Vanilla Options Research Articles

Multi-Curve Pricing of Non-Standard Tenor Vanilla Options

In this paper we analyse the pricing of non-standard tenor caps and swaptions in the presence of multiple curves for Libor forecasting and discounting. Pricing of non-standard tenor vanilla options usually resorts to volatility information available for standard-tenor options. However, when applying this information to non-standard tenor derivatives interest rate spreads and volatility basis between tenors need to be taken into account.We aim at disentangling the effects of interest rate spreads and volatility basis for the pricing of non-standard tenor vanilla options. This allows us to justify the constant basis point volatility approach for swaptions. Moreover, we propose a volatility transformation for caplets that consistently takes into account deterministic tenor basis and de-correlation of interest rates which drives the volatility basis.

Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging

We study option pricing and hedging with uncertainty about a Black-Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta-vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas, and volgas of the non-traded and the liquidly traded options.

Robust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options

We improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of multidimensional vanilla options in the Black–Scholes framework which emphasizes the numerical problems of integrating non-smooth functions. In high dimensions, this new algorithm is used as a control variate after a dimension reduction based on principal component analysis. Numerical tests are performed on the Genz package, on the pricing of basket, put on minimum and digital options in dimensions up to ten.

Estimating the Parameters of Stochastic Volatility Models Using Option Price Data

This article describes a maximum likelihood method for estimating the parameters of the standard square-root stochastic volatility model and a variant of the model that includes jumps in equity prices. The model is fitted to data on the S&P 500 Index and the prices of vanilla options written on the index, for the period 1990 to 2011. The method is able to estimate both the parameters of the physical measure (associated with the index) and the parameters of the risk-neutral measure (associated with the options), including the volatility and jump risk premia. The estimation is implemented using a particle filter whose efficacy is demonstrated under simulation. The computational load of this estimation method, which previously has been prohibitive, is managed by the effective use of parallel computing using graphics processing units (GPUs). The empirical results indicate that the parameters of the models are reliably estimated and consistent with values reported in previous work. In particular, both the volatility risk premium and the jump risk premium are found to be significant.

Proposal of new outperformance certificates in agricultural market

Th e paper analyses new structured fi nancial products in agricultural market with the aim to gain in the declining trends. Th ere are presented the reverse outperformance and the capped reverse outperformance certifi cates from the point of view of their investors and provided detailed descriptions of the profi t functions in the analytical form. It is shown that the payoff of these certifi cates is an engineered from a combination of the traditional fi nancial instrument and derivative products, especially the vanilla options. Further, there are developed formulas for pricing of these certifi cates and specifi ed the conditions under which the issuer is profi table in the primary market. Also the profi tability for the individual investor at the future trade date is presented. Several certifi cates for these types of certifi cates associated with the soybean futures contracts are designed and compared with the best results for the investor. Th e presented approach is based on the soybean options contracts traded on the Chicago Board of Trade.

Stochastic Volatility Double Jump-Diffusions Model: The Importance of Distribution Type of Jump Amplitude

This research examines if there exists an appealing distribution for jump amplitude in the sense that with this distribution, the stochastic volatility double jump-diffusions (SVJJ) model would potentially have a superior option market fit while keeping a sound balance between reality and tractability. We provide a general methodology for pricing vanilla options via Fourier cosine series expansion method, in the setting of Heston's SVJJ (HSVJJ) model that may allow a range of jump amplitude distributions. Example applications include the normal (N) distribution, the exponential (E) distribution and the asymmetric double exponential (Db-E) distribution, regarding to analytical tractability for options and economical interpretation. An illustrative example examines the implications of HSVJJ model in capturing option 'smirks'. This example highlights the impacts on implied volatility surface of various jump amplitude distributions, through both extensive model calibrations and carefully designed implied-volatility impacting experiments. Numerical results show that, with the Db-E jump distribution, the HSVJJ model not only captures the implied volatility smile and smirk, but also the 'sadness'.

