Option Pricing under Randomised GBM Models

Pricing Continuously Monitored Barrier Options Under the Sabr Model: A Closed-Form Approximation

This study proposes an artificial neural network (ANN) based option pricing framework under the SABR (stochastic alpha beta rho) and free boundary SABR volatility models. Unlike previous research, we do not directly apply the ANN technique to train and predict option implied volatilities. Instead, we use the technique to train and predict the differences between the implied volatilities calculated by asymptotic approximation and those computed by Monte Carlo simulation. Since the analytical solution for an option written on an underlying asset price in the SABR model is intractable, approximation techniques are used to derive an approximate closed-form solution in practice. However, these approximation formulas worsen as maturity increases and the underlying asset's volatility is high. The accuracy decreases rapidly in wings, which represents deep in-the-money and out-of-the-money cases. By combining the approximation formulas and ANN framework, our new option pricing method offers the following improvements: (1) the training becomes more robust, and the predictions produce more stable and accurate results: (2) it significantly speeds up the training procedure: and (3) the accuracy and efficiency of approximation are improved in the wings without sacrificing performance.

The aim of this research was to measure the risk of the IHSG stock data using the Value at Risk (VaR). IHSG stock index data typically indicates a jump. However, Geometric Brownian Motion (GBM) model can not catch any of the jumps. To view the jumps, it is necessary that the model was then developed into a Geometric Brownian Motion (GBM) model with Jumps. On the GBM model with Jumps, returns the data are discontinuous. To determine the value of VaR, the value of return to perform the simulation model of GBM with Jumps is required. To represent processes that contain jumps, discontinuous Poisson process using the Peak-Over Threshold is required. To determine the parameters of model, calibration of historical data using the Maximum Likelihood Estimation (MLE) method is performed. VaR value for GBM model with Jumps with a 95% and 99% confidence level are -0,0580 and -0,0818 while VaR value for GBM model with a 95% and 99% confidence level are -0,0101 and -0,0199. VaR for GBM model with Jumps with a confidence level of 95% and 99% show greater than the model VaR for GBM.

Model of stock price movements that follow stochastic process can be formulated in Stochastic Diferential Equation (SDE). The exact solution of SDE model is called Geometric Brownian Motion (GBM) model. Determination the optimal portfolio of three asset that follows Multidimensional GBM model is to be carried out in this research.Multidimensional GBM model represents stock price in the future is affected by three parameter, there are expectation of stock return, risk stock, and correlation between stock return. Therefore, theory of portfolio Markowitz is used on formation of optimal portfolio. Portfolio Markowitz formulates three of same parameter that is calculated on Multidimensional GBM model. The result of this research are optimal portfolio reaches with the proportion of fund are 39,38% for stock BBCA, 59,82% for stock ICBP, and 0,80% for stock INTP. This proportion of fund represents value of parameters that is calculated on modelling stock price.

Investing in the Malaysian stock market can be overwhelming due to the abundance of options, which necessitates informed decision-making to navigate the volatile market. This study addresses a common problem faced by investors venturing into the stock market, where instability and fluctuations pose significant risks, leading to financial losses stemming from inadequate knowledge about suitable stocks for investment. Unlike many studies that focus on long-term forecasting methods, this research adopts the Geometric Brownian Motion (GBM) model for short-term investment analysis. The study aims to identify the most effective volatility measurement model, develop a forecasting model using GBM based on the chosen volatility model, and evaluate the accuracy of the GBM model using Mean Square Error (MSE), Mean Absolute Percentage Error (MAPE), and Mean Absolute Deviation (MAD). Four volatility models, which include simple, log, high-low, and high-low-closed volatility are analysed to determine the most effective volatility measurement model. Four months of daily stock data were collected to ensure accuracy excluding factors such as seasonality, politics, natural disasters, and wars. Findings indicate that the simple volatility model is the most suitable for forecasting stock market trends using the GBM model, demonstrating high accuracy based on MSE, MAPE and MAD. These results suggest that employing the simple volatility model within GBM model can offer a practical and accurate approach for short-term market analysis in Malaysia, potentially aiding investors in mitigating risks and optimizing their trading strategies.

