The information frown in option prices

In the S&P 500 options market, the information content of implied volatilities differs by strike in a frown pattern that is a rough mirror image of the implied volatility smile. Implied volatilities calculated from moderately high strike price options are both unbiased and efficient predictors of future volatility. Implied volatilities calculated from low and at-the-money strikes are biased and less efficient. This bias cannot be explained by market imperfections but is consistent with the hedging pressure argument of Bollen and Whaley [J. Finan. 59 (2004) 711] and Ederington and Guan [J. Derivat. 10 (2002) (Winter) 9]. We also find that a serious estimation bias results when the relations are estimated using panel data.     

On the modelling of option prices

Dilip B Madan discusses how even good financial models often fail to perform when applied to real economic data. To overcome market volatility, he suggests, the right model must successfully reduce dimension.     

Tight bounds on American option prices

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.     

Risk sharing in a financial market with endogenous option prices

This article investigates a financial market in which investors may trade in risk-free bonds, stock and put options written on the stock. In each period, stock and option prices are simultaneously determined by market clearing. While the introduction of put options will decrease the systematic risk in the financial market, it will increase the price of risk. Investors with mean-variance preferences will generally hold portfolios containing the primary asset and the put option and may use the option to increase the risk in their wealth position in exchange for higher returns. Aggregate wealth is unaffected by an option market when there are no spillover effects on stock prices, and it is shown that short selling of options will increase the volatility of individual wealth positions. Investors with erroneous beliefs may on average be better off not trading in put options.     

Partial differential equations for Asian option prices

In this article, we consider fixed and floating strike European style Asian call and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelièvre have derived partial differential equations with one state variable, with the stock price as numeraire, for the option prices. In this paper, we derive a whole family of partial differential equations, each with one state variable with the stock price as numeraire, from which Asian options can be priced. Any one of these partial differential equations can be transformed into any other. This family includes four partial differential equations which have a particularly simple form including the three found by Rogers and Shi, Vecer, and Dubois and Lelièvre. Our analysis includes the case of a dividend yield; we also include the case of in progress Asian options with floating strike, whereby we discuss the new equation proposed by Vecer, which uses the average asset as numeraire. We perform an error analysis on the four special partial differential equations and Vecer’s new equation and find that their truncation errors are all of the same order. We also perform numerical comparisons of the five partial differential equations and conclude, as expected, that Vecer’s equations and that of Dubois and Lelièvre do better when the volatility is low but that with higher volatilities the performance of all five equations is similar. Vecer’s equations are unique in possessing a certain martingale property and as they perform numerically well or better than the others, must be considered the preferred choice.     

Assessing models of individual equity option prices

Review of Quantitative Finance and Accounting - This study sheds light on risk exposures of cooperative banks in Austria, Germany and Italy. We investigate how major risk elements of banks in these...     

Bounds for option prices and expected payoffs

This article sharpens Lo's upper bounds for option prices using an alternative approach with the assumption that the underlying asset price is continuously distributed. The increased sharpness is obtained using additional information about the distribution of the underlying assets. It is shown in this article that a large portion of Lo's upper bounds is the maximum of our bounds over all possible distributions.     

Biases in option prices : Evidence from the foreign currency option market

This paper examines a European call model of option pricing over a data set which does not suffer from the early exercise problems that have plagued earlier studies of call options on common stocks. We specifically examine a data set of American call prices on spot foreign exchange for which it is plausible to apply an adjusted version of the Garman-Kohlhagen (1983) and Grabbe (1983) European call option model. We make adjustments for interest rate risk and find that the model is nearly unbiased in the valuation of foreign currency options. We conclude that the Geske-Roll (1984) conjecture about dividend uncertainty creating biases in stock option prices holds analogously in the foreign currency option market. Interest rate differential risk (analogous to risky dividends) thus appears to be an important element in the valuation of foreign currency options.     

Exact solutions for bond and option prices with systematic jump risk

A variety of realistic economic considerations make jump-diffusion models of interest rate dynamics an appealing modeling choice to price interest-rate contingent claims. However, exact closed-form solutions for bond prices when interest rates follow a mixed jump-diffusion process have proved very hard to derive. This paper puts forward two new models of interest-rate dynamics that combine infrequent, discrete changes in the interest-rate level, modeled as a jump process, with short-lived, mean reverting shocks, modeled as a diffusion process. The two models differ in the way jumps affect the central tendency of interest rates; in one case shocks are temporary, in the other shocks are permanent. We derive exact closed-form solutions for the price of a discount bond and computationally tractable schemes to price bond options.     

Anticipated information releases reflected in call option prices

This study captures the ex ante information content of a financial reporting event (the annual earnings announcement) by examining the behavior of call option prices on dates leading up to and passing through the disclosure date. This approach differs from most previous empirical security price research which has been ex post in nature. The hypothesis that investors anticipate that the future release of annual earnings numbers will affect security prices is empirically confirmed. In particular, systematic changes in variance rates implied by the Black-Scholes option pricing model are demonstrated.     

