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Foad Shokrollahi - One of the best experts on this subject based on the ideXlab platform.

  • the evaluation of geometric asian power options under time changed mixed fractional brownian motion
    Journal of Computational and Applied Mathematics, 2018
    Co-Authors: Foad Shokrollahi
    Abstract:

    Abstract In this paper, the geometric Asian option pricing problem is investigated under the assumption that the underlying stock price is assumed following a mixed fractional subdiffusive Black–Scholes Model, and the geometric average Asian option pricing formula is derived under this assumption. We then apply the results to value Asian power options on the stocks that pay constant dividends when the payoff is a power function. Finally, lower bound of Asian options and some special cases are provided.

  • Subdiffusive fractional Black–Scholes Model for pricing currency options under transaction costs
    Taylor & Francis Group, 2018
    Co-Authors: Foad Shokrollahi
    Abstract:

    A new framework for pricing European currency option is developed in the case where the spot exchange rate follows a subdiffusive fractional Black–Scholes. An analytic formula for pricing European currency call option is proposed by a mean self-financing delta-hedging argument in a discrete time setting. The minimal price of a currency option under transaction costs is obtained as time-step $$\Delta t = {\left({{{{t^{\alpha - 1}}} \over {\Gamma (\alpha )}}} \right)^{ - 1}}{\left({{2 \over \pi }} \right)^{{1 \over {2H}}}}{\left({{k \over \sigma }} \right)^{{1 \over H}}}$$, which can be used as the actual price of an option. In addition, we also show that time-step and long-range dependence have a significant impact on option pricing

  • the evaluation of geometric asian power options under time changed mixed fractional brownian motion
    arXiv: Pricing of Securities, 2017
    Co-Authors: Foad Shokrollahi
    Abstract:

    The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes Model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stocks that pay constant dividends when the payoff is a power function. Finally, lower bound of Asian options and some special cases are provided.

Saikat Nandi - One of the best experts on this subject based on the ideXlab platform.

  • a closed form garch option valuation Model
    2000
    Co-Authors: Steven L Heston, Saikat Nandi
    Abstract:

    This paper develops a closed-form option valuation formula for a spot asset whose variance follows a GARCH(p,q) process that can be correlated with the returns of the spot asset. It provides the first readily computed option formula for a random volatility Model that can be estimated and implemented solely on the basis of observables. The single lag version of this Model contains Heston's (1993) stochastic volatility Model as a continuous-time limit. Empirical analysis on S&P500 index options shows that the out-of-sample valuation errors from the single lag version of the GARCH Model are substantially lower than the ad hoc Black-Scholes Model of Dumas, Fleming and Whaley (1998) that uses a separate implied volatility for each option to fit to the smirk/smile in implied volatilties. The GARCH Model remains superior even though the parameters of the GARCH Model are held constant and volatility is filtered from the history of asset prices while the ad hoc Black-Scholes Model is updated every period. The improvement is largely due to the ability of the GARCH Model to simultaneously capture the correlation of volatility with spot returns and the path dependence in volatility.

  • a closed form garch option valuation Model
    Review of Financial Studies, 2000
    Co-Authors: Steven L Heston, Saikat Nandi
    Abstract:

    This paper develops a closed-form option valuation formula for a spot asset whose variance follows a GARCH(p, q) process that can be correlated with the returns of the spot asset. It provides the first readily computed option formula for a random volatility Model that can be estimated and implemented solely on the basis of observables. The single lag version of this Model contains Heston's (1993) stochastic volatility Model as a continuous-time limit. Empirical analysis on S&P500 index options shows that the out-of-sample valuation errors from the single lag version of the GARCH Model are substantially lower than the ad hoc Black-Scholes Model of Dumas, Fleming and Whaley (1998) that uses a separate implied volatility for each option to fit to the smirk/smile in implied volatilities. The GARCH Model remains superior even though the parameters of the GARCH Model are held constant and volatility is filtered from the history of asset prices while the ad hoc Black-Scholes Model is updated every period. The improvement is largely due to the ability of the GARCH Model to simultaneously capture the correlation of volatility, with spot returns and the path dependence in volatility. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

  • a closed form garch option pricing Model
    1997
    Co-Authors: Steven L Heston, Saikat Nandi
    Abstract:

    This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The Model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single factor (one lag) version of this Model contains Heston's (1993) stochastic volatility Model as a diffusion limit and therefore unifies the discrete GARCH and continuous-time stochastic volatility literature of option pricing. The new Model provides the first option formula for a random volatility Model that is solely a function of observables; all the parameters can be easily estimated from the history of asset prices, observed at discreteintervals. Empirical analysis on S&P500 index options shows the single factor version of the GARCH Model to be a substantial improvement over the Black-Scholes (1973) Model. The GARCH Model continues to substantially outperform the Black-Scholes Model even when the Black-Scholes Model is updated every period while the parameters of the GARCH Model are held constant. The improvement is due largely to the ability of the GARCH Model to describe the correlation of volatility with spot returns. This allows the GARCH Model to capture strike price biases in the Black-Scholes Model that give rise to the skew in implied volatilities in the index options market.

