Exponential Ornstein-Uhlenbeck Model Research Articles

Power option pricing problem of uncertain exponential Ornstein–Uhlenbeck model

A power option is an exotic option whose nonlinear payoff is derived from the exponentiation of the price of the underlying asset. Due to its high-leverage, the power option is widely traded in the contemporary financial derivatives markets. This study examines the formulas for power call and put options based on the uncertain exponential Ornstein–Uhlenbeck model. Then the algorithms to calculate the power call and put option are designed. Furthermore, an analysis of the price sensitivity of power options is conducted.

Prediction of the Stock Prices at Uganda Securities Exchange Using the Exponential Ornstein–Uhlenbeck Model

We use the exponential Ornstein–Uhlenbeck model to predict the stock price dynamics over some finite time horizon of interest. The predictions are the key to the investors in a financial market because they provide vital reference information for decision making. We estimated all the parameters of the model (mean reversion speed, long‐run mean, and the volatility) using the data from Stanbic Uganda Holdings Limited. We used the parameters to forecast the stock price and the associated mean absolute percentage error (MAPE). The predictions were compared against those by the ARMA‐GARCH model. We also found the 95% prediction intervals before and during the COVID‐19 pandemic. Results indicate that the exponential Ornstein–Uhlenbeck stochastic model gives very accurate and reliable predictions with a MAPE of 0.4941%. All the forecasted stock prices were within the prediction region established. This was not the case during the COVID‐19 pandemic; the predicted stock prices are higher than the actual prices, indicating the severe impact COVID‐19 inflicted on the stock market.

Option pricing formulas for uncertain exponential Ornstein–Uhlenbeck model with dividends

Uncertain finance is an application to the finance of the uncertainty theory which provides an alternative analysis method from the probability theory under the circumstance that few samples are available. Under the research paradigm of uncertain finance, some financial assets are investigated and modeled with various tools in order to describe the price fluctuation accurately. This paper models the prices of stocks under uncertain finance based on the exponential Ornstein–Uhlenbeck model with periodic dividends. Through the $$\alpha $$ -path method, the pricing formulas are derived for European call and put options whose underlying assets follow the proposed stock model. In addition, some numerical algorithms to calculate the option prices are designed.

Optimal Trading with a Trailing Stop

Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first derive the optimal liquidation strategy prior to a given trailing stop, and prove the optimality of using a sell limit order in conjunction with the trailing stop. Our analytic results for the liquidation problem is then used to solve for the optimal strategy to acquire the asset and simultaneously initiate the trailing stop. The method of solution also lends itself to an efficient numerical method for computing the the optimal acquisition and liquidation regions. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein-Uhlenbeck model.

Optimal Dynamic Futures Portfolio in a Regime-Switching Market Framework

We study the problem of dynamically trading futures in a regime-switching market. Modeling the underlying asset price as a Markov-modulated diffusion process, we present a utility maximization approach to determine the optimal futures trading strategy. This leads to the analysis of the associated system of Hamilton-Jacobi-Bellman (HJB) equations, which are reduced to a system of linear ODEs. We apply our stochastic framework to two models, namely, the Regime-Switching Geometric Brownian Motion (RS-GBM) model and Regime-Switching Exponential Ornstein-Uhlenbeck (RS-XOU) model. Numerical examples are provided to illustrate the investor's optimal futures positions and portfolio value across market regimes.

Lookback option pricing problem of uncertain exponential Ornstein–Uhlenbeck model

A lookback option is an exotic option that allows investors to “look back” at the underlying prices occurring over the life of the option, and exercises the right at asset’s optimal point. This paper mainly investigates the lookback call and put option pricing formulas based on the uncertain exponential Ornstein–Uhlenbeck model and designs the algorithms to calculate those prices numerically. Several numerical examples are given to illustrate the effectiveness of the proposed model.

Optimal Trading with a Trailing Stop

Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first derive the optimal liquidation strategy prior to a given trailing stop, and prove the optimality of using a sell limit order in conjunction with the trailing stop. Our analytic results for the liquidation problem is then used to solve for the optimal strategy to acquire the asset and simultaneously initiate the trailing stop. The method of solution also lends itself to an efficient numerical method for computing the the optimal acquisition and liquidation regions. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein-Uhlenbeck model.

