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A few questions and my answers, to be sure I understand everything

Question 1

Suppose A and B agree on a forward contract: maturity

  • $T = 1Y$
  • spot at $t=0$: $S_0=100$
  • forward price $K = 120$.

Suppose B wants to sell this contract to C at $t = 6M$ (so that the contract will be between A and C). Suppose that $S_{6M}=135$, then C will buy the contract at 135-120=15 ?(so C buys the mark-to-market forward price)

Question 2

The value of a forward contract at $t$ = 0 equals

$$V_{0}=S_{0}-Kexp(-rT)$$

with $K=S_{0}exp(rT)$.

At $t = 6M$, the value is

$$V_t=S_t-Kexp(-r(T-t))=S_t-S_{0}exp(rT)exp(-r(T-t))=S_t-S_0exp(rt).$$

So again C will buy the contract at this price (?)

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  • $\begingroup$ If there are no dividends and the risk-free rate $r$ is contant, your last equations are spot on although in that case $S_0 \exp(rt) \ne S_0 \exp(rT) = 120$, hence $V_t \ne 135-120=15$. Put differently, it costs nothing to set up a "fresh" forward contract between two parties here $A$ and $B$ ($V_0=0$ at $t=0$). But as time passes $0 < t \leq T$ the contract will gain/lose value, hence $B$ will either win/lose money when selling it to $C$ depending on the MTM of the contract at $t$ ($V_t \ne 0$ in general). Note that for $A$ nothing changes. $\endgroup$
    – Quantuple
    Commented Aug 10, 2017 at 13:17
  • $\begingroup$ I don't understand your sentence: "Put differently [...] here A and B". Also, it seems that both reasoning are right. So C will buy the contract from B at which price ? The price given by Question 1 or 2 ? $\endgroup$
    – glork
    Commented Aug 10, 2017 at 17:01
  • $\begingroup$ See @Gordon complete answer sorry if I got you confused. $\endgroup$
    – Quantuple
    Commented Aug 10, 2017 at 17:04

1 Answer 1

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Assuming zero dividend and a constant interest rate $r$, the 1y forward price is then \begin{align*} 120 = K = S_0 e^r = 100\, e^r. \end{align*} Consequently, $e^r = 1.2$. The fair value of the forward contract, at 6M, is given by \begin{align*} e^{0.5 r} E\left(\frac{S_{1Y}-120}{e^{r}} \mid \mathcal{F}_{6M} \right) &= e^{0.5 r}\left(\frac{S_{6M}}{e^{0.5 r}} -\frac{120}{e^r}\right)\\ &= S_{6M} - 120 e^{-0.5 r}.\tag{1} \end{align*}

Then, for your Question 1, C is willing to buy the forward contract with the price 135-120 = 15, given that the value in $(1)$ is greater than 15.

For your second question, since \begin{align*} S_t - S_0 e^{rt} &= S_{6M} - S_0 e^{0.5r}\\ &= S_{6M} - S_0 e^r e^{-0.5r}\\ &=S_{6M}- 120 e^{-0.5 r}, \end{align*} C is indifferent for buying the forward contract with this price.

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  • $\begingroup$ Q1: for me it's a bit weird. C should buy the contract to the FV no ?(If it was a call option: for me B will sell it to C at the fair value ie at $Call_t$.) Do you think C will make the calculus and compare the FV and the mark-to-market price ? Q2: what do you mean by " C is indifferent..." I'm still confused. To sum up, why do you need to compare ? Thanks $\endgroup$
    – glork
    Commented Aug 18, 2017 at 9:11
  • $\begingroup$ For Q1, as the value of the forward contract is strictly greater than 15, C is willing to buy at 15. For Q2, as the value of the forward contract is the same as the offering price, there is no immediate profit for C unless C needs to hedge, then it is indifferent. $\endgroup$
    – Gordon
    Commented Aug 18, 2017 at 12:59

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