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Our study deals with the process of beta estimation and focuses on companies which are subject to European network regulation. Our most important conclusions are: (1) Sudden beta increases or decreases occur that often last only short periods of time and may therefore cause a significant misestimation of the future beta. (2) Three- and especially five-year betas are much more stable than one-year betas. (3) The choice between purely local, European or global betas may matter considerably. (4) Weekly or daily betas seem to be better than monthly ones. (5) Vasicek and Blume adjustments towards one lead to beta predictions that are too high.

We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(\alpha -\beta X_t)\,dt+\gamma \,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $\alpha$ and $\beta$ in the non-ergodic case (when $\beta, Comment: Published at https://doi.org/10.15559/19-VMSTA140 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)

In this dissertation the main focus area was to use one factor interest rate models in India to obtain the descriptive nature and risk profile of money markets in India. For the period of 2012-2020 we found that CIR (1985) model fits and describes the interest rate path for India much better than Vasicek model. Both of these models were calibrated to make each of their parameters pseudo time varying and maximum likelihood estimation (MLE) was used to find optimal model parameters. We used daily weighted average of call money market rates as input data for calibration. Under the assumptions of constant relative risk aversion (CRRA) and decreasing absolute risk aversion (DARA) of CIR model, we used the closed form solutions to obtain bond prices and yields on 3 month T-bill. Average 10 year yield curve for the period 2012-20 was also forecasted and it was observed that Indian yield curve has a hump shape, with higher yields on longer dated bonds. We find that there exists higher term premium for longer dated bonds, for example, term premium on 10 year G-Sec has increased at a faster rate, relative to shorter maturity bonds, with the trend getting stronger since 2016 with an exception in year 2019. For risk profiling, the paper uses Expected-shortfall (ES) and Value-at-Risk (VaR) on 3 month bonds. It was found that expected shortfall normalized for yield stood at 25% on average for the period 2012-20. This means that traders in the money market need to constantly look out for price risk, to hedge for this, we propose that the trader uses call options on bonds. We find that fair prices of call options for CIR distribution function under Blacks-Scholes model will be an expensive hedging strategy. But that said, the Greeks for these call options (Rho, Delta and Gamma) show that the call option will not be sensitive to interest rate and bond price changes and hence will be a stable hedging strategy. The paper concludes that this is because of very high speed of mean reversion and low volatility in interest rate paths. This result can be associated with credibility, transparency and clear policy decisions made by the Indian central bank and newly formed Monetary Policy Committee (MPC). It was also seen that MPC has been successful to lower volatility by 9 times since its inception while maintaining the same high levels of speed of reversion in interest rates.

We consider the option value of cash when nominal interest rates are no longer constrained by the zero lower bound. We provide a general valuation principle and solve for the value of cash in semi-closed form under Vasicek (1977) dynamics for the nominal short rate. In the absence of a zero lower bound, cash can have substantial option value and becomes a powerful recessionary hedge. However, a significant fraction of the value derives from the ability to hold cash over an extremely long horizon, and the risk of early redemption decreases the value and hedging performance of cash.

This paper analyzes immunization strategies in the mean-variance framework. We characterize the efficient portfolio allocations and identify the minimum variance immunization strategy. We show that the efficient allocations can be superior or inferior to the minimum variance allocation as time passes. Consequently, the efficiency of duration-based (Macaulay or stochastic) immunization strategies can be transient, which helps explain their mitigated performance documented in empirical studies. The minimum variance immunization strategy, characterized explicitly in the Vasicek model, appears robust to real yield curve fluctuations extrapolated from U.S. data from 1977 to 2020.

We consider two aspects of Vasicek interest rate models arising from chance-risk classification of German pension products. First, we show that the two-factor Vasicek model can explain significantly more effects that are observed at the market than its one-factor variant. Among them are humped shapes independent of the interest rate level and the occurrence of dipped yield curves. We further introduce a general change of measure framework for the Monte Carlo simulation of the Vasicek model under a subjective measure. In chance-risk classification it can then be used to avoid the occurrence of a far too high frequency of inverse yield curves with growing time.

