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How do I calculate the delivery price for a forward contract?

In the time interval $[0,t]$ the interest rate is $r_1$ and in the time interval $[t,T]$ the interest rate is $r_2$. Determine the delivery price $F(0,T)$ for a forward contract on an underlying asset whose price at time $0$ is equal to $S(0)$.

idknuttin
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    My guess would be $F(0,T) = S(0) \exp(r_1 (t - 0) + r_2 (T - t)) = S(0) \exp((r_1 - r_2) t + r_2 T)$. – Andrew Feb 28 '16 at 20:58
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    under the usual set of assumptions, the present value of the forward price is just the spot price. So if $d_i$ is the discount factor associated with rate $r_i$ over their respective periods $F=\frac {S}{d_1d_2}$ – lulu Feb 28 '16 at 20:59
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    @AndrewMiloradovsky You're assuming continuous compounding? – BCLC Mar 01 '16 at 18:07

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Denote $r_2^C$ and $r_1^C$ to be the continuously compounded interest rate equivalent of $r_2$ and $r_1$, resp.

Discount $F(0,T)$ from time $T$ to time $t$:

$$F(0,T)e^{-r_2^C(T-t)} \tag{*}$$

Discount $F(0,T)e^{r_2^C(T-t)}$ from time $t$ to time $0$:

$$F(0,T)e^{-r_2^C(T-t)}e^{-r_1^C(t-0)}$$

The above term is supposed to be equal to $S_0$.

Thus we have:

$$F(0,T) = S_0e^{r_2^C(T-t)}e^{r_1^C(t-0)}$$

I think $(*) = S_t$ and $S_T = F(0,T)$

BCLC
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    $e^{r_2(T-t)}$ this means compounded continuously correct? We can assume this since the question isn't specific on what type of interest to use, or is continuous interest always used when finding the delivery price of a forward contract? – idknuttin Mar 01 '16 at 17:21
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    @idknuttin oh shoot. you're right. edited – BCLC Mar 01 '16 at 18:06
  • I am confused by the notation, they give us $r_1$ and $r_2$, so you are saying $r_{1}^C$ and $r_{2}^C$ are the equivalent continuous interest rates to the interest rates the question gives us? – idknuttin Mar 01 '16 at 18:17
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    @idknuttin ummmmmm yes? – BCLC Mar 01 '16 at 18:20