Pricing Vanilla Options with Cash Dividends

The pricing of vanilla options on underliers with cash dividends is a surprisingly contentious and active research subject, for both European or American exercise style. Neither on the listed options side (calls and puts) nor on the flow/structured side of longer-term vanillas or light exotics are market participants in agreement on what model to use, nor on what an efficient practical implementation of the chosen model would be. The modeling problem boils down to the question of what a proper generalization of the Black-Scholes model to the case of cash dividends is, i.e. what should replace simple geometric Brownian motion (GBM).We discuss this question with the aim of taking a first step towards a rationalization and normalization of the equity volatility market. We compare the two main classes of models in use, namely the "spot model" (piecewise GBM) and several "hybrid models" (shifted GBM). We are interested in consistency, simplicity, speed, and generality (covering all traded vanilla options, dividend and borrow rate assumptions, as well as easy modeling of business time, events, term-structure, credit, light exotics, etc). We also discuss the calibration problems that market participants face in some detail.We show that: (i) all hybrid models are closely related on a mathematical level -- despite qualitatively different financial properties -- with simple and accurate relationships between calibrated parameters (borrow costs and volatilities) for both European and American options with cash dividends; (ii) all hybrid models allow accurate and very fast pricing of vanilla options using fine-tuned tree methods; (iii) some hybrid models have essentially all the desired properties outlined above; in particular, we describe a hybrid model closely related to the spot model, motivated by the spot-strike adjustment idea of Bos and Vandermark.

Assessment of Accuracy of Finite Difference Methods in Evaluation of Options within Heston Model

The recent analytical closed-form result (http://ssrn.com/abstract=2549033) discovered by Market Memory Trading L.L.C. (“MMT”) for the probability density function of the European style options with stochastic volatility, considered within the Heston model, has allowed for the first time an opportunity to assess the approximate nature of various numerical methods used for evaluation of these options. Our goal is to investigate the accuracy of the popular Crank-Nicolson Method (CNM), a two-dimensional variant of the finite difference method, applied to the vanilla options with stochastic volatility. Exact results for option prices for wide ranges of parameters of the underlying return and its variance were supplied by MMT. For brevity, studies reported are restricted to call options.

Deriving Option Prices from Characteristic Function

We describe an optimal numerical procedure for computing Fourier inversion integrals necessary for pricing vanilla options in models where the characteristic function is known in closed form. Our approach is motivated by a need for a procedure that works across all model parameters and all maturities. The procedure consists of regularizing the pricing integrand to render it finite in the region where the characteristic function is known to be analytic and then choosing the integration contour which is optimal for inverting Fourier integrals. Regularization of the pricing integrand is achieved by subtracting the Black-Scholes model with a specifically chosen volatility parameter.

Static hedging of chained-type barrier options

This paper concerns barrier options which are chained together. When the underlying asset price hits a certain barrier level, another barrier option is given to a primary option holder. Then, if the asset price hits another barrier, a third barrier option is given, and so on. The paper studies the hedging problem for these chained-type barrier options. We use the (double) reflection principle and propose a static replication portfolio of vanilla options for hedging of these options in the Black–Scholes model. The Monte Carlo simulation results for vanilla options with adjusted payoffs are provided to demonstrate the accuracy of the hedging strategies. A comparison between static hedging and delta hedging for a chained barrier option shows static hedge performs better than delta hedge.

Accurate & Efficient Pricing of Arithmetic Average Asian Options within the Hull-White Method

