Brian Shay

Complete Analytical Solution of the SABR Model for Fixed-Income Option Pricing and Value-at-Risk Problems - A Probability Density Function Approach

The first ever explicit formulation of the concept of an option’s probability density function 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). In this paper we report similar unique results for pricing options in the presence of stochastic volatility (SABR model), enabling complete analytical resolution of all problems associated with options considered within the SABR Model. Analogous results for the Heston Model for stochastic volatility have been reported earlier (see links: http://ssrn.com/abstract=2549033; http://ssrn.com/abstract=2609143 and http://ssrn.com/abstract=2605948).Our discovery of the probability density function for options with stochastic volatility within the SABR model enables exact analytical closed-form representations of their expected values (prices) for the first time without depending on approximate numerical methods. We demonstrate by means of these reference prices that approximate numerical methods introduce substantial errors, even as high as 200-300% in some ranges of parameters. Expected value is the first moment. All higher moments are as easily represented in analytical closed-form based on our probability density function, but are not calculable by extensions of any numerical methods now used to represent the first moment. Our formulation of the density function for options with stochastic volatility of the underlying securities within the SABR model is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results Value-At-Risk (“VAR”) characteristics. These VAR characteristics are probabilities that options or portfolios of options together with the underlying securities will be below or above any set of thresholds at termination or at any time prior to termination. Such assessments are absolutely out of reach of current published methods for treating options within the SABR model.All numerical evaluations based on our analytical closed-form results are practically instantaneous and absolutely accurate.

A Complete Analytical Resolution of the Double Barrier Options’ 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.

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.

Breakthrough Technology to Resolve Problems in Annuities Hedging

Hedging of illiquid financial instruments is carried out with liquid instruments that, as a rule, have simpler payoff functions. For example, hedging of Asian or long-dated put options is carried out with vanilla puts, hedging of Bermuda swaptions is done with vanilla swaptions, etc. This kind of hedging implies replication of the complex illiquid instrument in terms of simpler and liquid instruments. Such a replication is known in mathematics as linear regression. The major inputs in carrying out any regression scheme associated with hedging of complex illiquid financial instrument with simple and liquid financial instruments are correlations between the illiquid regressor and the liquid regressands as well as correlations between regressands themselves. These correlations can be found either empirically from time series for regressor and regressands or analytically from the joint probability density functions for regressor and regressands as well as for regressands themselves. The time series approach is impractical since the regressor is, as a rule, not tradable and, therefore, no time series exists for it and the regressands are often thinly traded, making their time series sparse enough to undermine any correlation estimate. Evidently, the only suitable approach is to use the probability density functions implied by payoff functions of regressor and regressands. To the best of our knowledge Market Memory Trading L.L.C. (“MMT�?) is the only group that has analytical closed-form solution for the joint probability density functions of a wide range of financial instruments that allow evaluation of correlations between these instruments with absolute accuracy and practically instantaneously. In particular, MMT has analytical closed-form expressions for joint probability density functions of various exotic options. In this brief report the focus is on the joint probability density function of any two put options. This case is directly applicable to hedging annuities.

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: 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 publication “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” (see link http://ssrn.com/abstract=2489601). •In this paper we report similar unique results for Asian Options, enabling complete analytical resolution of all problems associated with Asian Options. •Our discovery of the Asian Option probability density function enables exact closed-form analytical results for its expected value (price) for the first time without depending on Inverse Laplace or Fourier transforms that only abbreviate complex numerical integration procedures. •Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as Inverse Laplace or Fourier transforms, now used to represent the first moment. •Our formulation of the Asian Option probability density function is general enough to cover interesting and practical cases that are not addressed in the literature at all: for example, cases for which the averaging period is a terminal subset of the contract period, e.g. the last 3 months of the option lifetime. •Our formulation of the Asian Option probability density function 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 an Asian Option will be below or above any set of thresholds at any future time before or at termination. Such assessments are absolutely out of reach of current published methods for treating Asian Options. •Resolution of the Asian Option Pricing problem enables knowledge of their linear correlation with Vanilla Options and, therefore, hedging in terms of Vanilla Options. Moreover, an opportunity to construct “synthetic” Asian Options from Vanilla Options for a given termination becomes a trivial exercise. •All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

