This article will examine the way in which an options trader incorporates
the options Greeks described in "Derivatives Explained" to manage a
portfolio of options and cash positions. The method presented here is commonly
employed in the foreign exchange options markets. Naturally, different traders
use different techniques.
In this case, we will focus upon the options "Stepladder" report,
generated by most options risk management systems. When managing a portfolio of
options, it is inconvenient to think of them on an individual basis. Indeed, a
commercial bank foreign exchange options trader may have hundreds or thousands
of options positions with different maturities in his portfolio at any given
time. In addition to his options position, the options trader will have cash
positions as well. He needs a mechanism for describing the risk in the position
at any given point in time and for a given underlying or spot rate.
There are three kinds of risk to which the options portfolio is
exposed at a primary level: movements in the spot rate, convexity and implied
volatility. From our earlier discussion of dynamic hedging, we know
that we can insure the local exposure of an individual option to small changes
in the spot rate by delta hedging. We also know
that the delta for an individual option will change as the spot rate changes
because of the convexity inherent in the way the
option's price reacts to changes in the underlying spot rate. This is given to
us by the gamma. We also know that the option's value will vary with changes in
the implied volatility the market assigns to a
particular maturity and strike.
In the case of a foreign exchange option, we are talking about options on a
forward. A forward obligates the buyer to exchange one currency for another at
a pre-set rate for a particular delivery date. For example, a large consulting
firm might enter into a contract on January 5 that pays it $10 million US
dollars for delivery into its US dollar account for value February 2. They may
want to lock in the current rate of exchange the market is using for February
2. If the spot rate is 1.50, this might imply a rate for February 2 of 1.5010.
The difference, 0.0010, is called the forward premium. It is determined by
interest rate arbitrage and it is sensitive to the difference between interest
rates in Canada and the United States. The consulting firm sells US dollars for
value February 2 at 1.5010 to ABC Bank. Then, on February 2, it must deliver
$10 million US dollars into ABC Bank's US dollar account in exchange for which
ABC Bank will deliver $15,010,000 Canadian dollars into the consulting firm's
Canadian dollar account for value February 2, regardless of the prevailing spot
Canadian dollar exchange rate.
A
foreign exchange option is an option on a forward because it gives the holder
the right but not the obligation to exchange, in this case, $10 million US
dollars for value February 2 at a rate (or strike price) of 1.5010. If on
February 1, the Canadian dollar is weaker than 1.5010 (i.e. the exchange rate
is greater than 1.5010), the consulting firm can let the option lapse and
exchange its US dollars at the prevailing spot rate for value February 2. (Note
that the Canadian dollar has one day of settlement between the transaction and
delivery). In this case, February 1 is the option's maturity date.
Because a foreign exchange option is an option on a forward, it is
sensitive to changes in the interest rate differential, as well. The options
dealer will generate a Stepladder report that looks like the following to
characterize his portfolio's exposure to changes in the spot rate, assuming
that the interest rate differential for all maturities stays the same and that
implied volatilities stay the same. Spot is at 1.50. The report is generated
over a horizon of one day. Profit and Loss (P/L), Delta, Gamma and Vega are all
denominated in US dollars.
| Spot |
P/L |
Delta |
Gamma |
Vega |
| 1.5200 |
$67,256 |
$9,257,650 |
$3,240,445 |
$49,010 |
| 1.5150 |
$41,995 |
$7,111,889 |
$2,145,761 |
$46,789 |
| 1.5100 |
$18,554 |
$6,325,789 |
$786,100 |
$44,258 |
| 1.5050 |
$1,401 |
$4,976,111 |
$1,349,678 |
$37,337 |
| 1.5000 |
($11,256) |
$3,120,556 |
$1,855,555 |
$36,112 |
| 1.4950 |
($18,752) |
$125,778 |
$2,994,778 |
$32,145 |
| 1.4900 |
($17,895) |
($10,156,123) |
$10,281,901 |
$31,247 |
| 1.4850 |
($15,443) |
$556,741 |
($10,712,864) |
$34,125 |
| 1.4800 |
($16,742) |
($3,214,748) |
$3,771,489 |
$36,544 |
How do we interpret this Stepladder report? First, let's
consider the fact that this report is generated over a horizon of one day. We
know straight away that if spot stays at 1.50 without moving at all over the
next trading day, we will have lost $11,256. This number is the time decay for
the options portfolio. It includes the net change in value of all of the
options in the portfolio attributable to their maturities being shorter by one
day. It also incorporates any change in the value of the forward portfolio
attributable to their maturities being shorter by one day. And it includes the
cost of funding our positions. (If we borrow money to buy options, we must pay
interest on these balances).
The position has a delta of $3,120,556 at a spot rate of 1.50.
If spot trades higher, say up to 1.5100 over the trading day, the portfolio
will get longer US dollars. We could dynamically rebalance the delta hedge of
the portfolio at 1.51 by selling $6,325,789, making us delta neutral at 1.5100.
Should spot subsequently dip back down to 1.5000, the portfolio will now be
short $3,205,233 (the difference between $6,325,789 and $3,120,556). Buying
back $3,205,233 makes us delta neutral again.
At the end of the day, we compare the P/L number implied by the closing spot
rate (e.g. ($16,742) at a spot rate of 1.4800) to the spot trading P/L we have
earned by dynamically rebalancing the hedge. Hopefully, the net number is
positive.
There are different ways of reporting the gamma. In this Stepladder report,
the reported gamma is the difference between the current spot position and the
portfolio's spot position for a spot rate that is 0.0050 higher. It appears as
if we are long an option expiring tomorrow with a strike somewhere between
1.4950 and 1.4900. We suspect this because of the discrete jump in the spot
position by more than $10,000,000 between those two spot levels. It is offset
by our short position in another option expiring tomorrow with a strike between
1.4900 and 1.4850, suggested by the discrete jump in the spot position of more
than $10,000,000 between those two spot levels. This phenomenon is referred to
as strike risk or pin risk. When we have two options expiring on the same day
with similar but not identical strikes, it can be very challenging to manage
the net delta position at expiry if spot is near either of the two strikes.
Finally, there is the vega. At 1.5000, if the Canadian dollar implied
volatility curve shifted up in a parallel fashion by 1 vol (i.e. by 1
annualized standard deviation), then the portfolio would make $36,112, even if
spot did not move. Of course, if spot is not moving, then we are more likely to
see implied volatility move lower which compounds the portfolio's time decay
problem.
Article by Chand Sooran, Principal, Victory
Risk Management Consulting, Inc. |