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

I. Characterization **of** SDF **Monotonicity**In this section we consider three progressively stronger senses in which **the** SDF is decreasingin **the** price (or return) **of** an asset, **and** characterize **the**se in terms **of** **expected** returns**of** trading strategies.We show that each form **of** SDF monotonicity is equivalent to **the**monotonicity **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 set**of** 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) r**and**omvariable 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, **the**n **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 **the**re isno arbitrage. 4For example, if **the**re 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 ), **the**n **the** first-order condition for optimalityimplies that **the**re 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, st**and**ard aggregation arguments imply monotonicity**of** 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 **the**orem. 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 homo**the**tic preferences, **and** scalar multiples **of**5

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 forms**of** SDF monotonicity in terms **of** **the** slopes **of** various ”regressions” **of** a terminal stock priceon **the** SDF **and** **the**n characterize each in terms **of** **the** monotonicity **of** **expected** **option**strategy returns with respect to shifts in **the** strike price.This characterization forms **the**basis **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** **the**riskless 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.Pro**of**. See **the** appendix.consumption with translation-invariant preferences. A deterministic investment opportunity set **and** homo**the**ticpreferences 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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- Page 41 and 42: Brown, David, and Jens Carsten Jack
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- Page 45 and 46: Shive, S., and T. Shumway, 2006, Is
- Page 47 and 48: Table IIAverage Return Differences
- Page 49 and 50: Table IVOption Elasticities for AIG