WARRANT PRICING USING OBSERVABLE VARIABLES

The classical warrant pricing formula requires knowledge of the firm value and of the firm‐value process variance. When warrants are outstanding, the firm value itself is a function of the warrant price. Firm value and firm‐value variance are then unobservable variables. I develop an algorithm for pricing warrants using stock prices, an observable variable, and stock return variance. The method also enables estimation of firm‐value variance. A proof of existence of the solution is provided.     

Adjusting for Non-Linear Age Effects in the Repeat Sales Index

A true constant quality real estate price index should measure the general change in price level free from any change in quality over time. In recent years, the repeat-sales method has been widely used to construct constant quality property price indices. Since buildings depreciate over time, a simple repeat-sales index would underestimate the growth in property prices. The major problem of controlling the effects of age constant in a repeat-sales model arises from the exact multicollinearity between the age variable and the time dummy variables. In this study, we derive a solution that is theoretically sound and practical by allowing the age effects to be non-linear. In case of leasehold properties, we further incorporated interest rates into the model because the effects of age on real estate prices depend theoretically on interest rates. A sample of residential units in Hong Kong sold more than once from Quarter 2 of 1991 to Quarter 1 of 2001 (more than 11,000 repeat sales pairs) are used for the empirical analysis.     

Valuation of Callable Warrants

A simple valuation model for callable warrants is derived and tested. The model is expressed in closed form except for one term which can be evaluated numerically. Predictions of 78 warrant prices are compared to market prices and the average error is -.224 percent. By contrast, the Black-Scholes model applied to the same warrants produces an average error of 31.44 percent. Thus the callability feature cannot safely be ignored in determining warrant values.     

Warrant price responses to credit spread changes: Fact or fiction?

We use a new approach to analyze the relationship between warrant prices and issuers’ credit spreads. This approach allows us to gain insights into whether issuers use their credit risk systematically to increase their profits. In a post‐Lehman sample, we find strong support for a systematic use since issuers decrease prices less after cred it spread increases than they increase prices after credit spread decreases. Credit spread decreases are accompanied by price increases on several successive days. This sluggish adjustment in prices can be explained by the fact that retail investors’ purchase decisions depend on product prices.     

CONVERTIBLE BOND VALUATION: AN EMPIRICAL TEST

This research applies the options pricing model to the valuation of convertible bonds. A numeric algorithm is used to obtain theoretical values for a sample of 103 convertible bond issues. When market prices are compared with model valuations, the means are not significantly different, and 90 percent of model predictions are within 10 percent of market values. As a further test, the sample is divided on the basis of whether the model prices are (1) greater or (2) less than market prices. Returns are compared over a subsequent three-year holding period. The results indicate that without risk adjustment, the returns for the subsample identified by the model as “undervalued” (model prices exceed market prices) are significantly greater than returns for the subsample identified by the model as “overvalued” (market prices exceed model prices).     

On market price of risk in Asian capital markets around the Asian flu

This study investigates a contemporaneous relationship between realized market risk premia, and conditional variance and covariance in nine Asian markets and the US. The time period for this study is before, during, and after the Asian financial crisis. A contemporaneous state-dependent capital asset pricing model (CAPM) that allows for negative and positive market prices of variance and covariance risk is investigated. In the light of significant upstate and downstate reward to local and world variance risk for all markets and all periods, we conclude that a market return-generating process is a piecewise function of local and world variance over time. Furthermore, a cross-sectional analysis of upstate and downstate market prices of variance and covariance risk indicates that reward to risk is a mix of reward to local and world variance, depending on the ever-changing correlation with the world market. Our findings are consistent with the one-factor conditional international CAPM.     

How Crashes Develop: Intradaily Volatility and Crash Evolution

This paper explores whether affine models with volatility jumps estimated on intradaily S&P 500 futures data over 1983 to 2008 can capture major daily outliers such as the 1987 stock market crash. Intradaily jumps in futures prices are typically small; self‐exciting but short‐lived volatility spikes capture intradaily and daily returns better. Multifactor models of the evolution of diffusive variance and jump intensities improve fits substantially, including out‐of‐sample over 2009 to 2016. The models capture reasonably well the conditional distributions of daily returns and realized variance outliers, but underpredict realized variance inliers. I also examine option pricing implications.     

Empirical Testing of Real Option-Pricing Models

This research is the first to examine the empirical predictions of a real option-pricing model using a large sample of market prices. We find empirical support for a model that incorporates the option to wait to develop land. The option model has explanatory power for predicting transactions prices over and above the intrinsic value. Market prices reflect a premium for the option to wait to invest that has a mean value of 6% in our sample. We also estimate implied standard deviations for individual commercial property prices ranging from 18 to 28% per year.     

Consistent Variance Curve Models

We introduce a general approach to model a joint market of stock price and a term structure of variance swaps in an HJM-type framework. In such a model, strongly volatility-dependent contracts can be priced and risk-managed in terms of the observed stock and variance swap prices. To this end, we introduce equity forward variance term structure models and derive the respective HJM-type arbitrage conditions. We then discuss finite-dimensional Markovian representations of the fixed time-to-maturity forward variance swap curve and derive consistency results for both the standard case and for variance curves with values in a Hilbert space. For the latter, our representation also ensures non-negativity of the process. We then give a few examples of such variance curve functionals and briefly discuss completeness and hedging in such models. As a further application, we show that the speed of mean reversion in some standard stochastic volatility models should be kept constant when the model is recalibrated.     

