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Suppose you have a lease agreement where the functional/domestic currency is RUB and the currency on which the lease is written USD. Let $S$ be the USD/RUB exchange rate (# of rubles per 1 dollar). The lessee pays $NS_{t}$ RUB at each time $t$ the lease payment is due, where $N$ is some fixed amounts in USD. This exposure to $S_{t}$ creates an embedded derivative which must be "bifurcated" and valued separately on the lessee's balance sheet.

The derivative can be viewed as a short position on a USD/RUB forward contract (sell USD, buy RUB), where the strike is determined by the forward rate curve at inception of the lease agreement. This is straight-forward to value periodically on future dates.

Now suppose this agreement also has a floor $\underline{S}$ and a cap $\overline{S}$.

The cap is an asset to the lessee, since it limits the downside associated with a weakening RUB, while the floor is a liability since it limits the upside gain associated with a strengthening RUB. In other words, in terms of embedded derivatives, the cap is a long position in a USD/RUB call option struck at $\overline{S}$ and the floor is a short position in a USD/RUB put option struck at $\underline{S}.$

My question is this: Should we value this embedded derivative as the sum of the values of the cap + floor + FX forward? This seems logical, and what you would do if this arrangement was an actual OTC derivative contract. However, being that this is an embedded derivative, is it appropriate to value the optionality features like they were options? Something keeps nagging at me as if including the time-value of the optionality features is inappropriate - that only the intrinsic value is what is important in this case.


Update

Let $L$ be the lease agreement, $D$ the embedded derivative, and $B$ the bifurcated lease payoffs at time $t$, $K$ the strike for the time $t$ cash flow as determined from the forward curve at time $t=0$ (inception).

Then $L=D+B$ and we assume the USD notional is $N=1$ for convenience.

The terms of the lease agreement imply $$L=\left\{\begin{array}{ll}-\overline{S},&S_{t}>\overline{S}\\-S_{t},&\underline{S}\leq S_{t}\leq\overline{S}\\-\underline{S},&S_{t}<\underline{S}.\end{array}\right.$$

To make $B$ riskless, we simply put $$B:=-K.$$ Then, $$D=L-B=\left\{\begin{array}{ll}K-\overline{S},&S_{t}>\overline{S}\\K-S_{t},&\underline{S}\leq S_{t}\leq\overline{S}\\K-\underline{S},&S_{t}<\underline{S}.\end{array}\right.$$

But one verifies that then $$D=\max(S_{t}-\overline{S},0)-\max(\underline{S}-S_{t},0)-(S_{t}-K)=C-P-F$$ where $C$ is an $\overline{S}$ struck USD/RUB call, $P$ is an $\underline{S}$ struck USD/RUB put, and $F$ is a $K$ struck USD/RUB.

I think this analysis answers my question in the affirmative.

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  • $\begingroup$ 1. What GAAP are you talking about? IFRS or Russian? $\endgroup$ – GWD Mar 15 '15 at 10:54
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Don't look at the structure as consisting of 3 parts (i.e. a forward plus a cap plus a floor) look at it as 2 options one bought with the Floor as Strike1 and one sold with the Cap as Strike2. That way the time value changes of bought and sold option should offset - which by the way they will already do right now even wit the forward since that does not have time value. Additionally structuring the valuation the way you describe above would expose you to a flaw because as long as you move between the two strikes the bought option and the forward position would double up. So coming to your question: you should model the transactions like a bull spread (http://en.wikipedia.org/wiki/Bull_spread#/media/File:BullSpreadCalls.jpg) and not as a bull spread in combination with a forward.

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  • $\begingroup$ I'm a little bit confused. If you're correct, then it should be a bear spread (the position loses money if USD/RUB goes up - so the "cap" is really a floor and the "floor" is really a cap). In any event, if I plot the payoffs from the short forward, the long call (with $\overline{S}>\underline{S}$) and the short put (with $\underline{S}$), the net payoff looks like the correct payoff for the embedded derivative. Hence, its value at the time of the cash flow should be the value today, hence valuing it as a long call + short put + short forward. $\endgroup$ – Sargera Mar 15 '15 at 23:58
  • $\begingroup$ I'm not quite sure what you mean by "doubling up." Once $S_{t}>\overline{S}$, the payoff is $K-\overline{S}$ (the sum of the call + forward payoff) - i.e. the call's value starts compensating for the forward's loss beyond beyond $S_{t}=\overline{S}$. Similar for the short put/short forward position. $\endgroup$ – Sargera Mar 16 '15 at 0:04
  • $\begingroup$ Bythe way, thank you for your answer - I don't mean to sound combative or anything. But had one other thing to mention - there are some leases with just a floor and some with just a cap. How would I value those using your framework? It seems clear to me that you need to have the forward in there somewhere to get the correct payoff. $\endgroup$ – Sargera Mar 16 '15 at 0:22
  • $\begingroup$ Your argumentation would imply that below and above the strike prices there are FX effects since the forward would only compensate between these two. Also the payoffs could only offset if you were long the USD from cap to floor and at the same time shorting the USD forward. You have to keep in mind that once you bifurcate there is no more FX effect left in the lease contract, leaving you with a RUB lease contract with a fixed exchange rate (as of inception date) and a payoff between the 2 strikes (of cap and floor - see graph). No more forward necessary to compensate for anything. $\endgroup$ – GWD Mar 16 '15 at 0:40
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    $\begingroup$ Coming from the Put/Call parity side $+F = +C -P$ you are basically replicating the short Call ($-C$) position I am talking about via the relation $-C = -P -F$. Sorry about the "double up" error- that was a misunderstanding of your first explanation from my side. But based on your Update I hope I could make myself clearer now. $\endgroup$ – GWD Mar 16 '15 at 21:41

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