Think of the FRA as two parts:
Fixed: Payoff at 0.5 = 0.3% x 1000000 x 0.25
Floating: Payoff at 0.5 = -r(0.25,0.5) x 1000000 x 0.25
Your portfolio will replicate both parts.
Fixed part: This is straightforward, as P(0,0.5) has payoff of $100 at 0.5 then simply purchase/go long (0.3% x 1000000 x 0.25)/100 of P(0,0.5) to replicate this payoff.
Floating part: P(0,0.25) will have value $100 at 0.25 which will grow to $100(1+0.25r(0.25,0.5)) by 0.5. So selling/going short (1000000)/100 of P(0,0.25) will be worth:
-1000000(1+r(0.25,0.5)0.25)= -r(0.25,0.5) x 1000000 x 0.25 – 1000000
at 0.5. So, we are nearly there, I will leave you to determine how to adjust the portfolio to get a payoff of -r(0.25,0.5) x 1000000 x 0.25….
Your final portfolio will replicate the payoff of the FRA and so the current value of the FRA will be the current value of the portfolio. If the fixed rate is chosen to be the forward rate then the value is zero, any other rate will lead to a positive or negative value.
c) and d) adapt your portfolio from part b) and use that the value of the portfolio is equal to the value of the FRA.
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