  • a closed form garch option pricing Model
    1997
    Co-Authors: Steven L Heston, Saikat Nandi
    Abstract:

    This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The Model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single-factor (one-lag) version of this Model contains Heston's (1993) stochastic volatility Model as a diffusion limit and therefore unifies the discrete-time GARCH and continuous-time stochastic volatility literature of option pricing. The new Model provides the first readily computed option formula for a random volatility Model in which current volatility is easily estimated from historical asset prices observed at discrete intervals. Empirical analysis on S&P 500 index options shows the single-factor version of the GARCH Model to be a substantial improvement over the Black-Scholes (1973) Model. The GARCH Model continues to substantially outperform the Black-Scholes Model even when the Black-Scholes Model is updated every period and uses implied volatilities from option prices, while the parameters of the GARCH Model are held constant and volatility is filtered from the history of asset prices. The improvement is due largely to the ability of the GARCH Model to describe the correlation of volatility with spot returns. This allows the GARCH Model to capture strike-price biases in the Black-Scholes Model that give rise to the skew in implied volatilities in the index options market.

Stephen Figlewski - One of the best experts on this subject based on the ideXlab platform.

  • assessing the incremental value of option pricing theory relative to an informationally passive benchmark
    Journal of Derivatives, 2002
    Co-Authors: Stephen Figlewski
    Abstract:

    In modern finance, we measure the value of an active investment strategy by comparing its performance against the benchmark of passively holding the market portfolio and the riskless asset. We can evaluate the marginal contribution of a theoretical derivatives pricing Model in the same way, by comparing its performance against an “informationally passive” alternative Model. All rationally priced options must satisfy a number of conditions to rule out profitable static arbitrage. The Black-Scholes Model, and others like it, are obtained by assuming an equilibrium that also forecloses profitable dynamic arbitrage opportunities. The passive Model we consider incorporates only the fundamental properties of option prices that must hold to avoid static arbitrage, but has no theoretical content beyond that. We review different measures of Model performance, and apply them to several versions of the Black-Scholes Model and our passive Model. As with active portfolio management, it turns out to be not that easy for an active Model to do a lot better than a well-designed passive alternative. For example, the classic Black-Scholes turns out to be less accurate than the passive benchmark.

  • assessing the incremental value of option pricing theory relative to an informationally passive benchmark
    Social Science Research Network, 2002
    Co-Authors: Stephen Figlewski
    Abstract:

    In modern finance, the value of an active investment strategy is measured by comparing its performance against the benchmark of passively holding the market portfolio and the riskless asset. We wish to evaluate the marginal contribution of a theoretical derivatives pricing Model in the same way, by comparing its performance against an 'informationally passive' alternative Model. All rationally priced options must satisfy a number of conditions to rule out profitable static arbitrage. The Black-Scholes Model, and others like it, are obtained by assuming an equilibrium in which there are no profitable dynamic arbitrage opportunities either. The passive Model we consider incorporates only the fundamental properties of option prices that must hold to avoid static arbitrage, but has no theoretical content beyond that. We review different measures of Model performance and apply them to several versions of the Black-Scholes Model and our passive Model. As with active portfolio management, it turns out to be not that easy for an 'active' Model to do a lot better than a well designed passive alternative. For example, the 'classical' Black-Scholes Model turns out to be less accurate than the passive benchmark.

C C Heyde - One of the best experts on this subject based on the ideXlab platform.

  • a risky asset Model with strong dependence through fractal activity time
    Journal of Applied Probability, 1999
    Co-Authors: C C Heyde
    Abstract:

    The geometric Brownian motion (Black–Scholes) Model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator Model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

  • ito s formula with respect to fractional brownian motion and its application
    Journal of Applied Mathematics and Stochastic Analysis, 1996
    Co-Authors: W Dai, C C Heyde
    Abstract:

    Fractional Brownian motion (FBM) with Hurst index 1/2 < H < 1 is not a semimartingale. Consequently, the standard It calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 < H < 1. In this paper we derive a version of It’s formula for fractional Brownian motion. Then, as an application, we propose and study a fractional Brownian Scholes stochastic Model which includes the standard Black-Scholes Model as a special case and is able to account for long range dependence in Modeling the price of a risky asset.

Mirko Dohnal - One of the best experts on this subject based on the ideXlab platform.

  • qualitative phase portrait of modified black scholes Model
    Expert Systems With Applications, 2010
    Co-Authors: Jiři Konecný, Tomas Vicha, Mirko Dohnal
    Abstract:

    A qualitative Model is based on only three values - positive (increasing), zero (constant) and negative (decreasing). This paper gives a simplified interpretation of some basic concepts, to eliminate the necessity of an extensive study of qualitative reasoning. The generally accepted theory of option pricing is based on the Black-Scholes Model. The Black-Scholes Model is transferred into a set of qualitative relations and two additional variables. Mood on stock market and risk aversion are incorporated into the qualitative Model. The resulting Model has 1250 solutions, and there are 16,416 transitions among them. A Chaos related interpretation of the results is presented. An example of a prediction based on modified Black-Scholes Model is demonstrated below.