In these lectures’ notes I would like to introduce forwards, futures and options, and to review some results on Black-Scholes-73 and Black-76 models for positive prices, and also on alternatives models for negative prices for option valuation of futures contracts. I will focus on the first model introduced by Louis Bachelier in 1900, and on other ones, including Ornstein-Uhlenbeck model (1930) and Vasicek model (1977). For these models I will present some results related to the problem and project. In the end of this lecture notes there are three Appendices, A-C, which contain brief introductions to martingales, stochastic calculus (including Ito formula) and simulation. I’ll start with the References (they are also in the end of the lectures) to pay students’ attention to main literatures

In a market with stochastic interest rates, we consider an investor who can either (i) invest all of his wealth in a money market account or (ii) purchase zero-coupon bonds and invest the remainder of his wealth in the money market account. The indifference price of the zero-coupon bond is the price at which the investor could achieve the same expected utility under both strategies. In an affine term structure setting, we show that the indifference price of the zero-coupon bond is the root of an integral equation, when the investor's utility function is of exponential or power form. As an example, we compute the indifference price and the corresponding indifference yield curve in the Vasicek model and conduct sensitivity analysis to study the impact of various parameters on the yield curve. Furthermore, we discuss the choice of numeraire and its impact on the indifference prices.

This paper estimates the drift parameters in the fractional Vasicek model from a continuous record of observations via maximum likelihood (ML). The asymptotic theory for the ML estimates (MLE) is established in the stationary case, the explosive case, and the boundary case for the entire range of the Hurst parameter, providing a complete treatment of asymptotic analysis. It is shown that changing the sign of the persistence parameter changes the asymptotic theory for the MLE, including the rate of convergence and the limiting distribution. It is also found that the asymptotic theory depends on the value of the Hurst parameter.

The over-the-counter (OTC) market sees the majority of trading volume in the finance industry, yet it remains vastly unregulated. Inherent to its nature are issues with transparency, efficiency and security, which have been barely addressed by legislators post the 2008 crisis. We propose that the utilization of Smart Contracts in the OTC market and creation of “smart derivatives”, solves the majority of issues that are associated with the market. While other researchers have debated on this matter and proposed the use of Smart Contracts in OTC markets, there are no papers offering an example of practical implementation. We selected interest rate swaps (IRS) as the test-case derivative for implementation in the form of a smart contract. Having unpacked the mathematical and theoretical framework behind structuring and pricing IRS, we produced a Vasicek Simulated Yield Curve from the Ten Year Constant Maturity Treasury Rates and found that the curve was inverted, which, combined with a characteristic LIBOR Simulated Yield Curve pointed to an increasingly negative swap spread for longer maturities that was demonstrated with a plotted Swap Spread. The parameters of the model were derived from live market data through Maximum Likelihood Estimation with Monte Carlo Simulation of Euler Scheme discretized Vasicek instantaneous rates. We concluded that short term interest rate swap contracts were preferable and implemented a one-year Interest Rate Swap by instantiating the DAML Ledger Sandbox. The implementation offers the likely innovation of merging a Swap Execution Facility and a Swap Data Repository into one platform, ensuring end-to-end automation, as well as providing the benefits of visibility, while preserving anonymity, improving efficiency and security in the market.

As a result of the global financial crisis, the credit risk of OTC derivatives became a more important issue in finance industry. In contrast to exchange traded markets, OTC markets lack the advantage of a central clearing house ensuring that the counterparties fulfill their obligations. The risk that the promised payments are not made is called counterparty or default risk. Derivatives subject to counterparty risk are called vulnerable derivatives. Since the counterparty risk cannot be ignored, it should be considered in the valuation of OTC derivatives. This dissertation addresses the valuation of European and American options which are traded on OTC markets. Both European and American options exhibit unilateral counterparty risk, since these contracts constitute an obligation only for the option writer. For vulnerable European options, the valuation models of Klein (1996), Klein and Inglis (2001) as well as Liu and Liu (2011) prevail in the literature. Based on an extended Black-Scholes world, they use the structural approach of Merton (1974) to price European options subject to counterparty risk. In this dissertation, these models are combined in a general model which incorporates their key characteristics. Moreover, the mentioned models are extended to a stochastic interest rate framework. In addition, valuation models for vulnerable American options are set up using the core ideas of Klein (1996), Klein and Inglis (2001) as well as Liu and Liu (2011).

