Carlo techniques, or Fourier analysis. We show that there exist semi-closed-form expressions of their prices as well as of the hedging parameters.

AN EFFICIENT BINOMIAL TREE METHOD FOR CLIQUET OPTIONS

This work proposes a binomial method for pricing the cliquet options, which provide a guaranteed minimum annual return. The proposed binomial tree algorithm simplifies the standard binomial approach, which is problematic for cliquet options in the computational point of view, or other recent methods, which may be of intricate algorithm or require pre- or post-processing computations. Our method is simple, efficient and reliable in a Black-Scholes framework with constant interest rates and volatilities.

Liquidity Options

The financial crisis of 2008 revealed how critically important liquidity is to investors in times of turmoil. When investors and financial institutions were facing huge demands for cash that they either had to meet or else declare bankruptcy, security valuation that was derived from theoretical models became completely irrelevant. Tradable securities had to be dumped on the market and sold for whatever prices they could fetch. Looking back, one wonders: Can’t anyone figure out some way to hedge this risk using derivatives? Golts and Kritzman think they have. In this article, they propose the creation of a new kind of derivative contract they call a “liquidity option.” They describe how such options could be structured with payoffs resembling those of a cliquet option tied to some broad market index, and show how they should be priced to be consistent with the market’s current valuation of liquid versus illiquid instruments. <b>TOPICS:</b>Options, accounting and ratio analysis, VAR and use of alternative risk measures of trading risk

Pricing cliquet options by tree methods

This paper focuses on the problem of pricing the cliquet options which provide a guaranteed minimum annual return. The tree method which we propose simplifies the standard binomial Cox–Ross–Rubinstein approach which, in this context, is problematic from a computational point of view. Our technique provides very efficient and reliable evaluations in a Black-Scholes framework with piecewise constant interest rates and volatilities.

Pricing Forward Start Options in Models Based on (Time-Changed) Levy Processes

Options depending on the forward skew are very popular. One such option is the forward starting call option - the basic building block of a cliquet option. Widely applied models to account for the forward skew dynamics to price such options include the Heston model, the Heston-Hull-White model and the Bates model. Within these models solutions for options including forward start features are available using (semi) analytical formulas. Today exponential (subordinated) Levy models being increasingly popular for modelling the asset dynamics. While the simple exponential Levy models imply the same forward volatily surface for all future times the subordinated models do not. Depending on the subordinator the dynamic of the forward volatility surface and therefore stochastic volatility can be modelled. Analytical pricing formulas based on the characteristic function and Fourier transform methods are available for the class of these models. We extend the applicability of analytical pricing to options including forward start features. To this end we derive the forward characteristic functions which can be used in Fourier transform based methods. As examples we consider the Variance Gamma model and the NIG model subordinated by a Gamma Ornstein Uhlenbeck process and respectively by an Cox-Ingersoll-Ross process. We check our analytical results by applying Monte Carlo methods. These results can for instance be applied to calibration of the forward volatility surface.

An Accelerating Quasi-Monte Carlo Method for Option Pricing Under the Generalized Hyperbolic Lévy Process

In this paper, we develop a simple and yet practically efficient algorithm for simulating high-dimensional exotic options. Our method is based on an extension of Imai and Tan's linear transformation method, which is originally proposed in the context of simulating a Gaussian process. By generalizing this method to other stochastic processes and exploiting the numerical inversion method of Hormann and Leydold, this method can be used to enhance quasi-Monte Carlo method in a wide range of applications. We demonstrate the relative efficiency of our proposed simulation technique using exotic option examples including Asian, lookback, barrier, and cliquet options for which the underlying asset price follows an exponential generalized hyperbolic Levy process. We also illustrate the impact of our proposed method on dimension reduction.

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.

Cliquet Options: Pricing and Greeks in Deterministic and Stochastic Volatility Models

This paper presents a method to determine the price of a cliquet option, as well as its sensitivity to changes in the market, the Greeks, for deterministic (also incorporating skews) and stochastic (Hestonian) volatility and, lognormal and jump-diffusion asset price - processes, with almost machine precision in a fraction of a second. In the pricing algorithms we make use of a new Laplace transform inversion technique, which guarantees fast and numerically stable pricing. The computation of the Greeks is based on a Girsanov type drift adjustment.