The recovery theorem (RT) is a tool developed by the author to determine the predictive content of market prices. The RT enables the user to disentangle the future equity return distribution and pricing kernel from option prices, which has been considered impossible using only state prices. Other RT applications include forecasting long-run equity premiums, forecasting the probability of market upheaval, and testing the efficient market hypothesis.
What’s Inside?
State prices are a function of risk aversion and the natural return probability. The author’s central result is that under certain assumptions, state prices can predict the probability and pricing kernel uniquely.
The volatility surface is a 3-D graph that shows option-implied volatilities, maturity, and moneyness—that is, the relationship between the strike price of an option and the market price of its underlying. Using the volatility surface information of S&P 500 Index options, the author finds contingent forward prices. Next, using his recovery theorem (RT), he calculates risk aversion and the market’s anticipated future return probabilities. He applies the RT to data from a selected date, extends his results to multinomial processes, and discusses applications. He concludes by discussing extensions and limitations.
How Is This Research Useful to Practitioners?
In fixed-income markets, forward rates help practitioners infer the market’s prediction of future spot rates. By contrast, such forward rate predictions are missing in equity markets, but backward-looking historical data analysis and surveys are common.
The author studies option prices to determine market expectations. This step initially seems counterintuitive because Black–Scholes option prices do not depend on the underlying’s expected return. Furthermore, forward rates include market participants’ risk aversion embedded in the risk premium. So, how can option prices determine the market’s expected future returns? The author presents the RT as his solution.
This research could be useful in passive equity management, risk management, and asset allocation. Equity prices may become more transparent to practitioners, possibly influencing equity and derivatives portfolio strategies. Consultants and researchers can use the RT to forecast market volatility, the probability of upheavals, and the long-term equity risk premium for determining optimal asset allocation.
One limitation is that the author uses data for only one selected date to demonstrate how the RT can help determine the market’s anticipated distribution of returns. More data would illuminate how predicted theoretical results compare with actual results and further determine the usability of the RT.
How Did the Author Conduct This Research?
The author uses a mathematical approach to calculate risk aversion and the natural probability distribution of returns from option prices. He uses these assumptions:
- Markov process. State variables have underlying Markov stochastic processes whose state depends on the previous state alone; probabilities are time constant.
- Transition independence. The kernel is a function of the final state, normalized by the starting state. A representative agent with a separable intertemporal additive utility function satisfies this critical assumption.
- Irreducible pricing matrix. A nonnegative square matrix is irreducible if and only if its digraph is strongly connected. For the pricing matrix, this means the state variable can shift to any state from any other state.
- No arbitrage, complete markets, and discrete time. Discrete time enables a finite number of states. With no arbitrage, fundamental asset pricing theorems ensure nonnegative Arrow–Debreu (AD) state prices.
AD securities are idealized securities paying one unit when a specific state occurs. Risk-neutral probabilities can be viewed as forward AD prices. Under no arbitrage, the risk-neutral probabilities sum to 1.
The core of the problem is that just knowing the risk-neutral probabilities does not allow practitioners to determine the unique values of the market’s prediction of returns and the risk aversion. The author applies the restricting conditions described and uses the Perron–Frobenius theorem, which states, in part, that for an irreducible matrix, there exists a nonnegative eigenvector whose eigenvalue equals the matrix’s spectral radius. The problem of finding a unique solution is then reduced to finding these eigenvalues and eigenvectors, thus enabling the author to recover the natural probabilities.
Abstractor’s Viewpoint
The author’s thought-provoking research counters prevailing beliefs. Empirical testing against actual data would illuminate the RT’s potential. Extensions to other markets and research into different assumptions and conditions could deepen our understanding of how, when, and where the RT works, which may help stretch the bounds of how we use market price information.