In a tractable stochastic volatility model, we identify the price of the smile as the price of the unspanned risks traded in SPX option markets. The price of the smile reflects two persistent volatility and skewness risks, which imply a downward sloping term structure of low-frequency variance...

We develop a conditional capital asset pricing model in continuous-time that allows for stochastic beta exposure. When beta co-moves with market variance and the stochastic discount factor (SDF), beta risk is priced, and the expected return on a stock deviates from the security market line. The...

In this paper we develop a discrete-time pricing model for European options where the log-return of the underlying asset is subject to discontinuous regime shifts in its mean and/or volatility which follow a Markov chain. The model allows for multiple regime shifts whose risk cannot be hedge out...

This paper shows that the standard textbook formula for computing the present value of a future random cash flow – the discounted expected value – is formally incorrect and can generate significant errors when used to compute present values. The correct present value method is provided as...

Stocks with large increases in call implied volatilities over the previous month tend to have high future returns while stocks with large increases in put implied volatilities over the previous month tend to have low future returns. Sorting stocks ranked into decile portfolios by past call...

We document that the implied volatility skew of S&P 500 index puts is non-decreasing in the disaster index and risk-neutral variance, contrary to the implications of a broad class of no-arbitrage models. The key to the puzzle lies in recognizing that, as the disaster risk increases, customers...

The area of derivatives is arguably the most fascinating area within financial economics during the past thirty years. This chapter reviews the evolution of derivatives contract markets and derivatives research over the past thirty years. The chapter has six complementary sections. The first...

This paper reviews recent developments in macro and finance on the relationship between financial risk and the real economy. We focus on three specific topics: the term structure of uncertainty, time variation - and specifically the long-term decline - in the variance risk premium, and time...

We measure investors' short- and long-term stock-return expectations using both options and survey data. These expectations at different horizons reveal what investors think their own short-term expectations will be in the future, or forward return expectations. While contemporaneous short-term...

Single period risks acceptable to the market at zero cost are modeled by a convex set of random variables leading to bid and ask prices that are trade size dependent. The theory of nonlinear expectations is employed to construct dynamically consistent sequences of bid and ask unit size prices...