In this thesis we study the caps market. Caps are a contract where the interest rates are capped at some fixed value r. Purchasers of caps pay the prevailing interest rate if it is below r, but pay the interest rate r if the prevailing rate is above r. In the latter case the sellers of caps pay the difference. Therefore, the purchasers are able to prevent risks associated with the future change in interest rates. Caps consist of caplets which are European options on the forward rates called LIBOR (the London Inter-Bank Offer Rates). Caps are the derivatives of LIBOR. However, their relation is not so simple as the relation between a stock and its derivatives. One important difference is that the volatility in one market does not affect the volatility of the other market as much as in the stock and its derivative markets. This phenomenon is termed unspanned stochastic volatility (USV) and various research has been done. There are arguments supporting and against USV. This motivates further study of USV.In Chapter 2, we study the modeling and calibration of LIBOR caps. We use an unspanned stochastic volatility model to discuss the pricing of caps. We calibrate the cap prices using the model and compare them with the market caps. We choose the parameters of the model so that the difference between prices of the theoretical (the model based) caps and those of the market caps will be minimum. A goal of this chapter is to examine the effects of jumps in interest rates and in stochastic volatility. We compare the calibrations of USV models without jumps, with jumps in both interest rates and volatility, with jumps in interest rates only, and with jumps in volatility only. The calibration of Vasicek model is also performed for the sake of reference. Another goal is to obtain the calibration results based on a few days of data. So far most calibration results are based on one-day data. We find out that the calibrations with jumps give better results than without jumps and between the jumps in interest rates and volatility the jumps in interest rates give better calibration results. The results indicate that the more detailed study of effects of jumps is important.In Chapter 3, we examine the factors affecting the price of caps, so that accurate and efficient pricing and hedging of caps are possible. Since they are bond derivatives and the bonds are affected by the economic activities, it is interesting to examine how the economic activities influence the cap prices. We study empirically the effects of macroeconomic announcements and fed announcements on the implied volatility of difference caps. As mentioned earlier it has been observed that the prices of caps are driven by risk factors not spanned by the factors explaining LIBOR rates, even though caps are derivatives of LIBOR. We perform the regression analysis on the implied volatility of caps for all maturities and strike rates to see how the economic activities affect the cap prices. Using 21 series of macroeconomic announcements, first we do the event study to see which…

We provide a full classification of all attainable term structure shapes in the two-factor Vasicek model of interest rates. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results apply to both forward and yield curves and show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The key mathematical tool is the theory of total positivity, pioneered by Samuel Karlin and others in the 1950ies., major rewrite of the original paper

The Ornstein-Uhlenbeck (OU) process is one of the most popular stochastic system applied in many different fields of studies. It was introduced in 1930 and can be considered as a continuous extension of the autoregressive model of order one, AR(1). Furthermore, the OU process in finance it is known as the Vasicek model and is mainly used in interest rate modelling. Furthermore, it is deeply studied and its main properties are well known. However, many real data exhibit some properties of the OU process although they cannot be directly modelled with this classical system. This is in case when certain characteristics adequate to the OU process are visible in the data however other properties of the classical model change. In such case the subordination scenario can be considered. In general, the subordination it is a time change of the original process. In this paper we consider the Ornstein-Uhlenbeck process delayed (subordinated) by Gamma subordinator. The Gamma subordinator is Levy process of Gamma distribution. The main properties are studied, like the influence of the initial condition on the stationarity of the new subordinated process. Moreover, the formulas for the expected value and the autocovariance are derived. Furthermore, the simulation procedures and estimation algorithms are proposed.

In this paper, the pricing problems of variance swaps with discrete sampling times are studied, where the volatility of underlying assets follows a mean-reverting Gaussian (MRG in short) process, and the instantaneous interest rate is described by classical Vasicek model. By using measure transformation, Feynman–Kac formula and Fourier transform algorithm, a closed-form analytic pricing formula for variance swaps with the actual-return realized variance is presented.

This paper proposes a robust one-pass estimator that is easy to code: Justified by the market-model itself and using a prior that market-betas should not be less than –2 and more than +4, the market-model is run on daily stock rates of return that have first been winsorized at –2 and +4 times the contemporaneous market rate of return. The resulting “slope-winsorized” estimates outperform (all) other known estimators in predicting the future OLS market-beta (on R2 metrics). Adding reasonable age decay, suggesting a half-life of about 3 to 5 months, to observations entering the market-model further improves it. The estimates outpredict the Vasicek estimates by about half as much as the Vasicek estimates outpredict the OLS estimates.

