
Mark Ioffe  About mathematically correct definition of American option premium and using MonteCarlo method for its pricing
For the mathematically correct definition of the American option premium you must specify a sequence of points in time at which the conditions of the early expiration tests and the assumption that between two successive moments of testing option is a European. There is no reason to believe that using MonteCarlo method for American option pricing is in something superior to known binomial.


Mark Ioffe  About article “Unique Option Pricing Measure With Neither Dynamic Hedging nor Complete Markets” by Nassim Nicholas Taleb
The article presents the author's attempt to find a way to estimate the price of option other than the wellknown and common BlackScholes method, which reduces to the BlackScholes formula for European Call option and the numerical solution of the corresponding parabolic differential equations for the American option. The author believes that for derivation of the BlackScholes formula should be used not only dynamic hedging, but also some other economic assumptions. In fact, the Black Scholes formula can be easily obtained from the assumption that the process of change in stock price is described by the wellknown geometric random walk process with constant values of coefficient of trend and diffusion and without dynamic hedging and other economic assumptions.


Mark Ioffe  Pricing of Asian option with Matlab
The article refers to the calculation of the price of Asian option in Matlab. For reasons not completely understood, Matlab uses the socalled binomial option pricing model estimation. We show that for the calculation of the price of Asian option in Matlab we have to spend much less time using the method of LeviTurnbull than in case of using binomial method with no less accuracy.


Mark Ioffe  The effect on the Asian option price times between the averaging
The article refers to the calculation of the price of Asian option. In particular, we analyze the effect on the option price times between the averaging. Review shall be conducted for both Arithmetic and Geometric Mean in discrete and continuous averaging.


Mark Ioffe  Calculation of the integral required to calculate Asian option
The article refers to the calculation of the cost of Asian option. In particular, using the proposed M. Curran account for the fact that the Àrithmetic Mean is always no less than Geometric Mean. When calculating the integral occurs associated with a multidimensional normal distribution. The article presents a method for calculating the integral that is different from the one proposed by M. Curran and is much simpler.


Mark Ioffe  About M. Curran’s method for Asian options
The article refers to the calculation of the cost of Asian option. In particular, we show that the realization of proposed M. Curran’s method for Asian option [1] based on the fact that the Arithmetic Mean is always no less than the Geometric Mean can be greatly simplified. The article presents how it can be done.


Mark Ioffe  On the realization of the Butterworth filter in MATLAB
The analysis of time series is an essential tool of the financial analyst. In analysis time series digital filters and in particular low frequency play an important role. In this paper from mathematical point of view is considered lowpass Butterworth filter. Digital filter design, which consists in finding the coefficients of a finite difference scheme, is reduced to finding the coefficients of the continuous filter, for which the coefficients are the roots of the imaginary unit. To recalculate the cutoff frequency of the discrete to continuous time no additional parameters are required. As an example, compare the amplitude characteristics of the digital filter obtained by means of this method with the filter obtained in MATLAB function butter (n, f0). The reason of differences is that in MATLAB for frequency conversion from discrete to continuous one is using additional time parameter.


Mark Ioffe  European and American options with a single payment of dividends
The article provides a derivation of formulas for pricing European and American Call options with a single payment of dividends. For a European option derivation is based on the distribution of the stock price at the expiration time. For the American option is derived and numerically solved an equation for the option price as a function of the stock price and time until expiration. Numerical results are compared to calculations derived from calculations based on the formulas currently used methods. It turns out that the American option results to differ essentially from the method of Roll, Geske & Whaley.


Mark Ioffe  Binomial options pricing model.
Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple and obvious finite difference numerical method. Its basics parameters easily derived from general theory, without probabilistic considerations.


Mark Ioffe  Does the BlackScholes formula contain information regarding the rate of return of the stock?
It seems that vast popularity BlackScholes formula has become primarily because it contains no assumptions about nonrandom, predictable the rate of return of the stock changing. Our analysis shows that according to the formula BlackScholes 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 BlackScholes has quite specific information about the rate of return of the stock.


Mark Ioffe  Calculator for PublicPrivate Investment Fund.
This article describes calculator for the Geithner publicprivate 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 (lognormal). 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 DeltaHedging.
Deltahedging 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 (BlackSholes). 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 BlackScholes environment we show how to calculate weights of portfolio options and use derived formulas in spreadsheet for 6 barrier options: CallInDown, CallInUp, PutInDown, PutInUp, TouchInDown, TouchInUp.


Mark Ioffe  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 IssuerWeighted 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 23 year there may exist underestimation of risk capital up to 50%.


Mark Ioffe  Stability of explicit and implicit methods, used for pricing of the barrier option with 2 constant barriers.
BlackSholes’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.


Mark Ioffe  Pricing of window knock in/global knock out options
We describe method for pricing of firstthen barrier options. Method is based on solving of partial differential equation (PDE) by using Laplace transformation. For numerical integration we use Legendre method.
Submitted to RISK


Mark Ioffe  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 BlackSholes model while calculating premium of options, where payoffs are defined only by stock price at expiration.


Egar Technology  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 MonteCarlo method, flexibly accounting for the nuances of the statistical distribution.


Gena Ioffe, Mark Ioffe  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 CoxRossRubinstein 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.