Consistent pricing and hedging for a modified constant elasticity of variance model

Abstract

This paper considers a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP). It is shown that, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach.

Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the GOP. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding ‘risk neutral’ prices.     

Optimal investment strategies for general utilities under dynamic elasticity of variance models

This paper studies the optimal investment strategies under the dynamic elasticity of variance (DEV) model which maximize the expected utility of terminal wealth. The DEV model is an extension of the constant elasticity of variance model, in which the volatility term is a power function of stock prices with the power being a nonparametric time function. It is not possible to find the explicit solution to the utility maximization problem under the DEV model. In this paper, a dual-control Monte-Carlo method is developed to compute the optimal investment strategies for a variety of utility functions, including power, non-hyperbolic absolute risk aversion and symmetric asymptotic hyperbolic absolute risk aversion utilities. Numerical examples show that this dual-control Monte-Carlo method is quite efficient.     

What Is the Expected Return on a Stock?

We derive a formula for the expected return on a stock in terms of the risk‐neutral variance of the market and the stock's excess risk‐neutral variance relative to that of the average stock. These quantities can be computed from index and stock option prices; the formula has no free parameters. The theory performs well empirically both in and out of sample. Our results suggest that there is considerably more variation in expected returns, over time and across stocks, than has previously been acknowledged.     

THE SPEED OF ADJUSTMENT OF PRICES TO PRIVATE INFORMATION: EMPIRICAL TESTS

We estimate speeds of adjustment of individual stock prices to private information using daily data. We use a model in which private information gives rise to return variance and private information decays linearly over time. We find that, on average, about 85 percent to 88 percent of private information is incorporated into prices within one trading day, with variation depending upon the stock's trading volume and whether the stock is listed on an exchange. The findings support strong form market efficiency.     

The weekend effect on the distribution of stock prices: Implications for option pricing

Evidence of weekend effects on the distribution of security returns suggests that returns are generated by a process operating closer to trading time rather than calendar time. In contrast, accumulation of interest over the weekend follows a calendar-time process. Since both the variance of returns and the interest rate are important parameters of the Black-Scholes option pricing model, this paper suggests that the model be stated to account for this by utilizing a trading-time variance and a calendar-time interest rate. Empirical evidence indicates that this allows the model to better explain market option prices.     

Order Form and Information in Securities Markets

This paper examines the effects of price-contingent orders on security prices. We show that a market maker who knows the type and composition of trades will set larger spreads and adjust prices faster than if price-contingent orders were not allowed. Because traders have rational expectations over the book, we demonstrate that uncertainty over order type reduces the variance of prices but with a corresponding loss in price informativeness. We also show that the sequence property of price-contingent orders increases the probability of large price movements. This distinction between variance and episodic price volatility has important policy implications.     

Bond Pricing with a Time‐Varying Price of Risk in an Estimated Medium‐Scale Bayesian DSGE Model

New Keynesian model in which households have Epstein–Zin preferences with time‐varying risk aversion and the central bank has a time‐varying inflation target can match the dynamics of nominal bond prices in the U.S. economy well. The model generates a large steady‐state term spread and its fitting errors for bond yields are comparable to those obtained from a nonstructural three‐factor model, and one‐third smaller than in models with a constant inflation target or risk aversion. Including data on interest rates has large effects on variance decompositions, making investment technology shocks much less important than found in other recent papers.     

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.     

Rational expectations and the role of monetary policy: A generalization

This paper investigates the effects of introducing the following two alterations into a multimarket, partial information, rational expectations model; (1) individuals in any market may sample currently more prices than just the current price on their own market; (2) they choose the amount of their current information so as to minimize the sum of the costs of getting information and of being off their full information demand and supply curves. Under those circumstances any change in the variance of aggregate excess demand shocks (whether caused by real or monetary elements) affects the equilibrium level of information in the economy, in addition to its other direct effects previously recognized in the literature. Among the issues dealt with are the implications of these alterations for the following: (1) the Lucas hypothesis on the slope of the Phillips curve; (2) the optimal monetary variance and the optimal money feedback rule; (3) the effect of the level of information and of aggregate variance on the distribution of relative prices and related issues.     

Options on realized variance by transform methods: a non-affine stochastic volatility model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Heston's model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.     

The Dynamics of Gasoline Prices: Evidence from Daily French Micro Data

Using millions of individual gasoline prices collected at a daily frequency, we examine the speed at which market prices of refined oil are transmitted to retail gasoline prices in France. For that, we estimate a reduced‐form model of state‐dependent pricing where thresholds triggering price changes are allowed to vary over time and depend on the duration since the last price change. We find that the degree of pass‐through of wholesale prices to retail gasoline prices is on average 0.77 for diesel and 0.67 for petrol and depend on local market characteristics. The duration for a shock to be fully transmitted into prices is about 10 days. There is no significant asymmetry in the transmission of wholesale price to retail prices. Finally, the duration since the last price change has a significant effect on thresholds triggering price changes but a large variance of idiosyncratic shocks on thresholds is also crucial to replicate the size distribution of price changes.