This study is based on a reduced model of convertible bonds pricing in which the determined factors of convertible bond prices are stock prices and stochastic interest rates. Firstly, the assumption is that the stochastic process of interest rates obeys Vasicek model, three short-term interest rate series of Shanghai Interbank Offered Rate are selected to fit Vasicek model. Secondly, analysis of time series is used to test the stationarity of the three series and linear regression is used to get the stochastic processes. Thirdly, a convertible bond is given to verify the practicability of the Martingale pricing under Vasicek model. In order to get a more accurate consequence, three stochastic processes are applied to calculate the convertible bond price and make comparisons. Finally, due to the advantages of the Martingale pricing under stationary interest rate series, a convenient way to determine the benchmark interest rate in pricing for convertible bonds is put forward.

In this article, we present that fractional Lévy processes which is very an important field in both probability theory and its application in recent years. The fractional Brownian motion is suggested as the fractional Lévy processes in this article. We will make parameters estimate of the Vasicek process driven by fractional Brownian motion, that represented the short memory parameter (0 < H < ½) and the long memory parameter (½ < H < 1). So, Our aim is to study the behavior of stochastitc Vasicek Interest driven by fractional Brownian motion. We use maximum likelihood to estimate the drift, diffusion and Hurst parameters and generally the fractional Lévy processes. We illustrate our methods, and show the behavior of stochastic parameters using simulation and real data (ISX60).

Considering stochastic behavior of interest rates in financial market, we construct a new class of interest models based on compound Poisson process. Different from the references, this paper describes the randomness of interest rates by modeling the force of interest with Poisson random jumps directly. To solve the problem in calculation of accumulated interest force function, one important integral technique is employed. And a conception called the critical value is introduced to investigate the validity condition of this new model. We also discuss actuarial present values of several life annuities under this new interest model. Simulations are done to illustrate the theoretical results and the effect of parameters in interest model on actuarial present values is also analyzed.

The main goal of this paper is to propose a novel numerical algorithm to price American options on bonds. For this purpose, we illustrate the performance of this method by means of the valuation of an American Put Option on a discount bond under the extended Vasicek model due to Hull and White (HW) and using the consistent forward rate curves. In particular, an implicit Crank-Nicolson (CN) scheme in time is applied obtaining a discretized linear complementarity problem (LCP) and then we introduce a direct LU based method to solve the LCP. Finally, we carry out numerical experiments to examine the convergence of this method and to testify the efficiency and effectiveness of this numerical scheme against other standard approaches.

In this paper we study a corporate bond-pricing model with credit rating migration and a stochastic interest rate. The volatility of bond price in the model strongly depends on potential credit rating migration and stochastic change of the interest rate. This new model improves the previous existing models in which the interest rate is considered to be a constant. The existence, uniqueness and regularity of the solution for the model are established. Moreover, some properties including the smoothness of the free boundary are obtained. Furthermore, some numerical computations are presented to illustrate the theoretical results.

This article considers the possibility of forecasting real effective exchange rate indices for leading world countries. A stable unified mean trajectory for a reliable forecast of the index cannot be constructed using data from 31 January 1994 to 30 April 2017; however, it is reasonable to use data of the last period for this purpose. To construct a short-term forecast of real effective exchange rates based on initial value, we propose using a simulation with a discrete approximation of stochastic differential equations of Merton, Vasicek, Dosen, Ogden, and Cox-Ingersoll-Ross, and a polynomial residues model. Simulations using discrete approximations of the Vasicek, Merton, Dosen, and Ogden equations did not allow constructing a reliable forecast of the specified index using data from April 2016 to March 2017 for the USA, UK, Eurozone countries, Japan, and Switzerland. Processing simulation results based on a discrete approximation of the stochastic Cox-Ingersoll-Ross equation and the polynomial residues model, and for the considered countries for most of the same time period, resulted in a 50 % confidence interval for the mean trajectory of observed values of effective exchange rates indexes. The quality of the forecast essentially depends on the selected time period and methods used to determine the numerical parameters of discrete approximations for the original stochastic equations.

