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# Monotonicity of the stochastic discount factor and expected option ...

Monotonicity of the stochastic discount factor and expected option ...

## I. Characterization

I. Characterization of SDF MonotonicityIn this section we consider three progressively stronger senses in which the SDF is decreasingin the price (or return) of an asset, and characterize these in terms of expected returnsof trading strategies.We show that each form of SDF monotonicity is equivalent to themonotonicity of expected returns of a set of asset/option trading strategies in a shift parameter(typically a strike price).Our main contribution in this section is to characterizewhat we term ”strict monotonicity” (a strictly decreasing SDF as a function of return).Inparticular, we characterize the set of trading strategies with expected return increasing instrike under strict monotonicity. In addition to long call and put option positions, 3 this setof trading strategies includes butterfly spreads and binary cash-or-nothing calls and puts.Because many mainstream asset pricing models predict strictly monotonicity of the SDF,this provides a useful complement to the test of Coval and Shumway (2001).Fixing some payout date T , an SDF, or pricing kernel, m is a (time-T measurable) randomvariable such that the current price p (X) of any time-T payout X satisfies p (X) = E (Xm).If the number of states is finite, then the state-s value of the SDF, m s , can be interpretedas the per-unit-probability price of a claim to one unit of state-s consumption.Technicalitiesaside, it is well known that a strictly positive SDF exists if and only if there isno arbitrage. 4For example, if there exists an agent with utility over current and futureconsumption given by U (c 0 , c T ) = u 0 (0, c 0 ) + u (T, c T ), then the first-order condition for optimalityimplies that there exists an SDF equal to the agent’s marginal rate of substitution,m = u c (T, c T ) /u c (0, c 0 ), and risk aversion is equivalent to m decreasing in c T .With somepreference and distributional assumptions, standard aggregation arguments imply monotonicityof m in aggregate time-T consumption and time-T wealth. 53 Coval and Shumway (2001) show that what we term ”weak monotonicity” (negative slopes of all truncatedregressions of the SDF on return) is equivalent to call and put option expected returns increasing instrike.4 In the finite-state setting, the result follows directly from the separating hyperplane theorem. Note thatthroughout we are considering the case of unconstrained trading (no short-selling constraints, for example).5 The conditions on preferences are generally strong. Gorman aggregation allows heterogeneity essentiallyonly with respect to additive shifts consumption with homothetic preferences, and scalar multiples of5

All asset-pricing models are essentially models of the SDF, and its monotonicity propertiesdetermine the risk premia properties of all traded assets.Below we define three formsof SDF monotonicity in terms of the slopes of various ”regressions” of a terminal stock priceon the SDF and then characterize each in terms of the monotonicity of expected optionstrategy returns with respect to shifts in the strike price.This characterization forms thebasis of the empirical tests in Section III.Define the (gross) expected return function Rby R (X) = E (X) /E (mX) for any (time-T ) payout X.Note that R (1) is the riskless return.Fix throughout some stock S with time-T price (with dividends reinvested) S T .Thedistribution function of S Tis denoted F (), and is assumed throughout (for simplicity) tohave a density function which is strictly positive on (0, ∞).We use the common notationx + = max (0, x) for any x ∈ R.Finally, we define the risk-neutral measure Q as usual viadQdP =mE (m) ,(that is, risk-neutral probabilities are obtained by scaling the SDF-weighted true probabilities). 6For comparison with our later results, the following proposition gives some well-knownequivalent characterizations of a positive risk premium (expected return in excess of theriskless return).Proposition 1 The following are equivalent:i) Cov (S T , m) < 0.ii) R (S T ) > R (1) .iii) E (S T ) > E Q (S T ) .iv) R (S T − K) and R (K − S T ) are strictly increasing in K.Proof. See the appendix.consumption with translation-invariant preferences. A deterministic investment opportunity set and homotheticpreferences imply a deterministic consumption to wealth ratio, from which we get monotonicity withrespect to wealth. See Section C for more detailed conditions under which an SDF is monotonic in an assetreturn.6 In the finite-state setting, with state-s probability π s , the state-s risk-neutral probability isπ s m s / ∑ π s m s , which represents the forward price per unit payout in state s.6

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