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5

The fourier transform is $$\hat{w}_c= \int_{-\infty}^\infty (\sqrt{y}-K)^+ e^{-i\phi y}dy = \int_{K^2}^\infty (\sqrt{y}-K) e^{-i\phi y}dy$$ Now do a change of variable with $t=\sqrt{i\phi y}$ and solve the resulting integral.

4

Assume you have a consumption $c$ and an asset with the payoff $x$. Cochrane states that if you add "a little bit of this asset" in your portfolio first you care about the correlation between the payoff of the asset and consumption and ONLY then you care about variance. How you can see this? Let's assume that you slighlty change your portfolio by $\xi$ (i.e....

3

We should be able to replicate the payoff exactly in each of the two regions $S_{T}\leq k_{1}$ and $S_{T}\geq k_{2}$. From the first, $$a_{0}+a_{1}S_{T}+a_{3}(k_{2}-S_{T}) =S_{T}$$ so, matching coefficients, $a_{0}+a_{3}k_{2}=0$ and $a_{1}-a_{3}=1$. From the second, $$a_{0}+a_{1}S_{T}+a_{2}(S_{T}-k_{1})=P_{0}$$ so, matching coefficients, $a_{0}-a_{2}k_{1}=P_{... 3 That simply means that a bond pays one unit of the currency in any state (regardless what happens in the future, i.e. there is no default risk about the payoff of a bond). So you will receive 1 in the next period (regardless what you paid for it). Of course, today you probably pay less than 1 due to time value of money... 2 A bond repays its notional face value (plus interest sometimes), not the original purchase price. Do not assume the the price you pay for a bond is its face value. Sometimes a law or a regulation (pretty useless, in my humble opinion:) does require that a bond newly issued in primary market be sold at exactly 100% par price (face value). Then the coupon ... 2 Somewhere must be a little error, here I used$r=0.02$and$\sigma=0.25$. In black you have the payoff and in red the current price of the portfolio. Note that as the time to maturity decreases, the red line converges towards the black line. In grey and and yellow (on the secondary axis), you can see the individual call option prices which form your ... 2 You can check my answer to this question for general details on how to solve this kind of problem. Let$C_X(S_T)$and$P_Y(S_T)$be a call and a put option with strikes$X$and$Y$respectively, then:$\$\begin{align} (\text{i}) \quad V_T &= (S_T-A)1_{\{A\leq S_T\leq B\}}+(B-A)1_{\{S_T>B\}} \\ &=(S_T-A)1_{\{S_T\geq A\}}+(B-S_T)1_{\{S_T\geq B\}} \\...

1

For question (i), you simply buy one EU call with strike A and sell one EU call with strike B - this is called a bull call spread. Try using the put-call-parity to construct the corresponding bull put spread yourself.

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