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Option pricing formulas under a change of numeraire

Attalienti Antonio, Bufalo Michele

Opuscula Mathematica

2020

Vol. 40, no. 4

451--473

We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numeraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.

Lower Precision calculation for option pricing

Ścibisz-Mordelska K., Nielek R.

Computer Science

2017

Vol. 18 (4)

429--446

The problem of options pricing is one of the most critical issues and fundamental building blocks in mathematical finance. The research includes deployment of lower precision type in two options pricing algorithms: Black-Scholes and Monte Carlo simulation. We make an assumption that the shorter the number used for calculations is (in bits), the more operations we are able to perform in the same time. The results are examined by a comparison to the outputs of single and double precision types. The major goal of the study is to indicate whether the lower precision types can be used in financial mathematics. The findings indicate that Black-Scholes provided more precise outputs than the basic implementation of Monte Carlo simulation. Modification of the Monte Carlo algorithm is also proposed. The research shows the limitations and opportunities of the lower precision type usage. In order to benefit from the application in terms of the time of calculation improved algorithms can be implemented on GPU or FPGA. We conclude that under particular restrictions the lower precision calculation can be used in mathematical finance.

A modified Corrado-Miller implied volatility estimator

Płuciennik P.

Fasciculi Mathematici

2007

Nr 38

115-124

The implied volatility, i.e. volatility calculated on the basis of option price is a very important parameter in financial econometrics. Usually, it is calculated from the Black-Scholes option pricing formula, but it doesn't have any analytical solution. There are many ways to find it numerically. Unfortunately, all fast estimators give non rigorous results for deep-in or deep-out-of-the-rnoney options. In this paper there are compared some estimators of implied volatility and there are estimated errors for many cases of option price, strike price and real volatility. Furthermore, to reduce error using least squares surface approximation, a new estimator basing on the Corrado-Miller estimator is constructed. There are shown some cases in which the modified Corrado-Miller estimator gives more exact results.