Simulation or other approximate methods have traditionally been required to price most exotic options as there have existed no closed-form expressions for their prices. By contrast, simulation of vanilla options without dividends is not necessary, as Black-Scholes expected prices are represented by closed-form expressions. Relying on analytical closed-form results developed by Market Memory Trading, L.L.C. (MMT), described in recent publication http://ssrn.com/abstract=2546430, errors of the widely used Hull-White Method (HWM), based on binomial trees, for valuing arithmetic average Asian options are studied here as functions of the number of time steps n, grid-spacing h, moneyness and other option parameters. It is demonstrated that: As expected, the HWM approximation error with respect to exact reference prices provided by MMT is a decreasing function of the number of time steps n for a reasonably small grid spacing h, e.g. for h = 0.01 and n = 200 the error is within 0.1% for In-The-Money (ITM) options, around 5% for At-The-Money (ATM) options and a few tens of percent for deeply Out-of-The-Money (OTM) options for a wide range of options parameters. The remarkable accuracy of the HWM approximation for ITM options is due to the fact that approximation errors are associated with estimates of the options’ time value. Time value of deeply ITM options is relatively small, making relative value of errors in its estimates even smaller. For ATM and, especially, for OTM options time value is the only value and errors are substantial. The availability of exact prices provided by MMT has allowed discovery of the important and surprising fact that for any number of time steps n, even as small as 20, there exists an optimal grid spacing h* which reproduces exact prices within just 0.5% error margin for the entire spectrum of moneyness and for any choices of expiry T, volatility σ and interest rate r. For the most liquid ATM and close to ATM options, an empirical expression for h* was found in a remarkably simple form.The opportunity provided by this formula for practitioners is that as few as n = 20 time steps can be used to price an ATM Asian option with error less than 0.5% within milliseconds for any choice of option parameters, rather than running extremely long simulations (say, 500 or more time steps) with sub-optimal grid-spacing. In this sense, knowledge of h* is practically equivalent to having the analytical closed-form pricing of Asian options. All computations were performed using the multi-threaded C code (8 threads) on a Windows 8.1, 2.4GHz (8 virtual cores) Intel i7 machine. With suboptimal grid-spacing h=0.01 evaluation of each price for n=200 time steps has taken on average 450 seconds. On a similar platform MMT produces exact prices in a millisecond per price for any set of option parameters.

A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also see that equivalent results can be obtained by using Functional Itô Calculus. Using the same generalizing ideas, we also extend to nonexponential models the alternative call option price decomposition formula written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both the anticipative and the nonanticipative cases.

Mark-to-Market Credit Index Option Pricing and Credit Volatility Index

Credit index options cannot be priced as simple vanilla options. In this paper we derive the pricing formula using both replication method and linear annuity mapping. We develop a model-free process to calculate a VIX-like credit volatility index based on observable credit index options market. We imply all model parameters from market traded data.

Risk management for international portfolios with basket options: A multi-stage stochastic programming approach

The authors consider the problem of active international portfolio management with basket options to achieve optimal asset allocation and combined market risk and currency risk management via multi-stage stochastic programming (MSSP). The authors note particularly the novel consideration and significant benefit of basket options in the context of portfolio optimization and risk management. Extensive empirical tests strongly demonstrate that basket options consistently have more clearly improvement on portfolio performances than a portfolio of vanilla options written on the same underlying assets. The authors further show that the MSSP model provides as a supportive tool for asset allocation, and a suitable test bed to empirically investigate the performance of alternative strategies.

A Complete Analytical Solution of the Asian Option Pricing within the Heston Model for Stochastic Volatility: A Probability Density Function Approach

The first ever explicit formulation of the concept of the option’s probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” and “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach.” See links: http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430.The first ever explicit formulation of the concept of the options’ probability density functions within the framework of stochastic volatility (Heston model) has been introduced in our publications “Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach”, “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” and “A Complete Analytical Resolution of the Double Barrier Option’s Pricing Within the Heston Model. A Probability Density Approach.” See links:http://ssrn.com/abstract=2549033 and http://ssrn.com/abstract=2554038 and http://ssrn.com/abstract=2605948.In this paper we report complete analytical closed-form results for the European style Asian Options considered within the Heston model for Stochastic Volatility (SV). Our discovery of the probability density function of the European style Asian Options with SV enables exact closed-form representation of its expected value (price) for the first time ever. Our formulation of the probability density function for the European style Asian Options with SV is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results for such Value-At-Risk characteristics as the probabilities that an Asian Option with SV will be below or above any threshold at any future time before or at termination. Such assessments are absolutely out of reach of the current published methods for treating Asian Options even in the framework of constant volatility.All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

A Complete Analytical Resolution of the Double Barrier Optionss Pricing within the Heston Model. A Probability Density Function Approach

