Masaaki Kijima

A Markovian framework in multi-factor Heath-Jarrow-Morton models

We consider the general n -factor Heath, Jarrow, and Morton model (1992) and provide a sufficient condition on the volatility structure for the spot rate process to be Markovian with 2 n state variables. The price of a discount bond is also Markovian with the same state variables and, hence, claims against the term structure can be efficiently priced using standard simulation techniques. Our results extend earlier works such as Ritchken and Sankarasubramanian (1995) where the one-factor model is treated, and Carverhill (1994), where the volatility structure is non-random. Numerical experiments show that our model can explain the volatility smile observed in the interest rate options market and also overcome the biases noted by Flesaker (1993).

A multi-quality model of interest rates

We consider a consistent pricing model of government bonds, interest-rate swaps and basis swaps in one currency within the no-arbitrage framework. To this end, we propose a three yield-curve model, one for discounting cash flows, one for calculating LIBOR deposit rates and one for calculating coupon rates of government bonds. The derivation of the yield curves from observed data is presented, and the option prices on a swap or a government bond are studied. A one-factor quadratic Gaussian model is proposed as a specific model, and is shown to provide a very good fit to the current Japanese low-interest-rate environment.

Stochastic orders and their applications in financial optimization

Abstract. Stochastic orders and inequalities are very useful tools in various areas of economics and finance. The purpose of this paper is to describe main results obtained so far by using the idea of stochastic orders in financial optimization. Especially, the emphasis is placed on the demand and shift effect problems in portfolio selection. Some other examples, which are not related directly to optimization problems, are also given to demonstrate the wide spectrum of application areas of stochastic orders in finance.

Monotonicities in a Markov Chain Model for Valuing Corporate Bonds Subject to Credit Risk

In recent years, it has become common to use a Markov chain model to describe the dynamics of a firms credit rating as an indicator of the likelihood of default. Such a model can be used not only for describing the dynamics but also for valuing risky discount bonds. The aim of this paper is to explain how the Markov chain model leads to the known empirical findings such that prior rating changes carry predictive power for the direction of future rating changes and a firm with low (high, respectively) credit rating is more likely to be upgraded (downgraded) conditional on survival as the time horizon lengthens. The model will also explain practically plausible statements such as that bond prices as well as credit risk spreads would be ordered according to their credit qualities. Stochastic monotonicities of absorbing Markov chains play a prominent role in these issues.

Credit Events and the Valuation of Credit Derivatives of Basket Type

Thispaper provides a simple model for valuing a credit derivativewhose payoff depends on the identity (or identities) of the first(or first two) to occur of a given list of credit events, suchas defaults. The joint survival probability of occurrence timesof credit events is formulated in terms of stochastic intensityprocesses under the assumption of conditional independence. Basedon the joint survival probability, we can easily obtain the pricingformulas of such credit derivatives under the risk-neutral valuationframework. When the default intensity processes follow the extendedVasicek model, closed-form solutions of the pricing formulasare given.

Valuation of a Credit Swap of the Basket Type

This article provides a simple model to value a credit swap ofthe basket type. Unlike the previous literature, we considerthe joint survival probability of occurrence times of creditevents in terms of stochastic intensity processes under the assumptionof conditional independence. Based on the joint survival probability,such a credit swap can be valued under the risk-neutral valuationframework. Assuming that the default intensity processes followthe extended Vasicek model with a correlation structure, an analyticexpression of the valuation formula is derived. Some numericalexample is given to demonstrate the usefulness of our model.

Evaluation of credit risk of a portfolio with stochastic interest rate and default processes 1

This paper proposes a new model for evaluating credit risk of a portfolio consisting of interest rate sensitive assets. Our model is distinguished from existing risk valuation models such as CreditMetrics™ or CREDITRISK+ by (1) the dynamics of the default-free interest rate as well as hazard rate pro- cesses of defaultable assets are described by stochastic differential equations; and (2) prices of individual assets are evaluated by the single risk-neutral valuation framework. It is then possible to evaluate not only credit risk but also market risk of the portfolio in a synthetic manner. It is shown that value at risk (VaR) of the portfolio is approximately evaluated as a closed form solution.

A Multivariate Extension of Equilibrium Pricing Transforms: The Multivariate Esscher and Wang Transforms for Pricing Financial and Insurance Risks

This paper proposes a multivariate extension of the equilibrium pricing transforms for pricing general financial and insurance risks. The multivariate Esscher and Wang transforms are derived from Buhlmann’s equilibrium pricing model (1980) under some assumptions on the aggregate risk. It is shown that the Esscher and Wang transforms coincide with each other when the underlying risks are normally distributed.

Pricing Equity Swaps in a Stochastic Interest Rate Economy

“A swap is a package of forward contracts, and the standard Òcost of carryÓ model for valuing forwards, like other-risk neutral valuation relationships, involves the riskless interest rate but not the expected price change on the underlying asset. Thus, in a basic equity swap, the current term structure of interest determines the swap rate, while the equity price process plays no role. It has been demonstrated that under non-stochastic interest rates, this result holds whether a swap has fixed or variable notional principal. Kijima and Muromachi introduce stochastic interest rates, and show that when notional principal is constant, the result is the same, but with variable notional principal, the stock price process does enter the swap valuation equation in the correlation between interest rate changes and stock returns. They derive valuation formulas for capped equity swaps in the same framework.”

An economic premium principle in a multiperiod economy

Abstract This paper considers a multiperiod economic equilibrium model for deriving the economic premium principle of Buhlmann [Astin Bull. 11 (1980) 52–60; Astin Bull. 14 (1983) 13–21]. To do this, we construct a consumption/portfolio model in which each agent is characterized by his/her utility function and income and seeks to invest his/her wealth in both insurance as well as a financial market so as to maximize the expected, discounted total utility from consumption. The state price density in equilibrium is obtained in terms of the Arrow–Pratt index of absolute risk aversion for the representative agent. As special cases, power and exponential utility functions are examined, and some comparative statics results are derived.