Arbitrage bounds in markets with noisy prices and the puzzle of negative option prices implicit in bonds

The term structure of interest rates has occupied economists for many years. This is testimony to the importance of the term structure (or, equivalently, valuation operator) and to the difficulty of obtaining reliable estimates of it. Until recently, it was not possible to reconcile the theoretical existence of a multiplicity of valuation operators in an incomplete bond market and the empirical nonexistence of even a single one. MacKay and Prisman [Estimating valuation operators in incomplete market with noises: Can noise complete the market. Working paper, 1996] tackle this problem. In this paper, an amendment to and a generalization of their methodology is presented. The amendment preserves the linearity of valuation operators and allows use of efficient linear programming techniques. Further, it facilitates an admissible consideration of the puzzle of negative option prices embedded in bonds. The empirical results presented in this paper were obtained using data on extendable Government of Canada bonds. The results indicate that although the gap between the lowest and the highest prices assigned to a cash flow by different valuation operators is significant, it is not sufficient to resolve the above-mentioned puzzle.     

Insider employee stock option trading and stock prices

The question of which factors determine corporate bonds pricing is investigated by analysing the spreads of eurobonds issued by major G-10 companies during the 1991–2001 period. Three main results emerge from the analysis. First, bond ratings appear as the most important determinant of yield spreads, with investors’ reliance on rating agencies judgments increasing over time. Second, the primary market efficiency and the expected secondary market liquidity are not relevant explanatory factors of spreads cross-sectional variability. Finally, rating agencies adopt a different, ‘through the cycle’, evaluation criteria of default risk with respect to the forward looking one adopted by bond investors.     

Minimum option prices under decreasing absolute risk aversion

We establish bounds on option prices in an economy where the representative investor has an unknown utility function that is constrained to belong to the family of nonincreasing absolute risk averse functions. For any distribution of terminal consumption, we identify a procedure that establishes the lower bound of option prices. We prove that the lower bound derives from a particular negative exponential utility function. We also identify lower bounds of option prices in a decreasing relative risk averse economy. For this case, we find that the lower bound is determined by a power utility function. Similar to other recent findings, for the latter case, we confirm that under lognormality of consumption, the Black Scholes price is a lower bound. The main advantage of our bounding methodology is that it can be applied to any arbitrary marginal distribution for consumption. This revised version was published online in June 2006 with corrections to the Cover Date.     

Market illiquidity and bounds on European option prices

The paper analyses the impact of illiquidity of a stock paying no dividends on the pricing of European options written on that stock. In particular, it is shown how illiquidity generates price bounds on an option on this stock, even in the absence of other imperfections, such as transaction costs and trading constraints, or the assumption of stochastic volatility. Moreover, price bounds are shown to be asymmetric with respect to the option price under perfect liquidity. This fact explains, under some conditions, the appearance of a smile effect when the implied volatility is estimated from the mid-quote.     

Term structure movements implicit in Asian option prices

In this paper we implement dynamic term structure models that adopt bonds and Asian options in the estimation process. The goal is to analyse the pricing and hedging implications of term structure movements when options are (or are not) included in the estimation process. We investigate how options affect the shape, risk premium and hedging structure of the dynamic factors. We find that the inclusion of options affects the loadings of the slope and curvature factors, and considerably changes the risk premium and hedging structure of all dynamic factors.     

Recovering risk aversion from option prices and realized returns

A relationship exists between aggregate risk-neutral and subjectiveprobability distributions and risk aversion functions. We empiricallyderive risk aversion functions implied by options prices andrealized returns on the S&P500 index simultaneously. Theserisk aversion functions dramatically change shapes around the1987 crash: Precrash, they are positive and decreasing in wealthand largely consistent with standard assumptions made in economictheory. Postcrash, they are partially negative and partiallyincreasing and irreconcilable with those assumptions. Mispricingin the option market is the most likely cause. Simulated tradingstrategies exploiting this mispricing show excess returns, evenafter accounting for the possibility of further crashes, transactioncosts, and hedges against the downside risk.     

The early exercise premium in American put option prices

This study estimates the value of the early exercise premium in American put option prices using Swedish equity options data. The value of the premium is found as the deviation of the American put price from European put-call parity, and in addition a theoretical estimate of the premium is computed. The empirically found premium is also used in a modified version of the control variate approach to value American puts. The results indicate a substantial value of the early exercise premium, where the premium derived from put-call parity is higher than the theoretical premium. The premium also increases with moneyness and time left to expiration, while the effect of interest rate and volatility depends on the moneyness of the option. The modified control variate technique works reasonably well relative to the theoretical models. In particular, for deep in-the-money options, this technique is superior.     

Fast approximations of bond option prices under CKLS models

A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.