This paper reviews 17 studies addressing dynamic option hedging in frictional markets through Deep Reinforcement Learning (DRL). Specifically, this work analyzes the DRL models, state and action spaces, reward formulations, data generation processes and results for each study. It is found that policy methods such as DDPG are more commonly employed due to their suitability for continuous action spaces. Despite diverse state space definitions, a lack of consensus exists on variable inclusion, prompting a call for thorough sensitivity analyses. Mean-variance metrics prevail in reward formulations, with episodic return, VaR and CvaR also yielding comparable results. Geometric Brownian motion is the primary data generation process, supplemented by stochastic volatility models like SABR (stochastic alpha, beta, rho) and the Heston model. RL agents, particularly those monitoring transaction costs, consistently outperform the Black–Scholes Delta method in frictional environments. Although consistent results emerge under constant and stochastic volatility scenarios, variations arise when employing real data. The lack of a standardized testing dataset or universal benchmark in the RL hedging space makes it difficult to compare results across different studies. A recommended future direction for this work is an implementation of DRL for hedging American options and an investigation of how DRL performs compared to other numerical American option hedging methods.

In this work, we present a multiple time-step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho (SABR) model. The proposed method is an extension of the one time-step Monte Carlo method that we proposed in Leitao et al. (Appl. Math. Comput. 293: 461–479, 2017). We call it the mSABR method. A highly efficient method results, with many interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time-step Monte Carlo method is especially useful for long-term or for exotic options. This paper is a short version of an already published paper (Leitao et al. On an efficient multiple time-step Monte Carlo simulation of the SABR model. Quantitative Finance.

The Stochastic Alpha Beta Rho (SABR) model by Hagan et al.(2002) is currently one of the most popular models in the interestrate derivative markets. In this paper, based on Henry-Labordere's (2008) previous work, we derive a new exact analytical solution for the normal SABR model and verify its correctness by numerical examples.

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

The calendar spread options (CSOs) on agricultural commodities, most notably corn, soybeans and wheat, allow market participants to hedge the roll-over risk of futures contracts. Despite the interest from agricultural businesses, there is lack of both theoretical and empirical research on pricing and hedging performances of CSOs. We propose to price and hedge CSOs under geometric Brownian motion (GBM) and stochastic volatility (SV) models. We estimate the model parameters by using implied state-generalized method of moments (IS-GMM) and evaluate the in-sample and out- of-sample pricing and hedging performances. We find that the average pricing errors of the SV model are 0.79% for corn, 0.75% for soybeans and 1.2% for wheat; the pricing and hedging performance of the SV model are mostly superior to the benchmark GBM model, both in and out of sample, with only one exception where the out-of-sample hedging error for the GBM model for market makers is slightly better than the SV model.

Modeling and forecasting stock prices of public corporates are important studies in financial analysis, due to their stock price characteristics. Stocks investments give a wide variety of risks. Taking a portfolio of several stocks is one way to minimize risk. Stochastic process of single stock price movements model can be formulated in Geometric Brownian Motion (GBM) model. But for a portfolio that consist more than one corporate stock, we need an expansion of GBM Model. In this paper, we use multidimensional Geometric Brownian Motion model. This paper aims to model and forecast two stock prices in a portfolio. These are PT. Matahari Department Store Tbk and PT. Telekomunikasi Indonesia Tbk on period January 4, 2016 until April 21, 2017. The goodness of stock price forecast value is based on Mean Absolute Percentage Error (MAPE). As the results, we conclude that forecast two stock prices in a portfolio using multidimensional GBM give less MAPE than using GBM for single stock price respectively. We conclude that multidimensional GBM is more appropriate for modeling stock prices, because the price of each stock affects each other.

This paper proposes a way to forecast the future closing price of small size companies in Bursa Malaysia by using geometric Brownian motion (GBM). Forecasting is restricted to short term investment because most of the investors aim to gain profit in short period of time. The reasons of choosing small size companies are because the asset prices are lower, hence the asset are affordable for all level of investors. In this paper, we suggest that GBM which involves randomness, volatility, and drift can help investor in making their investment decision wisely. This research shows the model is highly accurate model in forecasting stock prices and it is proven by the lower value of mean absolute percentage error (MAPE). Although it is highly accurate, we try to find the suitable volatility measurements that give the forecast value closer to the actual movement of stock prices. The result shows that by using highs-lows-close volatility, the forecast stock prices are closest to the actual prices. This volatility measurement and GBM model are suggested to the investor to forecast future prices for a maximum of two week investment.

Enhanced Valuation of European Options Under Jump Processes and Innovative Characterization of Implied Volatility Smile

This paper considers the introduction of stock options in an (dynamically) incomplete securities market made up of a riskless bond and the stock. The stock price follows a geometric Brownian motion with constant drift. However, there is incomplete information about the unknown stochastic volatility. The option price is determined by a uniform-price auction. Thus an option pricing formula results from the interaction of market participants relying on private information on the unknown stochastic volatility under an explicit market structure. This paper incorporates market microstructure considerations into an extended Black-Scholes model with incomplete information on the underlying volatility. It relies on the growing importance of auctionlike trading rules in financial markets.