https://polymer.bu.edu/hes/cox79py538.pdf

Option pricing: A simplified approach☆

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. ONE OPTION-PRICING PROBLEM that has hitherto remained unsolved is the pricing of a European call on a stock that has a stochastic volatility. From the work of Merton [12], Garman [6], and Cox, Ingersoll, and Ross [3], the differential equation that the option must satisfy is known. The solution of this differential equation is independent of risk preferences if (a) the volatility is a traded asset or (b) the volatility is uncorrelated with aggregate consumption. If either of these conditions holds, the risk-neutral valuation arguments of Cox and Ross [4] can be used in a straightfoward way. This paper produces a solution in series form for the situation in which the stock price is instantaneously uncorrelated with the volatility. We do not assume that the volatility is a traded asset. Also, a constant correlation between the instantaneous rate of change of the volatility and the rate of change of aggregate consumption can be accommodated. The option price is lower than the BlackScholes (B-S) [1] price when the option is close to being at the money and higher when it is deep in or deep out of the money. The exercise prices for which overpricing by B-S takes place are within about ten percent of the security price. This is the range of exercise prices over which most option trading takes place, so we may, in general, expect the B-S price to overprice options. This effect is exaggerated as the time to maturity increases. One of the most surprising implications of this is that, if the B-S equation is used to determine the implied volatility of a near-the-money option, the longer the time to maturity the lower the implied volatility. Numerical solutions for the case in which the volatility is correlated with the stock price are also examined. The stochastic volatility problem has been examined by Merton [13], Geske [7], Johnson [10], Johnson and Shanno [11], Eisenberg [5], Wiggins [16], and

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1540-6261.1987.tb02568.x

The Pricing of Options on Assets with Stochastic Volatilities

The stochastic behavior of common stock variances : value, leverage and interest rate effects

We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed-form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model.

https://www.jstor.org/stable/pdfplus/2329288.pdf

A Simple Approach to Valuing Risky Fixed and Floating Rate Debt

Most investment expenditures have two important characteristics: First, they are largely irreversible; the firm cannot disinvest, so the expenditures are sunk costs. Second, they can be delayed, allowing the firm to wait for new information about prices, costs, and other market conditions before committing resources. An emerging literature has shown that this has important implications for investment decisions, and for the determinants of investment spending. Irreversible investment is especially sensitive to risk, whether with respect to future cash flows, interest rates, or the ultimate cost of the investment. Thus if a policy goal is to stimulate investment, stability and credibility may be more important than tax incentives or interest rates. This paper presents some simple models of irreversible investment, and shows how optimal investment rules and the valuation of projects and firms can be obtained from contingent claims analysis, or alternatively from dynamic programming. It demonstrates some strengths and limitations of the methodology, and shows how the resulting investment rules depend on various parameters that come from the market environment. It also reviews a number of results and insights that have appeared in the literature recently, and discusses possible policy implications.

https://papers.ssrn.com/sol3/Delivery.cfm/nber_w3307.pdf?abstractid=226671&mirid=1

Irreversibility, Uncertainty, and Investment

http://quantlabs.net/academy/download/free_quant_instituitional_books_/[Journal%20of%20Financial%20Economics,%20Geske]%20The%20Valuation%20of%20Compound%20Options.pdf

The valuation of compound options

Contrary to economic theory and common sense, stock returns are negatively related to both expected and unexpected inflation We argue that this puzzling empirical phenomenon does not indicate causality

Instead, stock returns are negatively related to contemporaneous changes in expected inflation because they signal a chain of events which results in a higher rate of monetary expansion Exogenous shocks in real output, signalled by the stock market, induce changes in tax revenue, in the deficit, in Treasury borrowing and in Federal Reserve “monetization” of the increased debt Rational bond and stock market investors realize this will happen They adjust prices (and interest rates) accordingly and without delay

Although expected inflation seems to have a negative effect on subsequent stock returns, this could be an empirical illusion, since a spurious causality is induced by a combination of: (a) a reversed adaptive inflation expectations model and (b) a reversed money growth/stock returns model

If the real interest rate is not a constant, using nominal interest proxies for expected inflation is dangerous, since small changes in real rates can cause large and opposite percentage changes in stock prices

The Fiscal and Monetary Linkage between Stock Returns and Inflation

This paper applies the technique for valuing compound options to the risky coupon, bond problem. A formula is derived which contains n-dimensional multivariate normal intecjrals. It is shown that, for some compound option problems, the special correlation structure allows an application of an integral reduction which may simplify the numerical evaluation. The effects of various indenture restrictions on the formula are discussed, and a new formula for evaluating subordinated debt is presented.

The Valuation of Corporate Liabilities as Compound Options

An analytic solution to the American put problem is derived herein. The hedge ratio and other derivatives of the solution are presented. The formula derived implies an exact duplicating portfolio for the American put consisting of discount bonds and stock sold short. The formula is extended to consider put options on stocks paying cash dividends. A polynomial expression is developed for evaluating these formulae. Values and hedge ratios for puts on both dividend and nondividend paying stocks are calculated, tabulated, and compared with values derived by numerical integration and binomial approximation. As with European options, evaluating an analytic formula is more efficient than approximating the stock price process or the partial differential equation by binomial or finite difference methods. Finally, applications of this American put solution are discussed.

The American Put Option Valued Analytically

The purpose of this paper is to compare a variety of approximation techniques for valuing contingent contracts when analytic solutions do not exist. The comparison is made with respect to the differences in both the approximation theory and the efficiency of the computation algorithms. The focus of the computational comparison is upon binomial and finite difference methods applied to option valuation models with one stochastic variable. However, many of the results would generalize to pricing corporate securities, and also to certain aspects of problems involving multiple stochastic variables.

Robert Geske

Papers

The valuation of compound options

The Fiscal and Monetary Linkage between Stock Returns and Inflation

The Valuation of Corporate Liabilities as Compound Options

The American Put Option Valued Analytically

Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques