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.

/pdf/expected-stock-returns-and-variance-risk-premia-2qqzsna3ss.pdf

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

/pdf/variance-risk-premiums-2r5d548tv2.pdf

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 Log Contract

We consider the problem of finding a model-free uper bound on the price of a forward-start straddle with payoff |FT2 −FT1 |. The bound depends on the prices of vanilla call and put options with maturities T1 and T2, but does not rely on any modelling assumptions concerning the dynamics of the underlying. The bound can be enforced by a super-replicating strategy involving puts, calls and a forward transaction. We find an upper bound, and a model which is consistent with T1 and T2 vanilla option prices for which the model-based price of the straddle is equal to the upper bound. This proves that the bound is best possible. For lognormal marginals we show that the upper bound is at most 30% higher than the Black-Scholes price. The problem can be recast as finding the solution to a Skorokhod embedding problem with non-trivial initial law so as to maximise E|Bτ − B0|.

/pdf/robust-bounds-for-forward-start-options-46rmux6gi9.pdf

Robust bounds for forward start options

We measure the skew risk premium in the equity index market through the skew swap. We argue that just as variance swaps can be used to explore the relationship between the implied variance in option prices and realized variance, so too can skew swaps be used to explore the relationship between the skew in implied volatility and realized skew. Like the variance swap, the skew swap corresponds to a trading strategy, necessary to assess risk premia in a model-free way. We find that almost half of the implied volatility skew can be explained by the skew risk premium. We provide evidence that skew and variance premia are manifestations of the same underlying risk factor in the sense that strategies designed to exploit one of the risk premia but to hedge out the other make zero excess returns.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2181340_code367152.pdf?abstractid=1571700

The Skew Risk Premium in the Equity Index Market

In dealership markets disclosure of size and price of details of public trades is typically incomplete. We examine whether full and prompt disclosure of public-trade details improves the welfare of a risk-averse investor. We analyze a model of dealership market where a market maker first executes a public trade and then offsets her position by trading with other market makers. We distinguish between quantity risk and price revision risk. We show that if the market maker learns some information about the motive behind public trade, neither regime is unambiguously welfare superior. This is because greater transparency improves quantity risk sharing but worsens price revision risk sharing.

Anthony Neuberger

Papers

Option Prices, Implied Price Processes, and Stochastic Volatility

The Log Contract

Robust bounds for forward start options

The Skew Risk Premium in the Equity Index Market

Trade disclosure regulation in markets with negotiated trades