Arbitrage in fractional Brownian motion models

Cheridito, Patrick

doi:10.1007/s007800300101

Cited by 278 publications

(218 citation statements)

References 14 publications

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“…Under this model the volatility exhibits long-memory, meaning intuitively that the volatility today is correlated to past volatility values with a dependence that decays very slowly. In this way we introduce long-memory in our model, but not directly in the returns, as was suggested by Cheridito (2003, [5]), Rogers (1997, [23]) and Sottinen (2001, [25]). On the contrary, we assume that the returns of the stock price are independent, which is an assumption that is supported by empirical findings, and also allows us to remain in an arbitrage-free context in continuous time.…”

Section: Introductionmentioning

confidence: 99%

Estimation and pricing under long-memory stochastic volatility

Chronopoulou

Viens

2010

Ann Finance

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We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein-Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.

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Section: Introductionmentioning

confidence: 99%

Estimation and pricing under long-memory stochastic volatility

Chronopoulou

Viens

2010

Ann Finance

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“…Note that if this is zero, it means that we are hedging in continuous time. It has, however, been shown (see for example [14]) that in continuous time we face the problem of arbitrage. In order to avoid the problem of arbitrage, we only consider the case where (δt)≠0.…”

Section: Lie Symmetry Classification Of the Geometric Asian Option Prmentioning

confidence: 99%

“…For models encoded as differential equations, assumptions that translate into the invariance of underlying equations are particularly useful in the search of their solutions. This is the main thrust of Lie symmetry analysis of differential equations pioneered by Sophus Lie [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In such analysis, one algorithmically looks for infinitesimal transformations (i.e., transformations depending on a small parameter and enjoying additional properties [2,9,17] of both dependent and independent variables that does not change the underlying differential equation.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

Bf¹,

Pindza²,

Maré³

2017

J Appl Computat Math

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We provide a closed-form solution for the European and Asian option pricing models when the source of randomness is a fractional Brownian motion as opposed to the geometric Brownian motion. In addition to the source of randomness, transaction costs are considered to be non-negligible. For the case of the European option, proportional transaction costs hide in the volatility and do not change the form of the model. The construction of the solution is based on the symmetries of the model. The model for Asian options has an additional parameter that makes the volatility time-dependent, which complicates the solution process. However, we are still able to obtain solutions using Lie symmetry methods. Citation: Nteumagn BF, Pindza E, Mare E (2017) Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional BrownianNoises. J Appl Computat Math 6: 356. doi: 10.4172/2168-9679.1000356 Page 2 of 8Volume 6 of the option pricing equation relies heavily on the random process followed by the underlying stock price. In the case of the celebrated Black-Scholes equation, one assumes that the stock follows the classical geometric Brownian motion [5,24]. This process assumes independent increments, which makes it different from the fractional Brownian motion (fBm) where there is a serial correlation in the increments. This correlation allows for predictability in the model. The similarity between the two processes is the fact that they are both geometric.It has been shown [8] that random changes can be accurately measured by the use of a parameter known as the Hurst parameter. The construction of a dam on the Nile river in the early 1950s saw the birth of the Hurst exponent. The measure of the water levels of increase or decrease was known to follow a random walk. In other words, the next level of increase or decrease is independent of the previous one. However, Edwin Hurst, hydrologist in the Nile dam project, showed that the long term memory of the system indicates an autocorrelation of the time series of water levels, and the rate at which these decrease as the lag between pairs of values increase [8]. The movement of stock price is similar to this model. We therefore exploit the results by Leland [11] for the derivation of our model. Earlier in 2014, we achieved an accurate numerical solution of our model with exponential convergence ([19]) by applying spectral methods to analyze the problem. In this paper, we use Lie symmetries to determine the exact solution, which sets us a step closer to the calibration of our model.We first give some preliminaries on Lie symmetries of differential equations in Section 2. Secondly, we determine the symmetries of the European and Asian option pricing models and obtain their analytical solutions in Section 3. In Section 4 we apply the Lie symmetry technique to reduce the PDEs and provide solutions. Finally, we provide concluding remarks in Section 5. Lie Symmetries and Equivalence TransformationsOur aim in this section is to provide a brief introduction to ...

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“…These results might be avoided either by restricting the class of trading strategies [16], introducing transaction costs [17] or replacing pathwise integration by a different type of integration [18] [19]. However this is not free of problems because the Skorohod integral approach requires the use of a Wick product either on the portfolio or on the self-financing condition, leading to unreasonable situations from the economic point of view (for example positive portfolio with negative Wick value, etc.)…”

Section: Introductionmentioning

confidence: 99%

No-arbitrage, leverage and completeness in a fractional volatility model

Mendes

Oliveira

Rodrigues

2015

Physica A: Statistical Mechanics and its Applications

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Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and completeness properties of the models are studied.

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Arbitrage in fractional Brownian motion models

Cited by 278 publications

References 14 publications

Estimation and pricing under long-memory stochastic volatility

Estimation and pricing under long-memory stochastic volatility

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

No-arbitrage, leverage and completeness in a fractional volatility model

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