Investment option under CIR interest rates

We analyze extensively the characteristics of the solution to an irreversible investment decision when the only source of uncertainty comes from interest rates. They are assumed to be driven by the popular Cox–Ingersoll–Ross (CIR) stochastic process. Particular attention is paid to the impact that both CIR parameters and risk aversion have on the threshold rate.     

Pricing American options under stochastic volatility and stochastic interest rates

We introduce a new analytical approach to price American options. Using an explicit and intuitive proxy for the exercise rule, we derive tractable pricing formulas using a short-maturity asymptotic expansion. Depending on model parameters, this method can accurately price options with time-to-maturity up to several years. The main advantage of our approach over existing methods lies in its straightforward extension to models with stochastic volatility and stochastic interest rates. We exploit this advantage by providing an analysis of the impact of volatility mean-reversion, volatility of volatility, and correlations on the American put price.     

Pricing options with Green's functions when volatility,interest rate and barriers depend on time

We derive the Green's function for the Black–Scholes partial differential equation with time-varying coefficients and time-dependent boundary conditions. We provide a thorough discussion of its implementation within a pricing algorithm that also accommodates American style options. Greeks can be computed as derivatives of the Green's function. Generic handling of arbitrary time-dependent boundary conditions suggests our approach to be used with the pricing of (American) barrier options, although options without barriers can be priced equally well. Numerical results indicate that knowledge of the structure of the Green's function together with the well-developed tools of numerical integration make our approach fast and numerically stable.     

Pricing volatility options under stochastic skew with application to the VIX index

In recent years, there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at Chicago Board Options Exchange. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform. To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.     

The impact of jump risks on nominal interest rates and foreign exchange rates

This article investigates the relationship between the nominal interest rate and inflation and also the forward exchange rate under a general specification of the underlying processes govering the foreign exchange rate. There are three distinct risks that affect the relation between the real rate of interest and the nominal rate namely, consumption risk, diffusion risk, and the existence of jump risks of inflation. Jump risks lower the nominal interest rate because of jump hedging of a nominal bond. The forward exchange rate depends on the expected depreciation of the domestic currency as well as these three risks. As the domestic jump risks increase, the domestic nominal interest rate decreases and the forward exchange rate decreases.     

Pricing pension buy-outs under stochastic interest and mortality rates

Pension buy-out is a special financial asset issued to offload the pension liabilities holistically in exchange for an upfront premium. In this paper, we concentrate on the pricing of pension buy-outs under dependence between interest and mortality rates risks with an explicit correlation structure in a continuous time framework. Change of measure technique is invoked to simplify the valuation. We also present how to obtain the buy-out price for a hypothetical benefit pension scheme using stochastic models to govern the dynamics of interest and mortality rates. Besides employing a non-mean reverting specification of the Ornstein–Uhlenbeck process and a continuous version of Lee–Carter setting for modeling mortality rates, we prefer Vasicek and Cox–Ingersoll–Ross models for short rates. We provide numerical results under various scenarios along with the confidence intervals using Monte Carlo simulations.     

Generic pricing of FX,inflation and stock options under stochastic interest rates and stochastic volatility

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schöbel and Zhu [Eur. Finance Rev., 1999, 4, 23–46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327–343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.     

Valuation of FX barrier options under stochastic volatility

This paper describes European-style valuation and hedging procedures for a class of knockout barrier options under stochastic volatility. A pricing framework is established by applying mean self-financing arguments and the minimal equivalent martingale measure. Using appropriate combinations of stochastic numerical and variance reduction procedures we demonstrate that fast and accurate valuations can be obtained for down-and-out call options for the Heston model.     

On the valuation of fader and discrete barrier options in Heston's stochastic volatility model

We focus on closed-form option pricing in Heston's stochastic volatility model, where closed-form formulas exist only for a few option types. Most of these closed-form solutions are constructed from characteristic functions. We follow this closed-form approach and derive multivariate characteristic functions depending on at least two spot values for different points in time. The derived characteristic functions are used as building blocks to set up (semi-) analytical pricing formulas for exotic options with payoffs depending on finitely many spot values such as fader options and discretely monitored barrier options. We compare our result with different numerical methods and examine the computational accuracy.     

Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results

Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, though, including the efficacy of the standard alternating direction implicit (ADI) numerical methods for solving SV and LSV pricing problems. In general, the amount of required computations for these methods is very substantial. In this paper, we address some of these issues and propose a viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We also discuss various approaches to solving the corresponding pricing problems in a semi-analytical fashion. We use the fact that in the zero correlation case some of the pricing problems can be solved analytically, and develop a closed-form series expansion in powers of correlation. We perform a thorough benchmarking of various numerical solutions by using analytical and semi-analytical solutions derived in the paper.     

The representation of American options prices under stochastic volatility and jump-diffusion dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square-root process as used by Heston [Rev. Financial Stud., 1993, 6, 327–343], and by a Poisson jump process as introduced by Merton [J. Financial Econ., 1976, 3, 125–144]. Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalizes in an intuitive way the structure of the solution to the corresponding European option pricing problem obtained by Scott [Math. Finance, 1997, 7(4), 413–426], but here in the case of a call option and constant interest rates.     

Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility,stochastic interest rates and stochastic mortality via the componentwise splitting method

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.     