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

• It is not widely emphasized in the literature that derivatives are complex random quantities which should, by custom, be characterized by their probability density functions. • It is understood that Black-Scholes style of derivatives pricing represents an expected value, i.e. the derivatives’ first moment and nothing else. But it is clearly impossible to reliably characterize any random variable (in our case derivative) with just its first moment. • This lack of attention to higher moments is reckless in spite of extensive analysis of “greeks�?, which provide only more information about first moments. Probabilities of losing value cannot be addressed except through density functions, most certainly not through first moments and “greeks�?. The lack of focus of practitioners on such probabilities invites the next crisis situation. • Hedging of derivatives, as understood in the present literature, is very indirectly related to risk of capital loss; it simply takes account of sensitivity of derivatives’ first moments to a few underlying parameters. • In this paper we address all the above problems from higher ground, namely an analytical framework for assessing probability density functions of derivatives and deriving analytically their higher moments and probabilities of loss and gain. We provide numerous examples of the benefits of these analytics in studying vanilla options. • Confirmation of analytical results is presented in the form of Monte Carlo simulation.

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: 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”, “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach” and “Complete Analytical Solution of the Heston Model for Option Pricing and Value-At-Risk Problems. A Probability Density Function Approach.” Please see links http://ssrn.com/abstract= 2489601, http://ssrn.com/abstract= 2546430, http://ssrn.com/abstract= 2549033). In this paper we report unique analytical results for pricing American Style Options in the presence of both constant and stochastic volatility (Heston model), enabling complete analytical resolution of all problems associated with American Style Options considered within the Heston Model. Our discovery of the probability density function for American and European Style Options with constant and stochastic volatilities enables exact closed-form analytical results for their expected values (prices) for the first time without depending on approximate numerical methods. Option prices, i.e. their expected values, are just the first moments. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods now used to represent the first moment. Our formulation of the density functions for options with American and European Style execution rights with constant and stochastic volatility (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 different execution rights, with constant or stochastic volatility, will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of current published methods for treating options.All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Robust Detection of Regime Change

Demonstration that noise filtered correlation matrices can be used for early detection of a regime change in temporal behavior of securities. This demonstration was carried out for a portfolio of 40 S&P500 securities with just two, randomly chosen, securities undergoing a deliberately arranged regime change. Demonstration that the regime changes in the behavior of the selected two securities is manifested in very noticeable change of correlation pattern between these securities and all others, leaving correlations patterns of other securities between themselves unchanged.Demonstration that unfiltered correlation matrices don’t allow detection of regime change on the same time scale.

Dramatically Improved Portfolio Optimization Results with Noise-Filtered Covariance

Demonstration that in-sample Markowitz type mean-variance optimization, carried out with noise filtered covariance matrices, results in asset allocation that leads to 2-3 times increase of the Sharpe ratio compared to the same optimization carried out without noise filtering.Demonstration of 2-3 times increase of the Sharpe ratio due to asset allocation obtained via optimization, carried out with noise filtered covariance matrices, for two possible optimization scenarios – maximization of portfolio return at a fixed volatility and minimization of portfolio volatility at a fixed return.

Filtering Noise from Volatility (Portfolio Management, Risk Analysis, et al.)

Demonstration of the omnipresence of noise in volatilities of returns of financial instruments. Demonstration that more than 30% of SP500 securities can have percentage change in volatility of more than 10% as a result of noise filtering. In our white paper “Filtering Noise From Correlation Matrices�? we have described in detail the source of noise in the correlation matrices. It is natural to assume that the same noise is present in the covariance matrix too. In particular, variances (diagonal elements of the covariance matrix - squares of volatility) contain noise as well. Our noise-filtering procedure is capable of reducing noise contained in variances in a coherent way with the noise reduction in the initial correlation matrix.