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 Mark Ioffe - Does the Black-Scholes formula contain information regarding the rate of return of the stock?Mark Ioffe - Calculator for Public-Private Investment FundMark Ioffe - About Hedge and Delta-HedgingMark Ioffe - Static Hedging of Barrier OptionsIoffe Mark - Weekly option calculator. D. Petrov, M. Pomazanov. Direct Calibration of Maturity Adjustment FormulaIoffe M. Binomial approach for lookback options.EGAR's Model for pricing Weather HDD and CDD swaps and optionsIoffe G, Ioffe M., Application of finite difference method for pricing barrier options.
Ioffe M., Ioffe G., Krell D. Pricing of Binary Range and Double KO Options.
Ioffe G. Differences in Theoretical and Actual Prices of Double Knock-out and Binary Range FX Options.
Ioffe Mark. Digital FX option arbitrage
Ioffe Mark. About a finite different method for pricing of Equity Lookback options.
Ioffe Mark. Probability Distribution of Cox-Ingersoll-Ross Process
Ioffe Mark. Clique option pricing
Ioffe Mark. Variance swaps pricing
Ioffe Mark. american exchange option pricng
Ioffe Mark. Boost option pricing
Ioffe Mark. The Old VIX vs. New VIX
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EGAR'S RESEARCH AND WORKING PAPERS Mark Ioffe - Does the Black-Scholes formula contain information regarding the rate of return of the stock? It seems that vast popularity Black-Scholes formula has become primarily because it contains no assumptions about non-random, predictable the rate of return of the stock changing. Our analysis shows that according to the formula Black-Scholes average value of the rate of return of the stock equal to the interest rate r that is not dependent on the characteristics of a particular stock. Thus, the model of the Black-Scholes has quite specific information about the rate of return of the stock.
Mark Ioffe - Calculator for Public-Private Investment Fund. This article describes calculator for the Geithner public-private scheme to buy toxic assets. Principal problem for investor is to define whether it is good or bad investment, because nobody knows what assets are really worth because it depends on future events. We consider 2 stochastic models for future asset: discrete (Bernoulli) and continuous (log-normal). Using calculator we can calculate the value of P&L (Profit / Loss) for investor and FDIC (taxpayers) for different parameters values.
Mark Ioffe - About Hedge and Delta-Hedging. Delta-hedging is based on the necessity and possibility of calculating the coefficient of sensitivity option price to the stock's price (delta). Obviously, to calculate the delta requires a model that describes the option price (Black-Sholes). Meanwhile, for the same model can suggest another method to hedge, i.e., reduce the risk of adverse stock's price movements. Below we consider one of these possible methods of hedging.
Mark Ioffe - Static Hedging of Barrier Options. Static Hedging of Barrier Options called barrier's options replication by vanilla Call, Put and Digital. Possibility of barrier's options replication by vanilla Call and Put is based on the fact that knowing the theoretical prices barrier's options for multiple strikes, we can estimate the density distribution of Asset and use this to find the corresponding portfolio of vanilla Call, Put and Digital. In Black-Scholes environment we show how to calculate weights of portfolio options and use derived formulas in spreadsheet for 6 barrier options: Call-In-Down, Call-In-Up, Put-In-Down, Put-In-Up, Touch-In-Down, Touch-In-Up.
Ioffe Mark - Weekly option calculator. For options with a small time life of one to 5 days proposes pricing method based not on the model of Black – Scholes but on the histogram distribution closing price in the previous 30 trading days.
D. Petrov, M. Pomazanov - Direct Calibration of Maturity Adjustment Formula from Average Cumulative Issuer-Weighted Corporate Default Rates, Compared with Basel II Recommendations Of late years there is considerable progress in the development of credit risk models. Revised Framework on International Convergence of Capital Measurement and Capital Standards (2004) raised the standards of risk management on the new high level. At the same time the problem of theoretical investigation of probability of default time structure (and consequently maturity dependence of capital requirement, expected loss, etc.) rests actual. Basel Committee recommends to perform maturity adjustment in capital requirement. By its sense this adjustment is a penalty for exceeding one year maturity. However the direct procedure of receiving of proposed maturity adjustment rests undisclosed. In this article the authors propose a method of calculation of maturity adjustment directly from open data. In addition analytical expressions revealing probability of default time structure are received. The character of our results is close enough to Basel proposal. However it was discovered that for low probabilities of default (high ratings) and maturities of 2-3 year there may exist underestimation of risk capital up to 50%.
Ioffe Mark - Stability of explicit and implicit methods, used for pricing of the barrier option with 2 constant barriers. Black-Sholes’s partial differential equation (PDE) is the basis for fair option pricing. There are two numerical methods of resolving this PDE: explicit and implicit. By the example of the barrier option pricing we analyze stability of both methods.