Central bank intervention and the volatility of foreign exchange rates: evidence from the options market

This paper tests the effects of central bank intervention on the ex ante volatility of $/DM and $/yen exchange rates between 1985 and 1991. In contrast to previous research which employed GARCH estimates of conditional volatility, we estimate ex ante volatility using the implied volatilities of currency option prices. We also control for the effects of other macroeconomic announcements. We find little support for the hypothesis that central bank intervention decreases expected exchange rate volatility. Instead, central bank intervention is generally associated with a positive change in ex ante exchange rate volatility, or with no change.     

Conditional volatility persistence and volatility spillovers in the foreign exchange market

I investigate the magnitudes and determinants of volatility spillovers in the foreign exchange (FX) market, using realized measures of volatility and heterogeneous autoregressive (HAR) models. I confirm both meteor shower effects (i.e., inter-regional volatility spillovers) and heat wave effects (i.e., intra-regional volatility spillovers) in the FX market. Furthermore, I find that conditional volatility persistence is the dominant channel linking the changing market states of each region to future volatility and its spillovers. Market state variables contribute to more than half of the explanatory power in predicting conditional volatility persistence, with the model that calibrates volatility persistence and spillovers conditionally on market states performing statistically and economically better. The utilization of market state variables significantly extends our understanding of the economic mechanisms of volatility persistence and spillovers and sheds new light on econometric techniques for volatility modeling and forecasting.     

Testing the unbiasedness hypothesis of foreign exchange rates and the analysis of transformations

Recent researchers have utilized various functional forms for testing the hypothesis that the forward rate is an unbiased predictor of future spot rates in foreign exchange markets. We compare a large number of these functional forms for a similar time period and test their consistency with the data for five major currencies. Our results imply that certain functional form models may be inappropriate for some currencies. Researchers must, therefore, be cautious of misspecification due to erroneous functional forms when testing the unbiased forward rate hypothesis.     

Heterogeneous real-time trading strategies in the foreign exchange market

The foreign exchange (FX) market is worldwide, but the dealers differ in their geographical locations (time zones), working hours, time horizons, home currencies, access to information,transaction costs, and other institutional constraints. The variety of time horizones is large: from intra-day dealers, who close their positions every evening, to long-term investors and central banks. Depending on the constraints, the different market participats need different strategies to reach their goal, which is usually maximizing the profit, or rather a utility function including risk. Different intra-day trading strategies can be studied only if high-density data are available. Oslen & Associates (O & A) has collected and analysed large amounts of FX quotes by market makers around the clock (up to 5000 non-equally spaced prices per day for the German mark against US$). Based on these data, a set of real-time intra-day trading models has been developed. These models give explicit trading recommendations under realistic constraints. They are allowed to trade only during the opening hours of a market, depending on the time zone and local holidays. The models have been running real-time for more than three years, thus leading to an ex ante test. The test results, obtained with a risk-sensitive performance measure, are presented. All these trading models are profitable, but they differ in their risk behaviour and dealing frequency. If a certain profitable intra-day algorithm is tested with different working hours, its success can considerably change. A systematic study shows that the best choice of working hours is usually when the most important markets for the particular FX rate are active. All the results demonstrate that the assumption of a homogeneous 24-hour FX market with identical dealers, following an identical ‘rational expectation’, is far from reality. To explain the market dynamics, a heterogeneous model of the market with different types of dealers is more appropriate.     

Deep networks for predicting direction of change in foreign exchange rates

Trillions of dollars are traded daily on the foreign exchange (forex) market, making it the largest financial market in the world. Accurate forecasting of forex rates is a necessary element in any effective hedging or speculation strategy in the forex market. Time series models and shallow neural networks provide acceptable point estimates for future rates but are poor at predicting the direction of change and, hence, are not very useful for supporting profitable trading strategies. Machine learning classifiers trained on input features crafted based on domain knowledge produce marginally better results. The recent success of deep networks is partially attributable to their ability to learn abstract features from raw data. This motivates us to investigate the ability of deep convolution neural networks to predict the direction of change in forex rates. Exchange rates for the currency pairs EUR/USD, GBP/USD and JPY/USD are used in experiments. Results demonstrate that trained deep networks achieve satisfactory out‐of‐sample prediction accuracy.     

Pricing and static hedging of European-style double barrier options under the jump to default extended CEV model

This paper develops two novel methodologies for pricing and hedging European-style barrier option contracts under the jump to default extended constant elasticity of variance (JDCEV) model, namely: a stopping time approach based on the first passage time densities of the underlying asset price process through the barrier levels; and a static hedging portfolio approach in which the barrier option is replicated by a portfolio of plain-vanilla and binary options. In doing so, both valuation methodologies are extended to a more general set-up accommodating endogenous bankruptcy, time-dependent barriers and the commonly observed stylized facts of a positive link between default and equity volatility and of a negative link between volatility and stock price. The two proposed numerical methods are shown to be accurate, easy to implement and efficient under both the JDCEV model and the nested constant elasticity of variance model.     

A term structure model of interest rates with quadratic volatility

This study proposes a no-arbitrage term structure model that can capture the volatility of interest rates without sacrificing the goodness-of-fit to the cross-section and predictive ability about the level of interest rates. The key feature of the model is the covariance matrix of changes in factors, which is specified as quadratic functions of factors. The quadratic specification can capture intense volatility even with spanned factors, which is not the case for the affine specification. Furthermore, since the quadratic specification guarantees the positive definiteness of the covariance matrix without restricting the sign of factors, it allows for a flexible specification of the physical drift as does the Gaussian term structure model, contributing also to accurate level prediction.