Yukio Muromachi

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.

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.

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.”

Improved Estimation Methods for Value-at-Risk, Expected Shortfall and Risk Contributions with High Precision

The (marginal) risk contribution is very useful for analyzing the concentration risk in a portfolio. However, it is difficult to estimate the risk contributions for value-at-risk (VaR) and expected shortfall (ES) precisely, especially using a Monte Carlo simulation. We applied a saddlepoint approximation to estimate the distribution function, so that the difficulty of estimating the risk contributions for VaR was dissolved. In this paper, we propose new estimation methods for ES and the risk contributions for ES based on the conditional independence and a saddlepoint approximation. Numerical studies confirm that these new methods are much better than existing ones.

On the Risk Evaluation Method Based on the Market Model

This paper presents a risk evaluation model for interest-rate sensitive products within the no-arbitrage framework. A yield-curve dynamics is modelled, based on the results of the principal component analysis (PCA), to generate future scenarios of interest rates under the observed probability measure. The market model is adopted for the pricing of interest-rate derivatives under the risk-neutral measure by identifying market prices of risk that are consistent with the yield-curve model. Given the future scenarios of yield curve and the market prices of risk, the prices of interest-rate sensitive products are calculated at any future time. Risk measures such as Value-at-Risk (VaR) of portfolios with interest-rate sensitive products can be evaluated through simple Monte Carlo simulation. It is shown, however, that some market models often used in practice are not consistent with the no-arbitrage paradigm.