Dan Galai

The option pricing model and the risk factor of stock

Abstract In this paper a combined capital asset pricing model and option pricing model is considered and then applied to the derivation of equitys value and its systematic risk. In the first section we develop the two models and present some newly found properties of the option pricing model. The second section is concerned with the effects of these properties on the securityholders of firms with less than perfect ‘me first’ rules. We show how unanticipated changes in firm capital and asset structures can differentially affect the firms debt and equity. In the final section of the paper we consider a number of theoretical and empirical implications of the joint model. These include investment policy as well as the causes and effects of non-stationarity in the systematic risk of levered equity and risky debt.

A comparative analysis of current credit risk models

The new BIS 1998 capital requirements for market risks allows banks to use internal models to assess regulatory capital related to both general market risk and credit risk for their trading book. This paper reviews the current proposed industry sponsored Credit Value-at-Risk methodologies. First, the credit migration approach, as proposed by JP Morgan with CreditMetrics, is based on the probability of moving from one credit quality to another, including default, within a given time horizon. Second, the option pricing, or structural approach, as initiated by KMV and which is based on the asset value model originally proposed by Merton (Merton, R., 1974. Journal of Finance 28, 449‐470). In this model the default process is endogenous, and relates to the capital structure of the firm. Default occurs when the value of the firm’s assets falls below some critical level. Third, the actuarial approach as proposed by Credit Suisse Financial Products (CSFP) with CreditRisk+ and which only focuses on default. Default for individual bonds or loans is assumed to follow an exogenous Poisson process. Finally, McKinsey proposes CreditPortfolioView which is a discrete time multi-period model where default probabilities are conditional on the macro-variables like unemployment, the level of interest rates, the growth rate in the economy, ... which to a large extent drive the credit cycle in the economy. ” 2000 Elsevier Science B.V. All rights reserved.

Prototype risk rating system

Abstract This paper explores the traditional and prevalent approach to credit risk assessment – the rating system. We first describe the rating systems of the two main credit rating agencies, Standard & Poors and Moodys. Then we show how an internal rating system in a bank can be organized in order to rate creditors systematically. We suggest adopting a two-tier rating system. First, an obligor rating that can be easily mapped to a default probability bucket. Second, a facility rating that determines the loss parameters in case of default, such as (i) “loss given default” (LGD), which depends on the seniority of the facility and the quality of the gurantees, and (ii) “usage given default” (UGD) for loan commitments, which depends on the nature of the commitment and the rating history of the borrower.

Empirical tests of boundary conditions for CBOE options

Abstract In this paper the lower boundary conditions for traded options are derived and subjected to empirical testing. Two hypotheses are formulated based on the theoretical conditions and tested on data on call options traded on the Chicago Board Options Exchange. The first hypothesis argues that the stock and options markets are well synchronized so that simultaneous closing prices are within the theoretical boundaries. The evidence in the ex post test is inconsistent with this hypothesis. The second hypothesis claims the markets to be efficient. The tests are directed toward the question whether arbitrage profits could actually have been made on the Exchange by exploiting the violations of the dominance condition. The tests, carried out as ex ante tests, indicate that positive profits could have been exploited on the average, but the magnitude of the average was small relative to the dispersion of the yields.

A contingent claim analysis of a regulated depository institution

Abstract A model of financial intermediation to determine the market value of bank equity, deposits and deposit insurance is developed. The implicit equilibrium interest rate on deposits is derived and analyzed. Three types of risks are considered in the model: interest-rate risk, financial risk and default risk. The effect of different regulatory measures, such as capital adequacy, reserve and liquidity requirements, deposit insurance and interest rate ceilings is analyzed and their impact on the bank behavior is also assessed. Moreover, we investigate the interactions among these measures to determine which are dominant under alternative circumstances, and which are redundant.

Hedging Volatility in Foreign Currencies

Wide jluctuations in exchange rates are o f continual concern to traders and investors in foreign currencies. One o f the main issues not addressed by current risk management techniques is the phenomenon that volatility itsey gyrates widely. These changes in volatility may have an adverse efect on risk management strategies, which typically assume that volatility is constant. In this article we propose the creation o f new derivative instruments, namely, options (and futures) on volatility, designed to help in the management of unexpected volatility jluctuations. We describe the construction o f a volatility index to refrct the volatility o f a spec& exchange rate. The index, based on implied volatilities3om currency options, will serve as an underlying instrument on which derivative products can be written. We discuss a variety ofissues in valuing such options, and present a simple binomial example.

THE RISK-RETURN RELATIONSHIP AND STOCK PRICES

According to the current state of knowledge in finance, the expected rate of return adjusted for risk is independent of the stock price. The basic proposition of the capital asset pricing model (CAPM) is that the expected rate of return for each security is a function of the “risk†of that security, and that this risk is measured by the contribution of the security to the variability of the market portfolio. The implication of the CAPM is that knowing the price of a security perse will add nothing to predicting its expected rate of return.

The interaction between the financial and investment decisions of the firm: the case of issuing warrants in a levered firm

Abstract In this paper we analyze the value of warrants and stock in a firm that has both equity and debt in its capital structure, and assume that the proceeds from exercising the warrants are reinvested in the firm. The effect of potential future scale investment on the firms stock, debt and warrants are investigated. The warrants in our analysis are European. In this framework there are two sources for the nonstationarity of the rate of return on any kind of financial claim; the first stems from changes in the firms capital structure, and the second is due to the potential scale increase in the value of the firm should the warrants be exercised. The effect of a potential transfer of wealth from equity-holders to debt-holders at time of exercise is analyzed. When warrants are exercised and the proceeds are reinvested in a scale expansion project, the probability of default may decrease. It follows, therefore, that ceteris paribus debt is likely to appreciate in value at the expense of equity. The method of pricing by arbitrage, as proposed by Harrison and Kreps is used to derive values for claims on the firm. Problems associated with measuring the volatility of equity are discussed.

An economic assessment of capital requirements in the banking industry

Abstract The paper uses a two-period model to evaluate the capital requirements imposed on banks. In particular, the interaction between capital requirements and other regulatory measures is analyzed. It is shown that, in a competitive set-up, there is a tradeoff between the capital ratio and the liquidity of assets.

Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants

We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firms asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firms assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firms total assets. Since stock prices are observable, then for practical purposes, traders prefer to use the stock as the underlying instrument, we concentrate on valuation models in terms of the stock price. Second, we derive an exact solution for the valuation in terms of the stock price of (i) a European call option on the stock of a levered firm, i.e. a European compound call option on the total assets of the firm, (ii) an equity warrant for an all-equity firm, and (iii) an equity warrant for a firm financed by equity and debt. Unfortunately, to compute these solutions we need to specify the function of the stock price in terms of the firms assets value. In general we are unable to specify this expression, but we propose tight bounds for the value of these options which can be easily computed as a function of the stock price. Our results provide useful extensions of the Black-Scholes model.