Measurement of a single interest rate curve is an important and well-studied inverse problem. To select plausible interest rate curves from the infinite set of possible interest rate curves, forward rates should be used in the regularization. By discretizing the interest rate curve it is shown that the inverse problem can be reformulated as a convex optimization model that can be efficiently solved using existing solvers. The convex optimization model can include bills, bonds, certificates of deposits, forward rate agreements and interest rate swaps using both equality constraints and inequality constraints that stem from bid/ask prices. The importance of an appropriate regularization and allowing for deviations from market prices to obtain stable forward rate curves is illustrated using market data.

This paper provides a two-step approach based on the stochastic differential equations (SDEs) to improve short-term prediction. In the first step of this framework, a Hull-White (HW) model is applied to obtain a baseline prediction model from previous days. Then, the extended Vasicek model (EV) is employed for modeling the difference between observations and baseline predictions (residuals) during an individual day. The parameters of this time-varying model are estimated at each sample using the residuals in a short duration of time before the time point of prediction; so it provides a real time prediction. The extracted model recovers the valuable local variation information during each day. The performance of our method in comparison with other methods improves significantly in terms of root mean squared error (RMSE), mean absolute error (MAE) and mean relative error (MRE) for real data from Tehran’s highways and the open-access PeMS database. We also demonstrate that the proposed model is appropriate for imputing the missing data in traffic dataset and it is more efficient than the probabilistic principal component analysis (PPCA) and k -Nearest neighbors ( k -NN) methods.

This paper studies the optimal consumption–investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption–investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton–Jacobi–Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption–investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results.

We consider the fractional Vasicek model of the form dXt = (α-βXt)dt + γdBHt, driven by fractional Brownian motion BH with Hurst parameter H ∈ (0,1). We construct three estimators for an unknown parameter θ=(α,β) and prove their strong consistency.

In this paper, we have constructed a model to price currency option. It contributes to releasing some assumptions the Black-Scholes' (1973) model makes. One of them is that the log price of asset doesn't follow a normal distribution any more. The other one is the interest rates in the domestic and foreign countries become stochastic. This general formula is first proposed by Amin and Jarrow (1991). Based on this model, we build extended normal distribution model [developed by Ki, Choi, Chang and Lee (2005)] under the assumption of stochastic interest rate economy. In numerical examples, our proposed model would be compared with Amin and Jarrow (1991) under CIR [Cox, Ingersoll and Ross (1985)] interest rate term structure. Furthermore, Monte Carlo simulation is used to provide another outcome to be another comparative example. Finally, we think that the proposed model provides more correct currency option prices when taking account of stochastic interests and extended normal distribution. The drawback of the Black-Scholes' formula which fails to catch the volatility smile effect is resolved by using the proposed model. The market participants can use the actual market data to calibrate the parameters of the proposed model and use the proposed model to price the currency options and derivatives accurately.

A parsimonious autoregressive model that is globally mean-reverting but locally driven by momentum is proposed. The local-momentum autoregression (LM-AR) model carries one extra parameter, and depending on the sign of this extra parameter, it can be either local momentum-preserving or momentum-building. The LM-AR model is motivated by observing US interest rate movement over many decades, which over a long time span seems to mean revert but over a period of several months or years can actually exhibit a momentum-like behavior. We use the LM-AR model with a stochastic central tendency factor as the dominant global risk factor in interest rates and add a local variation component of the standard mean-reverting type to create a 3-factor risk environment. We then derive its corresponding term structure model and empirically implement the model on US interest rates of seven maturities from January 1954 to December 2013 on a weekly frequency to establish the presence of local momentum building.

As an important macroeconomic variable and monetary policy tool, interest rate has been included in the core of the economic analysis for a long time. Reasonable interest rate is significant in the aspects of improving the social credit level and playing the economic leverage role, so the modeling approach of interest rate is our concern. This paper proposes a new interest rate model on the basis of exponential Ornstein–Uhlenbeck equation under the uncertain environment. Based on the model, the pricing formulas of the zero-coupon bond, interest rate ceiling and interest rate floor are derived through the Yao–Chen formula. In addition, some numerical algorithms are designed to calculate the prices of derivations according to the pricing formulas above.

While the literature on dynamic portfolio selection with stochastic interest rates only confines its investigation to the continuous-time setting up to now, this paper studies a multi-period mean-variance portfolio selection problem with a stochastic interest rate, where the movement of the interest rate is governed by the discrete-time Vasicek model. Invoking dynamic programming approach and the Lagrange duality theory, we derive the analytical expressions for both the efficient investment strategy and the efficient mean-variance frontier of the model formulation. We then extend our model to the situation with an uncontrollable liability.

