Pricing Options with Hybrid Stochastic Volatility Models

Are There Critical Levels of Stochastic Volatility for Early Option Exercise?

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

In the financial world, the thing that is interesting for investors today is to predict a reasonable Asian Call Option price that does not cause the seller or buyer of the option to experience losses. The constant interest assumption is more often used by previous research as an assumption in predicting the price of Asian Call Options. This is not in line with the fact that interest rate movements are not constant. The use of incorrect assumptions will affect the accuracy of prediction results. In overcoming this problem, this study aims to predict the price of Asian Call Options with non-constant interest rates using Monte Carlo simulation. The non-constant interest rate model used is Cox-Ingersoll-Ross (CIR) because this model produces positive interest rates. This study begins with a screening of Normally distributed interest rate data. The data is used to determine the parameters of the CIR interest rate model. Next, an Asian Call Option price simulation with non-constant interest rates was carried out and ended by analyzing a graph of the Asian Call Option price simulation results with non-constant interest rates. The simulated price result of the Asian Call Option with non-constant interest rates converged at the value of $170.82. Based on the price chart of Asian Call Options with different strike prices, the larger the strike price causes, the larger the Asian Call Option price, which corresponds to the price characteristics of the Asian Call Option. In addition, by extending the maturity period, the price of Asian Call Options with non-constant interest rates is greater according to the price characteristics of Asian Call Options.

The Indonesian stock market, especially PT Bank Central Asia (BCA) stock experiences high volatility. Therefore, this research is very important to provide a deeper understanding of the pricing of put and call options as a risk management instrument. This study aims to analyze the pricing of put and call options on PT Bank Central Asia (BCA) shares using the Black-Scholes model and identify factors that affect stock price fluctuations on option prices. This study aims to understand the changes in put and call option prices on PT Bank Central Asia (BCA) shares caused by fluctuations in stock prices. This study uses historical data of PT Bank Central Asia (BCA) shares, volatility, and current interest rates as the basis for analysis. The Black-Scholes method is used as a framework to analyze the pricing of put and call options by considering volatility and interest rates. The results of this study analyze the price changes of put and call options in the face of stock price fluctuations and the factors that affect option prices so as to provide insight into risk management and investment decision making. This research is expected to provide practical guidance for investors and risk managers in making decisions related to put and call options on PT Bank Central Asia (BCA) shares.

The thesis consists of three parts. In the first part, we establish a dual-curve market model. The new model accounts for the evolution of interest market after the 2007 credit crisis. The model can capture the spreads of swap rates between different tenors, and it can price caps, floors and swaptions in closed forms. In addition, we incorporate the stochastic volatility to the dual-curve market model, which allows us to capture volatility smiles or skews carried by interest rate caps and floors. In the second part, we create a market model for inflation rates using a Lévy process as driving process. An arbitrage-free condition is derived, which allows us to unite existing inflation-rate models developed under the approach of “foreign currency analogy”. Pricing formulae for inflation-linked swaps and options are developed, using moment generating function and fast Fourier transforms. In the third part, we develop two approximated option pricing formulas for Heston model. The first is based on generalized Hull-White pricing formula, while the second is obtained through weak Taylor expansions. The two methods can be utilized as well in time-dependent Heston model, where closed-form pricing is not available.

ABSTRACTOption‐pricing models that assume a constant interest rate may misprice futures options if the interest rate fluctuates significantly or if the price of the underlying asset is correlated with the interest rate. The futures option‐pricing model of Ramaswamy and Sundaresan allows for a stochastic interest rate and correlation of the underlying asset's price with the interest rate. Using a data set of daily closing prices for Comex gold futures options, this paper tests the Ramaswamy and Sundaresan model against a constant interest rate model. Results indicate that the stochastic interest rate model is a superior predictor of market prices.

This chapter introduces the no-arbitrage models for pricing market risk, assuming that default-free interest rates are non-random. This constant interest rate assumption restricts the model’s use to assets that are shortdated and whose price movements are uncorrelated to interest rates. Hence, the qualification in the chapter title to equities, foreign currencies (FX), and commodities. For real applications, since interest rates are stochastic, the assumption of constant interest rates needs to be relaxed. This will be done in the next chapter.Historically, the no-arbitrage approach for pricing assets under constant interest rates was first presented by Black Scholes Merton (BSM) [5], [52] in the context of a very simple evolution for the risky asset price process, called geometric Brownian motion. For the purposes of this chapter, although we maintain the constant interest rate assumption, we generalize the evolution of the asset price process. Because the model presented here is based on the original BSM paper, we call it the BSM methodology.

We apply weak split-step approximations of the Heston model for evaluation of put and call option prices in this model.

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

We consider a large market model of defaultable assets in which the asset price processes are modelled as Heston-type stochastic volatility models with default upon hitting a lower boundary. We assume that both the asset prices and their volatilities are correlated through systemic Brownian motions. We are interested in the loss process that arises in this setting and we prove the existence of a large portfolio limit for the empirical measure process of this system. This limit evolves as a measure valued process and we show that it will have a density given in terms of a solution to a stochastic partial differential equation of filtering type in the two-dimensional half-space, with a Dirichlet boundary condition. We employ Malliavin calculus to establish the existence of a regular density for the volatility component, and an approximation by models of piecewise constant volatilities combined with a kernel smoothing technique to obtain existence and regularity for the full two-dimensional filtering problem. We are able to establish good regularity properties for solutions, however uniqueness remains an open problem.

Correlation plays an important role in pricing multi-asset options. In this work we incorporate stochastic correlation into pricing quanto options which is one special and important type of multi-asset option. Motivated by the market observations that the correlations between financial quantities behave like a stochastic process, instead of using a constant correlation, we allow the asset price process and the exchange rate process to be stochastically correlated with a parameter which is driven either by an Ornstein–Uhlenbeck process or a bounded Jacobi process. We derive an exact quanto option pricing formula in the stochastic correlation model of the Ornstein–Uhlenbeck process and a highly accurate approximated pricing formula in the stochastic correlation model of the bounded Jacobi process, where correlation risk has been hedged. The comparison between prices using our pricing formula and the Monte-Carlo method are provided.

The conventional control variate method proposed by Kemna and Vorst for evaluating Asian options using the Black-Scholes model utilizes a constant control parameter.We generalize this method, applying it to a stochastic control process through the martingale representation of the conventional control. This generalized control variate has zero variance in the optimal case, whereas the conventional control can only reduce its variance by a finite factor. By means of option price approximations, the generalized control is reduced to a linear martingale control. It is straightforward to extend this martingale control to a non-linear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian options using stochastic volatility models.We demonstrate multiple choices of controls and test them with Monte Carlo and quasi-Monte Carlo simulations. Quasi-Monte Carlo methods perform significantly better after adding a control when using the two-step control variate method, while the variance reduction ratios increase to 320 times for randomized quasi-Monte Carlo methods, compared with 60 times for Monte Carlo simulations with a control.

In this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index.

Estimation of Jump-Diffusion Processes via Empirical Characteristic Functions