A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13     

Option Prices, Implied Price Processes, and Stochastic Volatility

This paper characterizes all continuous price processes that are consistent with current option prices. This extends Derman and Kani (1994), Dupire (1994, 1997), and Rubinstein (1994), who only consider processes with deterministic volatility. Our characterization implies a volatility forecast that does not require a specific model, only current option prices. We show how arbitrary volatility processes can be adjusted to fit current option prices exactly, just as interest rate processes can be adjusted to fit bond prices exactly. The procedure works with many volatility models, is fast to calibrate, and can price exotic options efficiently using familiar lattice techniques.     

Dynamic copula models for the spark spread

We propose a non-symmetric copula to model the evolution of electricity and gas prices by a bivariate non-Gaussian autoregressive process. We identify the marginal dynamics as driven by normal inverse Gaussian processes, estimating them from a series of observed UK electricity and gas spot data. We estimate the copula by modeling the difference between the empirical copula and the independent copula. We then simulate the joint process and price options written on the spark spread. We find that option prices are significantly influenced by the copula and the marginal distributions, along with the seasonality of the underlying prices.     

Optimal stopping under model uncertainty and the regularity of lower Snell envelopes

The analysis of American options in incomplete markets has motivated the development of robust versions of the classical Snell envelopes: The cost of superhedging an American option is characterized by the upper Snell envelope, while the infimum of the arbitrage prices is given by the lower Snell envelope. Lower Snell envelopes also appear in the problem of optimal stopping under model uncertainty. In this paper we focus on the lower Snell envelope. We construct a regular version of this stochastic process. To this end, we apply results due to Dellacherie and Lenglart on the regularization of stochastic processes and 𝒯-Systems.     

Random walk versus breaking trend in stock prices: Evidence from emerging markets

This paper investigates whether stock-price indexes of seventeen emerging markets can be characterized as random walk (unit root) or mean reversion processes. We implement a test that can account for structural breaks in the underlying series and is more powerful than standard tests. We find that for fourteen countries, stock prices exhibit structural breaks. Furthermore, for ten countries, the null hypothesis of a random walk can be rejected at the one or 5% significance level. Our results indicate that ignoring structural breaks that arise from the liberalization of emerging markets can lead to incorrect inference that these indices are characterized by random walks, and are consistent with the points made by Bekaert et al. [J. Int. Money Finan. 21 (2002) 295]. Our findings hold true regardless of whether stock indexes are denominated in US dollar terms, in local currencies terms, or in real terms.     

A MULTIFACTOR MODEL OF THE QUALITY OPTION IN TREASURY FUTURES CONTRACTS

The values of quality options in Treasury futures contracts are set relative to the prices of all coupon bonds in their respective deliverable sets. As a result, any model used to value the quality option should set its price relative to the set of observed bond prices. This requirement rules out the use of most simple equilibrium models that represent all bond prices in terms of a finite number of state variables. We use the two-factor Heath-Jarrow-Morton model, which permits claims to be priced relative to observable bond prices, to investigate the potential value of the quality option in Treasury bond and note futures. We show that the quality option has significantly more value in a two-factor interest rate economy than in a single-factor economy, and that ignoring it could lead to significant mispricing.     

Asymmetric temporary and permanent stock-price innovations

This paper estimates a nonlinear, structural bivariate threshold model in order to identify independent temporary and permanent stock-price innovations in positive-return and negative-return regimes. The model finds that temporary innovations account for as much as 68% of the innovations after negative returns and less than 4% of the innovations after positive returns. This paper shows that the negative-return regime is also the high-volatility regime, a result that links the excess-volatility and asymmetric-volatility literatures. This link provides evidence that temporary stock-price innovations are due primarily to time-varying expected returns in an efficient market and not to a market inefficiency.     

How Naïve Is the Market’s Use of Firm‐Specific Earnings Information?

Recent studies suggest the apparent delay in the stock-price response to earnings announcements (i.e., post-earnings announcement drift) is caused by investors who underestimate the autocorrelation of seasonally-differenced earnings (persistence). I present results that suggest: (1) a firm's future persistence is predictable on the basis of its past persistence; (2) the immediate stock-price response to earnings is positively related to historical persistence; (3) post-earnings-announcement drift is independent of historical persistence; and (4) consistent with (2) and (3), the difference between a firm's current observed persistence and that implied in stock prices is independent of its historical persistence. These results extend prior research by demonstrating that investors are aware not only that seasonally-differenced earnings are autocorrelated, but that investors recognize firm-specific differences in the magnitude of the autocorrelation.     

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.     

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.