Liuren Wu

Time-Changed Levy Processes and Option Pricing ⁄

Abstract The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-changed Levy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

A Tale of Two Indices

Pricing options involves a volatility input. Since volatility is not directly observable and it also varies stochastically over time, volatility risk is a significant concern for most option traders. Not surprisingly, then, there has always been considerable interest in the CBOEs VIX volatility index, although relatively little trading of derivatives based on it. The original VIX was a weighted average of Black-Scholes implied volatilities, which made it (in principle) a good estimator of future volatility, but hard to replicate with traded securities. The original VIX has been recently replaced by a new formula that is not model dependent. In this article, Carr and Wu review the old VIX (now called the VXO) and the new VIX and present a wide variety of results on their behavior. An interesting difference is that the new VIX, squared, is a good hedge for realized variance. This makes the new VIX a better underlier for volatility derivative contracts, which are now being launched into the marketplace by the CBOE. The article examines the performance of the two indices as forecasts of realized volatility, and shows how the VIX responds around an uncertainty diminishing information event, meetings of the Federal Open Market Committee. Carr and Wu also obtain interesting results on the markets volatility risk premium from direct measurement of risk neutral volatility in the VIX.

Asset Pricing under the Quadratic Class

We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector. We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application. We consider asset pricing in general and derivative pricing in particular under the quadratic class. We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi-closed form. We further illustrate how the quadratic model and the transform methods can be applied to more general settings.

The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments

This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock index.

Accounting for Biases in Black-Scholes

Prices of currency options commonly differ from the Black-Scholes formula along two dimensions: implied volatilities vary by strike price (volatility smiles) and maturity (implied volatility of at-the-money options increases, on average, with maturity). We account for both using Gram-Charlier expansions to approximate the conditional distribution of the logarithm of the price of the underlying security. In this setting, volatility is approximately a quadratic function of moneyness, a result we use to infer skewness and kurtosis from volatility smiles. Evidence suggests that both kurtosis in currency prices and biases in Black-Scholes option prices decline with maturity.

Predictable changes in yields and forward rates

We consider the patterns in the predictability of interest rates expectations hypothesis (EH), and attempt to account for them with affine models. We make the following points: (i) Discrepancies in the data from the EH take a particularly simple form with forward rates: as theory suggests, the largest discrepancies are at short maturities. (ii) Reasonable estimates of one-factor Cox-Ingersoll-Ross models imply regressions on the opposite side of the EH than we see in the data: regression slopes are greater than one (iii) Multifactore affine models can nevertheless approximate both departures from the EH and other properties of interest rates.

Are Interest Rate Derivatives Spanned by the Term Structure of Interest Rates

This is an investigation of whether the same finite-dimensional system spans both interest rates (the yield curve) and interest rate options (the implied volatility surface). The options market is found to exhibit factors seemingly independent of the underlying yield curve. While three common factors are adequate to capture the systematic movement of the yield curve, three additional factors are needed to capture the movement of the implied volatility surface. Simulation analysis confirms the robustness of these findings.

Design and Estimation of Multi-Currency Quadratic Models

To simultaneously account for the properties of interest-rate term structure and foreign exchange rates within an arbitrage-free framework, we propose a multi-currency quadratic model with an (m+n) factor structure. The m factors model the term structure of interest rates in both countries. The n factors capture the portion of the exchange rate movement that is independent of the term structure of either country. We estimate a series of multi-currency quadratic models using U.S. and Japanese LIBOR and swap rates and the exchange rate between the two countries.

A No-Arbitrage Analysis of Macroeconomic Determinants of the Credit Spread Term Structure

From a large array of economic and financial data series, this paper identifies three fundamental risk dimensions underlying an economy: inflation, real output growth, and financial market volatility. Furthermore, through a no-arbitrage model, the paper links the dynamics and market pricing of the three risk dimensions to the term structure of U.S. Treasury yields and corporate bond credit spreads. Model estimation shows that positive inflation shocks increase Treasury yields and widen credit spreads on corporate bonds across all maturities and credit-rating classes. Positive real output growth shocks also increase Treasury yields, but they suppress the credit spreads at low credit-rating classes, thus generating negative correlations between interest rates and credit spreads. The financial market volatility factor has a small and transient effect on the Treasury yield curve, but it exerts a strongly positive and persistent effect on the credit spread term structure. The paper provides a robust and internally consistent method for extracting systematic economic information from a large array of noisy observations and establishing how different risk dimensions of the fundamental economy interact with interest rate and credit risk.

Jumps and Dynamic Asset Allocation

This paper analyzes the optimal dynamic asset allocation problem in economies with infrequent events and where the investment opportunities are stochastic and predictable. Analytical approximations are obtained, with which a thorough comparative study is performed on the impacts of jumps upon the dynamic decision. The model is then calibrated to the U.S. equity market. The comparative analysis and the calibration exercise both show that jump risk not only makes the investors allocation more conservative overall but also makes her dynamic portfolio rebalancing less dramatic over time.