Trading professionals use a variety of tools in options portfolio management to deal with varying types of risk. These include the Stepladder report and the Greek options.
Options Portfolio Management
During options portfolio management, traders can incorporate the Greek options to manage a portfolio of options and cash positions. One of the methods commonly employed in the foreign exchange options markets, is the “Stepladder” report, generated by most options risk management systems.
Options portfolio management requires one not think of options 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. 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 U.S. dollars for delivery into its U.S. 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 U.S. dollars for value February 2 at 1.5010 to ABC Bank. Then, on February 2, it must deliver $10 million U.S. dollars into ABC Bank’s U.S. 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 U.S. 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 U.S. 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 U.S. 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 U.S. 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.
The use of options portfolio management by trading professionals is truly an excellent example of the multitude of analytical tools used to deal with varying types of risk.