• The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” and “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach” (see links http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430).• The first ever explicit formulation of the concept of an options’ probability density functions within the framework of stochastic volatility (Heston model) has been introduced in our publications “Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach” and “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” (see links http://ssrn.com/abstract=2549033 and http://ssrn.com/abstract=2554038).The probability density approach has allowed complete analytical resolution of not only all the pricing problems but for the first time complete analytical resolution of all the associated Value-At-Risk (VAR) problems by specifying probabilities of options, as stochastic quantities, to be below (above) of any threshold. • In this paper we report analogous results for pricing Double Barrier options in the presence of stochastic volatility (Heston model), even enabling complete analytical resolution of all problems associated with these options. • Our discovery of the analytical closed-form for the probability density function for the Double Barrier options with stochastic volatility enables exact results for pricing of these options for the first time without depending on approximate numerical methods. • Our formulation of the density function for the Double Barrier options with stochastic volatility within the Heston model is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with stochastic volatility will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of the current published methods for treating options within or outside the Heston model.• All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Commodity price risk management using option strategies

In the world of increasing price volatility, it is more important than ever to understand how to manage the price risk. Th e paper deals with the price risk management issues associated with commodities. Using options is performed by an analysis of hedging strategies in the commodity market. The authors focus on the application of the vanilla option strategies to risk management in order to point out the advantages and disadvantages of each hedging strategy. Based on the general expressions of selling price intervals, there are modelled various hedged scenarios of wheat. The authors look at the wheat option contracts traded on the Chicago Board of Trade. The comparative comparison of the option hedging strategies has shown the best results for the commodity seller who hedges against a price decline.

Stochastic elasticity of variance with stochastic interest rates

This paper aims to improve the implied volatility fitting capacity of underlying asset price models by relaxing constant interest rate and constant elasticity of variance and embedding a scaled stochastic setting for option prices. Using multi-scale asymptotics based on averaging principle, we obtain an analytic solution formula of the approximate price for a European vanilla option. The combined structure of stochastic elasticity of variance and stochastic interest rates is compared to the structure of stochastic volatility and stochastic interest rates. The result shows that of the two, the former is more appropriate to fit market data than the latter in terms of convexity of implied volatility surface as time-to-maturity becomes shorter.

Decomposing and valuing convertible bonds: A new method based on exotic options

We use exotic options to develop a complete decomposition method for analyzing callable convertible bonds (CCBs), and puttable callable convertible bonds (PCCBs) with credit risk. Since exotic options are path-dependent while vanilla options are not, exotic options can better address the risk exposure, and better replicate the payoff features of CCBs and PCCBs embedded with path-dependent options, than do vanilla options or warrants used by previous decomposition methods. Our method provides investors with an effective tool to analyze the effects and interactions of the different provisions contained in CCBs and PCCBs. This provides better insight into the valuation and analysis of CCBs and PCCBs.

Calibrating and Pricing with a Stochastic-Local Volatility Model

The constant volatility plain vanilla Black–Scholes model is clearly inadequate to reproduce even plain vanilla option prices observed in the market. Efforts to build a pricing model with modified dynamics that allow a better fit have mostly proceeded in one of two directions. One variant is the local volatility (LV) framework, which posits a volatility process that varies over time but is nonstochastic given the stock price. The other variant introduces a fully stochastic volatility (SV) process in which volatility has its own stochastic driving factor that is only imperfectly correlated with the returns process. Both types of models can be calibrated to market prices of at the money options. But the LV model’s ability to match today’s implied volatility surface exactly entails dynamics that can lead to large inaccuracies for exotic options, while SV models generally feature mean reversion in the volatility and tend to have trouble with far in- or out-of-the-money options. In this article, the authors propose a combined “stochastic-local volatility” model. The main structure comes from the Heston SV model, but in the returns equation, the volatility from the variance equation is multiplied by a “leverage factor” that allows the model to fit the volatility surface better. Fitting the model to the data presents some econometric challenges, but the authors show that the result works well. <b>TOPICS:</b>Options, quantitative methods