Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

We find that the difference between implied and realized variances, or the variance risk premium, is able to explain more than fifteen percent of the ex-post time series variation in quarterly excess returns on the market portfolio over the 1990 to 2005 sample period, with high (low) premia predicting high (low) future returns. The magnitude of the return predictability of the variance risk premium easily dominates that afforded by standard predictor variables like the P/E ratio, the dividend yield, the default spread, and the consumption-wealth ratio (CAY). Moreover, combining the variance risk premium with the P/E ratio results in an R 2 for the quarterly returns of more than twenty-five percent. The results depend crucially on the use of “modelfree”, as opposed to standard Black-Scholes, implied variances, and realized variances constructed from high-frequency intraday, as opposed to daily, data. Our findings suggest that temporal variation in risk and risk-aversion both play an important role in determining stock market returns.

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Expected Stock Returns and Variance Risk Premia

We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the risk-neutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premiums on five stock indexes and 35 individual stocks. (JEL G10, G12, G13) It has been well documented that return variance is stochastic. When investing in a security, an investor faces at least two sources of uncertainty, namely the uncertainty about the return as captured by the return variance, and the uncertainty about the return variance itself. It is important to know how investors deal with the uncertainty in return variance to effectively manage risk and allocate assets, to accurately price and hedge derivative securities, and to understand the behavior of financial asset prices in general. We develop a direct and robust method for quantifying the return variance risk premium on an asset using the market prices of options written on this asset. Our method uses the notion of a variance swap, which is an over-thecounter contract that pays the difference between a standard estimate of the realized variance and the fixed variance swap rate. Since variance swaps cost zero to enter, the variance swap rate represents the risk-neutral expected value of the realized variance. We show that the variance swap rate can be synthesized accurately by a particular linear combination of option prices. We propose to

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Variance Risk Premiums

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.

Option Prices, Implied Price Processes, and Stochastic Volatility

http://faculty.haas.berkeley.edu/ross_levine/Papers/2003_JFE_LawEndFin.pdf

Law, endowments, and finance

The virtue of an American option is that it can be exercised at any time. This right is particularly valuable when there is model uncertainty. Yet almost all the extensive literature on American options assumes away model uncertainty. This paper quantifies the potential value of this flexibility by identifying the supremum on the price of an American option when no model is imposed on the data, but rather any model is required to be consistent with a family of European call prices. The bound is enforced by a hedging strategy involving these call options which is robust to model error.

On the value of being American

The UK Pension Protection Fund (PPF), established by the 2004 Pensions Act to protect beneficiaries of defined benefit pension schemes when the sponsor becomes insolvent and the scheme is underfunded, is required to finance itself through a levy on participating schemes. In July 2005, the PPF issued a consultative document setting out its proposal for the structure of the levy. In this paper, we provide a critique of the proposal and, in particular, its heavy reliance on securing levy income from the weakest schemes. We propose an alternative structure for the levy that recognises the limits on capacity to pay and also mitigates some other undesirable features of the proposal.

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Pricing Pension Insurance: The Proposed Levy Structure for the Pension Protection Fund*

In the presence of proportional transactions costs, the tightest bounds that can be imposed on the price of a call option when the asset price follows a geometric diffusion are those imposed by static portfolio strategies. The price of a call is bounded above by the value of the asset and below by its intrinsic value. However, with a pure jump process it is possible to obtain much tighter arbitrage bounds on the value of a contingent claim, which converge to the no-transaction-cost valuation as transaction costs become small.

Option Replication with Transaction Costs: An Exact Solution for the Pure Jump Process

We identify a global risk factor that drives the cross-section of volatility excess returns in the foreign exchange market. We show that a zero-cost strategy that buys forward volatility agreements with downward sloping volatility curves and sells those with upward slopes - the volatility carry strategy - earns on average 5.15% per month. When we form slope-sorted portfolios, the covariation with volatility carry returns fully explains the cross-sectional variation of our portfolios. The lower the slope of the volatility curve, the more the forward volatility agreement is exposed to volatility carry risk. A standard no-arbitrage model of exchange rates with two types of factor - a set of country specifi c factors and a global one - provides intuition for the findings. The state variables determining the exposure to the global risk factor are empirically related to squared deviations of changes in economic growth. In the cross-section, the returns to volatility carry strategy are only weakly related to traditional currency risk factors, like carry, global imbalance, global volatility and global liquidity risk.

Anthony Neuberger

Papers

On the value of being American

Pricing Pension Insurance: The Proposed Levy Structure for the Pension Protection Fund*

Option Replication with Transaction Costs: An Exact Solution for the Pure Jump Process

The Cross-Section of Currency Volatility Premia

The cross-section of currency volatility premia