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2

You ought to compare the $t$-values of two self-financing strategies, under the assumption that there exists a risk-free money market account and that the dividend is deterministic but proportional to the random stock price. Strategy 1 - Entering a forward contract At inception ($t=0$), you do not pay anything by definition, $\Pi_1(0)=0$ At maturity ($t=T)$,...

4

Option 1 is not a steepener trade. It is an outright bearish trade that the 5y5y forward rate will move upwards. Option 2 is a steepener trade, if the dv01 is equal on the 5yr and 10yr legs. Ignoring discounting , you would need to pay fixed on 50mm 10yr versus receiving on 100mm 5yr to make it duration neutral, and thus a curve trade.

2

I guess generally what ATM means depends a lot on asset classes. FX vols are quoted as ATM DNS (delta neutral straddles). This in itself can be Spot, Forward, Spot premium adjusted, forward premium adjusted with the following formulas retrieved from the working paper FX volatility smile construction : However, based on your wording I assume you think 50D ...

3

We assume a single-curve environment. Let us recall that a floating LIBOR payment fixed at time $T$ and paid at time $T^\prime$ can be written in terms of zero-coupon bonds: $$L(t,T,T^\prime):=\frac{1}{T^\prime-T}\left(\frac{P(t,T)}{P(t,T^\prime)}-1\right)$$ Let $\mathcal{T}:=\{T_0,\dots,T_n,\dots,T_m\}$ be a schedule such that a spot 5y swap starts fixing ...

1

I am assuming that in option 1 you are entering into payer swap. If the curve is flat then option a) and b) are the same because you will get the same cashflows in both cases. Why? In option b) both the floating legs and fixed legs on 10Y swap and 5Y will cancel for the first 5 years i.e. the cashflows will be opposite sign, effectively making it 5y5y swap. ...

-1

When referring to a long straddle, atm means the 50 delta strike.

0

For straddles ATM usually implies 0 delta. In general, ATM is determined by the market conventions in question.

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