Forward Probability Measure Research Articles

Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

Valuation of an option using non-parametric methods

This paper provides a general valuation model to fairly price a European option using parametric and non-parametric methods. In particular, we show how to use the historical simulation (HS) method, a well-known non-parametric statistical method applied in the financial area, to price an option. The advantage of the HS method is that one can directly obtain the distribution of stock returns from historical market data. Thus, it not only does a good job in capturing any characteristics of the return distribution, such as clustering and fat tails, but it also eliminates the model errors created by mis-specifying the distribution of underlying assets. To solve the problem of measuring transformation in valuing options, we use the Esscher’s transform to convert the physical probability measure to the forward probability measure. Taiwanese put and call options are used to illustrate the application of this method. To clearly show which model prices stock options most accurately, we compare the pricing errors from the HS method with those from the Black–Scholes (BS) model. The results show that the HS model is more accurate than the BS model, regardless for call or put options. More importantly, because there is no complex mathematical theory underlying the HS method, it can easily be applied in practice and help market participants manage complicated portfolios effectively.

Semi-Markov migration process in a stochastic market in credit risk

In the present the idea of stochastic Market environment comes into play to express the changes in general economy, which affects any industry in small or great amounts of turbulence. We model the evolution of the Market among its possible ν-states as an F-inhomogeneous semi-Markov process. This idea leads us to modeling the migration process of defaultable bonds as ν different F-inhomogeneous semi-Markov process. The survival probabilities of a defaultable bond in every credit grade are found. The asymptotic behaviour of the survival probabilities is established under certain conditions. Also, it is proved under what conditions the convergence is geometrically fast. The stochastic foundation of the general stochastic discrete-time Market is provided, by proving that the market is viable, if and only if, there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price process for all possible states of the Market is a martingale. The term structure of credit spread and the change of real-world probability measure to forward probability measure are studied. In the form of a theorem it is proved that under certain conditions, changing the real probability measure to a forward probability measure, does not affect the inhomogeneous semi Markov process modeling the migration of defaultable bonds. That is, it is proved that it only changes the basic parameters and we provide a relation among the transition probabilities under the two measures. Finally, parameter estimation and calibration of the inhomogeneous semi-Markov chain in stochastic environment is being provided.

Fuzzy semi-Markov migration process in credit risk

We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We model the evolution of credit migration of a defaultable bond as an inhomogeneous semi-Markov process with fuzzy states. We study the effects of changing the real probability measure to a forward probability measure. In addition, we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default. Finally, we discuss parameter estimation and calibration of the inhomogeneous Markov chain with fuzzy states.

Asymptotic behaviour of the survival probabilities in an inhomogeneous semi-Markov model for the migration process in credit risk

We start with the stochastic foundation of the general discrete-time Market of defaultable bonds. We prove that the above market is viable, if and only if there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price is a martingale. Assuming that the migration process of defaultable bonds evolves as an inhomogeneous semi-Markov process, we study the asymptotic behaviour of survival probabilities. We provide a method of estimating real-world transition probability sequences for the semi-Markov process, and statistics for testing their homogeneity over time.

A note on calibration of Markov processes

AbstractUsing Malliavin calculus, we derive under the forward probability measure Ito's formula for anticipating processes in the multi‐dimensional case. We use it when the instantaneous volatility, the spot rate and the dividend yield of the stock price are Markov processes to approximate solutions of the Black type to the prices of call options as well as binary options. The stochastic dividend yield makes it possible to consider the pricing of contingent claims not only in the equity market but also in the commodities, fixed income and forex markets. We then derive the Greeks of the model, which will be used for the risk management of the exotic products, and approximate the implied volatility surface induced by the model. Copyright © 2010 Wilmott Magazine Ltd.