Small-Time Asymptotics under Local-Stochastic Volatility with a Jump-to-Default: Curvature and the Heat Kernel Expansion

We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $\rho < 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $\beta=1, \rho \le 0$. \bl{For uncorrelated stochastic volatility models, our results also include a SABR-type model with $\beta=1$ and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model.} We later augment the model with a single jump-to-default with intensity $\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $\rho=0$, log-moneyness $x > 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \& Spector \cite{AS12} for the case $\rho=0$.

Speculative Futures Trading under Mean Reversion

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, or exponential Ornstein–Uhlenbeck model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor’s timing option to enter or exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short.

Speculative Futures Trading Under Mean Reversion

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR), or exponential Ornstein-Uhlenbeck (XOU) model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing option to enter or exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short.

Option pricing formulas for uncertain financial market based on the exponential Ornstein–Uhlenbeck model

Uncertain finance is an application of uncertainty theory in the field of finance. This paper investigates the uncertain financial market based on the exponential Ornstein---Uhlenbeck model. European option pricing formulas and American option pricing formulas are derived via the $$\alpha $$ź-path method. Finally, some mathematical properties of the uncertain option pricing formulas are discussed.

The Right Time to Enter

We analyze the dependence of future returns on past drawdowns of the monthly time series of the S&P500 index, from 1967 to 2012. From historical data, it appears that when the drawdown increases in absolute value, future returns increase, both in mean as well as in distributional values. We then run a Monte Carlo simulation on three models used in asset pricing, namely the Black-Scholes model, the CEV model and the exponential Ornstein-Uhlenbeck model, to assess whether these models reproduce this drawdown effect. It turns out that, while the first two do not exhibit a dependence of future returns on past drawdowns, in the exponential Ornstein-Uhlenbeck future returns increase when past drawdowns increase in absolute value, consistently with the empirical findings on the S&P500 index.

The low volatility fluctuations regime of the exponential Ornstein – Uhlenbeck model

In this work the analytical characterization of the probability density of financial returns in the exponential Ornstein-Uhlenbeck model is addressed. Since prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of the mean-reverting process Y, an exact expression for the density is hard to be drawn. However, we derive the closed-form formula for the characteristic function and for the cumulants of the approximating linear model under the low volatility fluctuations regime. Theoretical results are confirmed numerically by use of Monte Carlo simulations. The effectiveness of the analytical predictions is tested on the German DAX30 Index, finding a good agreement between the empirical data and the theoretical description.

The probability distribution of returns in the exponential Ornstein–Uhlenbeckmodel

We analyze the problem of the analytical characterization of the probability distribution offinancial returns in the exponential Ornstein–Uhlenbeck model with stochastic volatility. Inthis model the prices are driven by a geometric Brownian motion, whose diffusioncoefficient is expressed through an exponential function of an hidden variableY governed by a mean-reverting process. We derive closed-form expressions for the probabilitydistribution and its characteristic function in two limit cases. In the first one the fluctuations ofY are larger than the volatility normal level, while the second one correspondsto the assumption of a small stationary value for the variance ofY.Theoretical results are tested numerically by intensive use of Monte Carlo simulations. Theeffectiveness of the analytical predictions is checked via a careful analysis of the parametersinvolved in the numerical implementation of the Euler–Maruyama scheme and is tested ona data set of financial indexes. In particular, we discuss results for the German DAX30 andDow Jones Euro Stoxx 50, finding a good agreement between the empirical data and thetheoretical description.

Option pricing under stochastic volatility: the exponential Ornstein–Uhlenbeckmodel

We study the pricing problem for a European call option when the volatility of theunderlying asset is random and follows the exponential Ornstein–Uhlenbeck model. Therandom diffusion model proposed is a two-dimensional market process that takes alog-Brownian motion to describe price dynamics and an Ornstein–Uhlenbeck subordinatedprocess describing the randomness of the log-volatility. We derive an approximate optionprice that is valid when (i) the fluctuations of the volatility are larger than its normal level,(ii) the volatility presents a slow driving force, toward its normal level and, finally, (iii) themarket price of risk is a linear function of the log-volatility. We study the resultingEuropean call price and its implied volatility for a range of parameters consistent withdaily Dow Jones index data.

Extreme times for volatility processes

Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.

Extreme Times for Volatility Processes

Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.