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

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

## To compute

To compute the variance, we first show thatCov (p (j) , p (i)) = − 1 V ar (p (j)) i ≠ j, i, j ∈ {1, . . . , k} .k − 1This is done by defining the random variable⎧⎪⎨ p (i) with probability (k − 1) /kx =⎪⎩ p (j) with probability 1/kNote that x is uniformly distributed on {1, . . . , k} and is independent of p (j).ThereforeUsing V ar (L t ) = V ar0 = Cov (p (j) , x) = k − 1k Cov (p (j) , p (i)) + 1 V ar (p (j)) .k(k∑jX j), we compute the variance:j=1V ar( k∑j=1jX j)=k∑j=1k∑ijCov (p (j) , p (i))i=1{ k∑= V ar (p (1))j=1⎧⎨ k∑= V ar (p (1))⎩j=1= V ar (p (1)) k2 (k + 1)12j 2 − 1k − 1j 2 − 1k − 1}k∑ ∑ijj=1i≠j⎛[ k∑] 2⎝ j − ∑j=1j⎞⎫⎬j 2 ⎠⎭Finally substituteV ar (p (1)) = 1 kk∑j=1(j − k + 1 ) 2= 1 (k + 1) (k − 1) ,2 1237

where we have used, for any k ∈ {1, 2, . . .},k∑j 2 =j=1k (k + 1) (2k + 1).638

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