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

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

## The intuition for

The intuition for the equivalence of (i) and (ii) is that if higher stock payouts are associatedwith less valuable payout states, in the sense that a linear regression of S T against mhas a negative slope, then the current price is more heavily discounted, resulting in a higherexpected return.The equivalence to (iii) follows because replacing the true probability foreach state by the forward price per unit payout in that state reduces the expected payoutif payouts are higher in low-value states.Finally, the equivalence to (iv) follows becauseincreased leverage in a long stock position (and decreased leverage in a short-stock/lendingposition) increases expected returns if and only if the risk premium of the stock is positive.A. Weak MonotonicityA stronger condition than negative covariance, or a negative regression slope, is what we termweak monotonicity with respect to S T , which we define by negative slopes for all regressionstruncated to a half-line.Definition 2 The SDF m is weakly monotonic with respect to S T ifCov (S T , m | S T ∈ [α, β]) < 0 for all 0 ≤ α < β with either α = 0 or β = ∞.The following proposition relates weak monotonicity to monotonicity of option expectedreturns, and, analogous to the unconditional case, characterizes weak monotonicity in termsof conditional risk premia and conditional risk-neutral expectations:Proposition 3 (weak monotonicity) The following are equivalent:i) m is weakly monotonic with respect to S T .ii) R ( (S T 1 {ST ∈[α,β]})> R 1{ST ∈[α,β]})for all 0 ≤ α < β with either α = 0 or β = ∞.iii) E (S T | S T ∈ [α, β]) > E Q (S T | S T ∈ [α, β]) for all 0 ≤ α < β with either α = 0 orβ = ∞.7

iv) (Coval and Shumway (2001)) R ( (S T − K) +) and R ( (K − S T ) +) are strictly increasingin K.Proof. See the appendix.Part (ii) says that weak monotonicity is equivalent to strict positivity of conditional riskpremia; that is, positive risk premia for all asset-or-nothing binary calls (puts) relative tocash-or-nothing binary calls (puts).Part (iv) is the Coval and Shumway (2001) characterizationin terms of traded option strategies, which they apply to testing weak monotonicityof m with respect to the S&P 500 index.Note that (iv) holds regardless of the stock-pricedistribution function F .B. Strict MonotonicityWe define strict monotonicity as monotonicity of the conditional expectation, or nonlinearregression of m on S T :.Definition 4 The SDF m is strictly monotonic with respect to S T if m (s) = E (m |S T = s)is strictly decreasing in s.While it might appear that weak monotonicity is close to strict monotonicity, the followingsimple example shows that weak monotonicity can hold despite the absence of strictmonotonicity for intermediate stock-price values. 7It is a highly stylized representation ofthe behavior of the SDF projected on S&P 500 returns.Example 5 Suppose a discrete setting with four equally-likely and equally-spaced possibleterminal stock prices: s 1 < s 2 < s 3 < s 4 . Then m is weakly monotonic if and only ifm (s 1 ) > m (s 4 ) and m (s 2 ) , m (s 3 ) ∈ (m (s 1 ) , m (s 4 )).That is, the price of consumptionin the lowest stock-price state exceeds that in the highest stock-price state, and the prices of7 A derivation is provided in the appendix.8

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