In this article, we investigate the pricing of European-style options under a Markovian regime-switching Hull–White interest rate model. The parameters of this model, including the mean-reversion level, the volatility of the stochastic interest rate, and the volatility of an asset’s value, are modulated by an observable, continuous-time, finite-state Markov chain. A closed-form expression for the characteristic function of the logarithmic terminal asset price is derived. Then, using the fast Fourier transform, a price of a European-style option is computed. In a two-state Markov chain case, numerical examples and empirical studies are presented to illustrate the practical implementation of the model.

This paper proposes a novel interest rate model that presents simple analytical pricing formulas for interest rate-based derivatives, including swaps, futures, swaptions, caps and floors. Exploring the regime-switching feature of Markov chains, the proposed model focuses on discrete changes in the central bank policy rates – the main driver of short-term rate fluctuations. An empirical analysis shows that the proposed model generally outperforms other standard short-term rate models in fitting cross-sections of options prices. Moreover, the explicit nature of policy rates, to some extent, enables the model to infer risk-neutral probabilities of the central-bank rate decisions.

There remains a lack of literature on a pairs-trading model that is able to capture the mean reversion and two different states of spreads. The purpose of this study is to combine the Markov regime-switching model and the Vasicek model to implement a pairs-trading strategy that utilizes the S&P 500 stock components from January 1, 2006, through September 28, 2012. We compare our model's performance with the performance of previous methods based on a variety of portfolios and trading periods. The empirical results show that the trading rule of the Markov regime-switching model with mean reversion has the best performance with a simple portfolio. Furthermore, the results show that shorter trading periods produce better performance than longer trading periods and that the trading rule performs strongly during the global financial crisis of 2008 to 2009.

We derive a closed-form expression for the price of a Euro- pean quanto call option when both foreign and domestic interest rates follow the Vasicek's short rate model. A quanto is a type of financial derivative whose pay-out currency differs from the natural denomination of its underlying financial variable. A quanto option is a cash-settled, cross-currency derivative whose underlying asset has a payoff in one currency, but the payoff is converted to another currency when the option is settled. For that reason, the correlation between underlying asset and currency exchange rate plays an important role in pricing quanto option. Quanto options in this paper have both the strike price and the underlying asset price denominated in foreign currency. At exercise, the value of the option is calculated as the options intrinsic value in foreign currency, which is then converted to the domestic currency at the fixed exchange rate. This allows investors to obtain exposure to foreign assets without the corresponding foreign exchange risk. Pricing quanto options based on the classical Black-Scholes (1) model, on which most of the research on quanto options has focused, has a weakness of assuming a constant volatility and constant interest rates. To overcome such weakness, in valuing quanto option, it is natural to consider a stochastic volatility or stochastic interest rate models. Despite its importance, very few researches have been done on finding analytic solutions of quanto option prices under a stochastic volatility model primarily due to the sophisticated stochastic processes and inability to obtain the general closed form. However, by assuming constant interest rates, Giese (4) provided a closed-form expression for the price of a quanto option in the Stein-Stein stochastic volatility model, and then Y. Lee et al. (5) got a closed-form expression for the price of a European quanto

SYNOPTIC ABSTRACTWe expand the Merton's structural credit risk model into a model that includes an underlying asset process with a non-Gaussian and serially correlated stochastic nature. Using a standard Edgeworth expansion, we arrive at closed-form analytic expressions for the probability of default, the distance to default, and the term structure of credit spread, allowing us to evaluate more accurately the credit risk of incorporating nonnormal asset returns. Moody's KMV (Kealhofer, McQuown, and Vasicek) analogous procedures are proposed for the estimation of the model parameters. Empirical applications for credit-risk evaluation are illustrated, which reveal the significant effects on credit risk due to the non-Gaussianity of the underlying firm's asset process.

We present an arbitrage-free non-parametric yield curve prediction model which takes the full discretized yield curve data as input state variable. Absence of arbitrage is a particularly important model feature for prediction models in case of highly correlated data as, for instance, interest rates. Furthermore, the model structure allows to separate constructing the daily yield curve from estimating its volatility structure and from calibrating the market prices of risk. The empirical part includes tests on modeling assumptions, out-of-sample back-testing and a comparison with the Vasiček (1977) short-rate model.

