Martingales and stochastic integrals in the theory of continuous trading

Harrison, J. Michael; Pliska, Stanley R.

doi:10.1016/0304-4149(81)90026-0

Cited by 2,356 publications

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“…In a finite state market, the fundamental theorem of arbitrage has two parts. The first part relates to existence of a risk neutral measure, while the second relates to the uniqueness of the measure (See [15], [16] & [17]):…”

Section: Http://journalsuobedubhmentioning

confidence: 99%

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with Transaction Cost for Pricing European Options and Hedging

Oyediran¹

2016

IJCTS

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Abstract:In this paper models with transaction costs for pricing of European options using mixed fractional Brownian motion (fbm) and Partial differential equation (PDE) model are considered. Investigation on price sensitivity to volatility and formulation of asymptotic strategy for replicating self financing assets are also made. Simulation experiments for the models are run to obtain the call and put prices using fbm with Hurst parameter 1 3H  and the Crank-Nicolson method for the numerical solution to the PDE model.It is found that stock prices increase with time to maturity dates, whereas, the call prices decrease steadily as time approaches maturity dates in conformity with theta hedging strategy.Keywords: Simulation, European option, fractional Brownian motion, stochastic volatility. Dedicated to evergreen memory of Late Professor M. S. Sasey Author's contributionsIn the recent times, development of financial instruments for pricing options are on increasing and the use of mixed fractional Brownian motion (fbm) for simulation of derivatives are not too common. In this paper, simulation experiments are designed and implemented using codes in MATLAB for the models considered. We obtained the call and put prices using fbm with Hurst parameter 1 3H  and the Crank-Nicolson method for the numerical solution to the PDE model using fbm realization. Investigation on price sensitivity to volatility and formulation of asymptotic strategy for replicating self financing assets are also made.

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Section: Http://journalsuobedubhmentioning

confidence: 99%

“…When the stock price process is assumed to follow a more general semi martingale process (see [15], [18]), then the concept of arbitrage opportunity is too strong, and a weaker concept with vanishing risk must be used to describe this opportunity in an infinite dimensional setting.…”

Section: Http://journalsuobedubhmentioning

confidence: 99%

with Transaction Cost for Pricing European Options and Hedging

Oyediran¹

2016

IJCTS

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“…Indeed, we find that the technical source of the problems in fitting the empirical data with our model is the very large cumulative contribution of the risk-free interest rate r(t). Recall that this contribution comes from the noarbitrage condition obtained under a change of probability measure from the objective to a 'risk-neutral' probability measure (a priori distinct from the objective one), which changes the price process from a semi-martingale into a martingale [8,9]. We thus propose to change the no-arbitrage condition, which is equivalent to requiring that the prices are martingales, by the condition that the prices follow a semi-martingale, with the drift of the semi-martingale taken proportional to the risk-free rate.…”

Section: Model 2: Weaker Semi-martingale Condition On Pricesmentioning

confidence: 99%

Fundamental factors versus herding in the 2000–2005 US stock market and prediction

Zhou

Sornette

2006

Physica A: Statistical Mechanics and its Applications

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We present a general methodology to incorporate fundamental economic factors to our previous theory of herding to describe bubbles and antibubbles. We start from the strong form of Rational Expectation and derive the general method to incorporate factors in addition to the log-periodic power law (LPPL) signature of herding developed in ours and others' works. These factors include interest rate, interest spread, historical volatility, implied volatility and exchange rates. Standard statistical AIC and Wilks tests allow us to compare the explanatory power of the different proposed factor models. We find that the historical volatility played the key role before August of 2002. Around October 2002, the interest rate dominated. In the first six months of 2003, the foreign exchange rate became the key factor. Since the end of 2003, all factors have played an increasingly large role. However, the most surprising result is that the best model is the second-order LPPL without any factor. We thus present a scenario for the future evolution of the US stock market based on the extrapolation of the fit of the second-order LPPL formula, which suggests that herding is still the dominating force and that the unraveling of the US stock market antibubble since 2000 is still qualitatively similar to (but quantitatively different from) the Japanese Nikkei case after 1990.

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“…(12) Note that the first term (J T |S t − 1 = s − ) will also have to be estimated separately. Intuitively, the estimator resembles the form of the general estimator derived in Fu and Hu (1992) …”

Section: η(T )mentioning

confidence: 99%

Sensitivity Analysis for Monte Carlo Simulation of Option Pricing

1995

Prob. Eng. Inf. Sci.

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Monte Carlo simulation is one alternative for analyzing options markets when the assumptions of simpler analytical models are violated. We introduce techniques for the sensitivity analysis of option pricing which can be efficiently carried out in the simulation. In particular, using these techniques, a single run of the simulation would often provide not only an estimate of the option value but also estimates of the sensitivities of the option value to various parameters of the model. Both European and American options are considered, starting with simple analytically tractable models to present the idea and proceeding to more complicated examples. We then propose an approach for the pricing of options with early exercise features by incorporating the gradient estimates in an iterative stochastic approximation algorithm. The procedure is illustrated in a simple example estimating the option value of an American call. Numerical results indicate that the additional computational effort required over that required to estimate a European option is relatively small.

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Martingales and stochastic integrals in the theory of continuous trading

Cited by 2,356 publications

References 24 publications

with Transaction Cost for Pricing European Options and Hedging

with Transaction Cost for Pricing European Options and Hedging

Fundamental factors versus herding in the 2000–2005 US stock market and prediction

Sensitivity Analysis for Monte Carlo Simulation of Option Pricing

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