17 March, 2015

Can a Derivative's Value Exceed the Underlying Notional Value?

On a recent project I valued some derivatives, the results of which the client balked at because the values exceeded the notional on which they were written.  So is it ever possible for a derivative's valuation to exceed its underlying notional?

The answer, of course, depends, and first we need to clarify what we mean by a derivative's value exceeding its notional.  A typical derivative (like an option or future) is written on some underlying asset with price $S_{t}$ and some quantity or notional $N$.  The term quantity is frequently used for assets like stocks and the term notional for assets like currencies - so in the latter case, if I have USD/EUR call option, then I view the asset as the US dollar (that I want to buy a call option on) priced in the European Euro (that is what the USD/EUR exchange rate is - the cost of a US dollar in Euros), with a notional (i.e. quantity) equal to (say) $\$100,000,000$ USD.

Notice that the quantity/notional has units in the underlying asset and that the spot price $S_{t}$ has units of value in the numeraire/settlement currency per 1 asset.  Hence, when we ask if a derivative's value can exceed its underlying notional, we are really asking whether the value at time $t$, denoted $V_{t}$, can exceed the quantity $NS_{t}$, which has units in the settlement currency (EUR in the above example).  In other words, we ask whether
can hold without introducing an arbitrage.

Essentially, the purpose of the above discussion was to express precisely what we mean for the valuation to exceed the underlying notional and moreover, to emphasize that derivative's valuation cannot be directly compared to the notional in order to answer the question, since since the units are not the same - the underlying notional $N$ needs to be multiplied by the spot price $S_{t}$ so that each has units in the valuation currency.

The classic counter-example to answering this question affirmatively in all instances comes from considering a call on option written on $N$ of some asset with price $S_{t}$.  If you value this option at time $t$, then it is clear that
for otherwise one could short a covered option at no cost and an arbitrage would exist.  But this argument no longer holds for instruments with payoffs that are not artificially bounded by some optionality mechanism.  Indeed, the example from my experience involved an FX forward on CUR1/CUR2 (to be generic).  For such a forward, let $K$ be the strike, $N$ the notional (denominated in CUR1), $D$ the discount factor and $\alpha$ the CUR2/CUR1 exchange rate ($1/S_{t}$).  Then if the entity is in the short position we have
$$-\alpha N\cdot(F-K)\cdot D>N$$
$$(F-K)<-\frac{1}{\alpha D}$$
$$F<K-\frac{1}{\alpha D}.$$
Hence, if the forward rate is sufficiently small (i.e. price of CUR1 declined) with respect to the inception strike, then the value of the forward will exceed the notional as an asset. Conversely,
$$-\alpha N\cdot(F-K)\cdot D<-N$$
$$(F-K)>\frac{1}{\alpha D}$$
$$F>K+\frac{1}{\alpha D}$$
Hence, if the forward rate is sufficiently large (i.e. the price of CUR1 increased) with respect to the inception strike, then the value of the forward will exceed the notional as a liability.

It is not difficult to come up with similar bounds for other basic instruments such as swaps either.