This article provides new developments in characterizing the class of regime-switching exponential affine interest rate processes in the context of pricing a zero-coupon bond. A finite-state Markov chain in continuous time dictates the random switching of time-dependent parameters of such processes. We present exact and approximate bond pricing formulas by solving a system of partial differential equations and minimizing an error functional. The bond price expression exhibits a representation that shows how it is explicitly impacted by the rate matrix and the time-dependent coefficient functions of the short rate models. We validate the bond pricing formulas numerically by examining a regime-switching Vasicek model.

We construct a time-consistent and arbitrage-free three-factor Vasicek model for long-term bonds. A new methodology based on a stochastic mean reversion rate which captures uncertainty in long-term bond yields is presented. To allow measurement errors to be accounted for in observed yields, the model is expressed in a state space form. Kalman filtering is then applied to filter uncertainty in the observed yields. An appropriate set of transition equations on state variables and measurement equations on observed yields are derived. Using historical market data from the US Treasury daily interest rates (March 2006 to June 2020), Germany Government bond yields (August 2000 to 15 January 2021) and Canada Government bond yields (16 January 2020 to 14 January 2021), parameters of one-, two- and three-factor models are estimated. The results indicate that the constructed Vasicek model can fit the US, Germany and Canada term structure of interest rates.

This paper examines the price of European put options with underlying asset zero-coupon bond and the interest rate that satisfied the Vasicek model with jump. The study was conducted by reconstructing the European put option pricing equation. The jump size is defined following a mixed-exponential distribution. By utilizing infinitesimal generator and martingale concepts, Laplace transform is constructed for the distribution of the Vasicek model with jump. Then, using the results of the Laplace transform for the distribution of the Vasicek model with jump and the concept of equivalent martingale measure, the European put option pricing equation with underlying asset zero-coupon bond is constructed.

Traditionally, futures Minimum Variance Hedge Ratios (MVHRs) are determined ex post. In this paper, we derive 3 increasingly realistic ex ante MVHRs, based on the carry cost and the Vasicek model. The hedging performance of the most realistic ex ante MVHR determination method is compared to, and found to be superior to, that of the traditional MVHR for the S&P 500 index, gold, and the EUR/USD exchange rate.

This paper introduces state uncertainty due to information-processing constraints into the Vasicek model to examine the impacts of rational inattention. By exploiting the term structure of interest rates, we derive closed-form solutions for the subjective bond price and the corresponding bond yield and find that uncertainty induced by informational frictions plays vital roles in undervaluing the bond price and overestimating the bond yield. Furthermore, we clarify the applications of interest rate dynamics under rational inattention and generate the following results: (i) there is an ambiguous relationship between the investor’s channel capacity and option price; (ii) an increase in state uncertainty via a change in the degree of channel capacity is likely to accelerate investment.

Stochastic models traditionally used to describe metals’ prices have proved not to be suitable to represent the dynamic behavior and time-related nature of metal markets. Rates of return are characterized by non-Gaussian and heterogeneous characteristics, which requires the use of properly adjusted models. In this paper, we introduce a stochastic model that takes under consideration the mentioned specific characteristics of the real data corresponding to the mineral commodity prices, namely the non-homogeneous character (time-dependent characteristics) and non-Gaussian distribution. The introduced model is in some sense the extension of the classical Ornstein–Uhlenbeck process (called also the Vasicek model) which was originally used to the interest rate data description. The proposed in this paper model, in contrast to the classical process, has the time-dependent parameters. This perfectly captures the time-dependent characteristics of the real data. Moreover, it is based on the general class of the skewed Student’s t-distribution (SGT), which is related to the non-Gaussian behavior of the real metals’ prices. This paper is a continuation of the authors’ previous research where the simpler model (Chan–Karolyi–Longstaff–Sander, CKLS) based on the SGT distribution was proposed. We demonstrate here the step-by-step procedure of the time-dependent parameters’ estimation and check its effectiveness by using the simulated data. Finally, based on the real-time series analysis, we demonstrate that the proposed stochastic model is universal and can be applied to metals’ prices description for the long-term prediction.