Ioffe Mark - Pricing of window knock in/global knock out options We describe method for pricing of first-then barrier options. Method is based on solving of partial differential equation (PDE) by using Laplace transformation. For numerical integration we use Legendre method.
Ioffe Mark - Binomial approach for lookback options While very respectful of the importance, usefulness, relative simplicity and popularity of the Binomial CCR Method and its authors, it is considered from a mathematical point of view as only one of countless numerical methods of solving partial differential equations and even not the best one. In particular, the Binomial CCR Method of solving Cauchy problem (the problem with the given initial conditions) or the problem with the boundary and initial conditions for parabolic equation of the second order, the specific (oblique) explicit finite difference scheme is used. Binomial Method we need to have both parabolic partial differential equation of the second order and initial and boundary conditions. Here we can see how the mentioned equation and initial conditions appear in a Black-Sholes model while calculating premium of options, where pay-offs are defined only by stock price at expiration.
EGAR's Model for pricing Weather HDD and CDD swaps and options This model uses the selection of an analytical distribution function using open statistical methods of distribution function feed and then calculates the price of the derivative using Monte-Carlo method, flexibly accounting for the nuances of the statistical distribution.
Ioffe Gena, Ioffe Mark - Application of finite difference method for pricing barrier options. In recent years a number of authors pointed out significant stability and convergence problems while using Cox-Ross-Rubinstein binomial method to price and hedge barrier options. Different modifications were suggested to improve the convergence and stability of the binomial method. However, as this article shows, lattice approach in general has limited stability factor when applied to barrier options. This paper explores the use of the implicit finite difference approach in the pricing of barrier options with one or two barriers. This method has excellent stability and convergence to the solution of the underlying differential equation. This paper illustrate the use of the implicit finite difference method and provide several numeric examples.
Ioffe Gena, Ioffe Mark, Krell David - Pricing of Binary Range and Double KO Options. This paper provides close form pricing formulas for double knock out and binary range options.
Ioffe Gena - Differences in Theoretical and Actual Prices of Double Knock-out and Binary Range FX Options. In this paper, Ioffe:
Ioffe Mark - Digital FX option arbitrage We found out a theoretical possibility to build arbitrage for FX options. This possibility based on drawback Black-Sholes model. As sample, we analyze digital option, but the same is true for other FX options. For practical decision we use FOCUS system
Ioffe Mark - About a finite different method for pricing of Equity Lookback options. We try to clarify what “finite difference method” means in the financial literature. A finite difference method in mathematics is a numerical method for solving a partial differential equation (PDE). It seems that in the financial literature it is something different. We analyze this issue using as an example the pricing of Equity Lookback options.
Ioffe Mark - Probability Distribution of Cox-Ingersoll-Ross Process The classical Cox-Ingersoll-Ross process is wide spread in theoretical finance literature. This process has the noncentral chi-square distribution. By solution of first order linear partial differential equation we calculated characteristic function of this process and compare it with known characteristic function of noncentral chi-square distribution. In general case this functions and distributions are different.
Ioffe Mark - Using matlab for select applied problems. Matlab software is used for solving across a wide spectrum of applied problems and has a specific character. This article presents a basic mathematical formulas and Matlab functions for computer aided tomography, when the process of reconstructing an image from projection through the image is simulated.
Ioffe Mark - Clique option pricing We show how can be calculated Clique option premium. If number of averaging dates enough great we use central limit theorem for stochastic variables and derived analytical formula for option price. For small number of averaging we use 2 methods: Monte Carlo and method based on Gauss Legendre formula for numerical integration.
Ioffe Mark - Variance swaps pricing This article describes potential practical approaches to volatility swap pricing. We analyze 3 possible pricing methods based on historic data. First method used historic volatilities, second method uses implied volatilities and third method uses direct option prices.
Ioffe Mark - American exchange option pricng We describe 3 potential methods for calculation American exchange option price. For European exchange option exists analytical formula for pricing (formula Margrabe). Unfortunately, there is no analytical formula for American option and we must use numerical method. We analyze 3 possible numerical methods. On our opinion, the best is Gauss-Hermite method of numerical integration.
Ioffe Mark - Boost option pricing We describe method for pricing of BOOST option. Method based on solving of partial differential equation (PDE) by using Fourier transformation. We reduce pricing problem to numerical integration. For numerical integration we use Simpson’s method.
Ioffe Mark - The Old VIX vs. New VIX In 1993, the Chicago Board Options Exchange® (CBOE®) introduced the CBOE Volatility Index® .VIX ® and it quickly became the benchmark for stock market volatility. The index calculations were based on paper by Professor Robert E. Whaley of Duke University. Accordingly this paper VIX ® is constructed from implied volatilities of the eight near –the-money, nearby, and second nearby OEX option series.
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