Multi-Currency Fast Stochastic Local Volatility Model

We consider a stochastic local volatility model with domestic and foreign stochastic interest rates and identify a bias with respect to the deterministic local volatility with deterministic rates. Relating the local volatility of our model to that of the forward price, we quantify that bias by equating the variance swap contract under the risk-neutral measure with that under the forward probability measure. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, the bias term simplifies and can easily be computed. We then describe a simple implementation of our model. In the discretised dynamics, the local volatility function being piecewise constant, we apply the bootstrapping technique to calculate its value by solving a quadratic equation at each maturity and strike. As a result, we get a fast and robust way of calibrating a stochastic local volatility model with stochastic rates to market prices.

A Note on Calibration of Markov Processes

Using Malliavin calculus, we derive under the forward probability measure Ito's formula for anticipating processes in the multi-dimensional case. We use it when the instantaneous volatility, the spot rate and the dividend yield of the stock price are Markov processes to approximate solutions of the Black type to the prices of call options as well as binary options. The stochastic dividend yield makes it possible to consider the pricing of contingent claims not only in the equity market but also in the commodities, fixed income and forex markets. We then derive the Greeks of the model, which will be used for the risk management of the exotic products, and approximate the implied volatility surface induced by the model.

An inhomogeneous semi-Markov model for the term structure of credit risk spreads

We model the evolution of the credit migration of a corporate bond as an inhomogeneous semi-Markov chain. The valuation of a defaultable bond is done with the use of the forward probability of no default up to maturity time. It is proved that, under the forward probability measure, the semi-Markov property is maintained. We find the functional relationships between the forward transition probability sequences and the real-world probability sequences. The stochastic monotonicity properties of the inhomogeneous semi-Markov model, which play a prominent role in these issues, are studied in detail. Finally, we study the term structure of credit spread, provide an algorithm for the estimation of the forward probabilities of transitions under the risk premium assumptions, and present an estimation method for the real-world probability sequences.

An inhomogeneous semi-Markov model for the term structure of credit risk spreads

We model the evolution of the credit migration of a corporate bond as an inhomogeneous semi-Markov chain. The valuation of a defaultable bond is done with the use of the forward probability of no default up to maturity time. It is proved that, under the forward probability measure, the semi-Markov property is maintained. We find the functional relationships between the forward transition probability sequences and the real-world probability sequences. The stochastic monotonicity properties of the inhomogeneous semi-Markov model, which play a prominent role in these issues, are studied in detail. Finally, we study the term structure of credit spread, provide an algorithm for the estimation of the forward probabilities of transitions under the risk premium assumptions, and present an estimation method for the real-world probability sequences.

A multicurrency extension of the lognormal interest rate Market Models

The Market Models of the term structure of interest rates, in which forward LIBOR or forward swap rates are modelled to be lognormal under the forward probability measure of the corresponding maturity, are extended to a multicurrency setting. If lognormal dynamics are assumed for forward LIBOR or forward swap rates in two currencies, the forward exchange rate linking the two currencies can only be chosen to be lognormal for one maturity, with the dynamics for all other maturities given by no–arbitrage relationships. Alternatively, one could choose forward interest rates in only one currency, say the domestic, to be lognormal and postulate lognormal dynamics for all forward exchange rates, with the dynamics of foreign interest rates determined by no-arbitrage relationships.

A Multicurrency Extension of the Lognormal Interest Rate
 Market Models

The Market Models of the term structure of interest rates, in which forward LIBOR or forward swap rates are modelled to be lognormal under the forward probability measure of the corresponding maturity, are extended to a multicurrency setting. If lognormal dynamics are assumed for forward LIBOR or forward swap rates in two currencies, the forward exchange rate linking the two currencies can only be chosen to be lognormal for one maturity, with the dynamics for all other maturities given by no-arbitrage relationships. Alternatively, one could choose forward interest rates in only one currency, say the domestic, to be lognormal and postulate lognormal dynamics for all forward exchange rates, with the dynamics of foreign interest rates determined by no-arbitrage relationships.