Ian Martin

What Is the Expected Return on the Market

I derive a lower bound on the equity premium in terms of a volatility index, SVIX, that can be calculated from index option prices. The bound implies that the equity premium is extremely volatile and that it rose above 20% at the height of the crisis in 2008. The time-series average of the lower bound is about 5%, suggesting that the bound may be approximately tight. I run predictive regressions and find that this hypothesis is not rejected by the data, so I use the SVIX index as a proxy for the equity premium and argue that the high equity premia available at times of stress largely reflect high expected returns over the very short run. I also provide a measure of the probability of a market crash, and introduce simple variance swaps, tradable contracts based on SVIX that are robust alternatives to variance swaps.

On the Valuation of Long-Dated Assets

I show that the pricing of a broad class of long-dated assets is driven by the possibility of extraordinarily bad news. This result does not depend on any assumptions about the existence of disasters, nor does it apply only to assets that hedge bad outcomes; indeed, it applies even to long-dated claims on the market in a lognormal world if the market’s Sharpe ratio is higher than its volatility, as appears to be the case in practice.

The Quanto Theory of Exchange Rates

We present a new, theoretically motivated, forecasting variable for exchange rates that is based on the prices of quanto index contracts, and show via panel regressions that the quanto forecast variable is a statistically and economically significant predictor of currency appreciation. We also test the quanto variables ability to forecast differential currency appreciation out of sample, and find that it outperforms predictions based on uncovered interest parity, on purchasing power parity, and on a random walk.

What is the Expected Return on a Stock

We derive a formula for the expected return on a stock in terms of the risk-neutral variance of the market and the stocks excess risk-neutral variance relative to the average stock. These quantities can be computed from index and stock option prices; the formula has no free parameters. The theory performs well empirically both in and out of sample. Our results suggest that there is considerably more variation in expected returns, over time and across stocks, than has previously been acknowledged.

Options and the Gamma Knife

Steve Ross, a towering figure in modern finance, died last spring. Since then, numerous memorials and special events have taken place, honoring his many achievements in economics and finance. This article takes a fresh look at one of Ross’s earliest papers on options, in which he showed that they have the ability to “complete” an incomplete market. The example in that 1976 paper showed how a set of options with different strikes can subdivide the range for the possible future price of a given stock to create Arrow–Debreu securities, each of which pays off on the occurrence of a single future stock price. In this article, the author reviews Ross’s paper and shows how it leads to the Breeden–Litzenberger methodology for extracting the risk-neutral density from options market prices. He then extends Ross’s one-stock model to the case with multiple risk factors and shows how options written on a single linear combination of their prices can still complete the higher dimensional market.