This paper considers a fractional version of the Vasicek interest rate model where the noise part of the model is modeled as fractional Brownian motion with Hurst index H ∈ ( 0 , 1 2 ) . We first show that the increments of the model have short-range dependence. Then, the efficiency of the proposed model versus the classical interest rate models is examined by employing the MLE calibration method and the Akaike Information Criterion (AIC). An analytic approximation formula for pricing zero-coupon bond when the dynamics of the interest rate governed by the model is derived. We also find a margin for the option price by using bid and ask formulas.

In this chapter we deal with marginal probability densities of diffusion type processes, generated by sixteen models of short-term interest rates, which allow us to obtain densities in analytical form. This family covers almost all currently used models of continuous time. Some densities (Vasicek, Cox–Ingersoll–Ross, geometric Brownian motion, Ahn–Gao) are well studied in the literature and are given here for convenience of comparison. Other densities are described for the first time. The main focus is on the analytical properties of densities and their four first moments (mathematical expectation, variance, asymmetry and kurtosis), which are most often of interest to practitioners. In the main, stationary densities and moments are considered, although several models generate non-stationary processes.

We address the so-called calibration problem which consists of fitting in a tractable way a given model to a specified term structure like, e.g., yield, prepayment or default probability curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possibly coupled with compound Poisson jumps, JCIR), are tractable processes but have limited flexibility; they fail to replicate actual market curves. The deterministic shift extension of the latter (Hull-White or JCIR++) is a simple but yet efficient solution that is widely used by both academics and practitioners. However, the shift approach is often not appropriate when positivity is required, which is a common constraint when dealing with credit spreads or default intensities. In this paper, we tackle this problem by adopting a time change approach. On the top of providing an elegant solution to the calibration problem under positivity constraint, our model features additional interesting properties in terms of implied volatilities. It is compared to the shift extension on various credit risk applications such as credit default swap, credit default swaption and credit valuation adjustment under wrong-way risk. The time change approach is able to generate much larger volatility and covariance effects under the positivity constraint. Our model offers an appealing alternative to the shift in such cases.

This paper considers the continuous-time portfolio optimization problem with both stochastic interest rate and stochastic volatility in regime-switching models, where a regime-switching Vasicek model is assumed for the interest rate and a regime-switching Heston model is assumed for the stock price.We use the dynamic programming approach to solve this stochastic optimal control problem. Under suitable assumptions, we prove a verification theorem.We then derive a closed-form solution of the associated Hamilton-Jacobi-Bellman (HJB) equation for a power utility function and a special choice of some model parameters. We prove the optimality of the closed-form solution by verifying the required conditions as stated in the verification theorem. We present a numerical example to show the optimal portfolio policies and value functions in different regimes.

Reinforcement learning has become one of the best approach to train a computer game emulator capable of human level performance. In a reinforcement learning approach, an optimal value function is learned across a set of actions, or decisions, that leads to a set of states giving different rewards, with the objective to maximize the overall reward. A policy assigns to each state-action pairs an expected return. We call an optimal policy a policy for which the value function is optimal. QLBS, Q-Learner in the Black-Scholes(-Merton) Worlds, applies the reinforcement learning concepts, and noticeably, the popular Q-learning algorithm, to the financial stochastic model of Black, Scholes and Merton. It is, however, specifically optimized for the geometric Brownian motion and the vanilla options. Its range of application is, therefore, limited to vanilla option pricing within financial markets. We propose MQLV, Modified Q-Learner for the Vasicek model, a new reinforcement learning approach that determines the optimal policy of money management based on the aggregated financial transactions of the clients. It unlocks new frontiers to establish personalized credit card limits or to fulfill bank loan applications, targeting the retail banking industry. MQLV extends the simulation to mean reverting stochastic diffusion processes and it uses a digital function, a Heaviside step function expressed in its discrete form, to estimate the probability of a future event such as a payment default. In our experiments, we first show the similarities between a set of historical financial transactions and Vasicek generated transactions and, then, we underline the potential of MQLV on generated Monte Carlo simulations. Finally, MQLV is the first Q-learning Vasicek-based methodology addressing transparent decision making processes in retail banking.

Using methods from the theory of total positivity, we provide a full classification of attainable term structure shapes in the two-factor Vasicek model. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The mathematical tools from total positivity can likely be applied to higher-dimensional generalizations of the Vasicek model and to other interest rate models as well.