- Text
- Returns,
- Monotonicity,
- Strict,
- Increasing,
- Binary,
- Strategies,
- Statistics,
- Stocks,
- Prices,
- Underlying,
- Stochastic,
- Discount,
- Factor

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

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, **the**n **the** current price is more heavily **discount**ed, resulting in a higher**expected** 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 **Monotonicity**A 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 ei**the**r α = 0 or β = ∞.The following proposition relates weak monotonicity to monotonicity **of** **option** **expected**returns, **and**, analogous to **the** unconditional case, characterizes weak monotonicity in terms**of** 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 ei**the**r 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 ei**the**r α = 0 orβ = ∞.7

iv) (Coval **and** Shumway (2001)) R ( (S T − K) +) **and** R ( (K − S T ) +) are strictly increasingin K.Pro**of**. 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 **the**y apply to testing weak monotonicity**of** m with respect to **the** S&P 500 index.Note that (iv) holds regardless **of** **the** stock-pricedistribution function F .B. Strict **Monotonicity**We 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 **of****the** 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 **of**7 A derivation is provided in **the** appendix.8

- Page 1: Monotonicity of the stochastic disc
- Page 4 and 5: 500 results, our empirical analysis
- Page 6 and 7: I. Characterization of SDF Monotoni
- Page 10 and 11: consumption in the two middle state
- Page 12 and 13: The first two examples below presen
- Page 14 and 15: tional to the slope of m (), as is
- Page 16 and 17: strict monotonicity because it requ
- Page 18 and 19: eturns, and the sample reduces to 1
- Page 20 and 21: skewness of option returns (most OT
- Page 22 and 23: money.Assuming only differentiabili
- Page 24 and 25: against the alternative hypothesis
- Page 26 and 27: in an increasing across strike grou
- Page 28 and 29: Further compounding the heterogenei
- Page 30 and 31: is equivalent to0 > E ( mS T 1 {ST
- Page 32 and 33: The inverse of the expected return
- Page 35 and 36: Using integration by parts:Cov (Y,
- Page 37 and 38: Equation B3 is the skewness adjuste
- Page 39 and 40: where we have used, for any k ∈ {
- Page 41 and 42: Brown, David, and Jens Carsten Jack
- Page 43 and 44: Huang, C., and R. Litzenberger, 198
- 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