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Multi-factor options

Key facts
The value of a multi-factor option is determined by the behaviour of two or more underlying assets and therefore by the correlation between those assets.
The main classes are rainbow and basket options.
These options are useful in managing complex combinations of risks although the details of their pricing and mark-to-market value are complex.
The assets involved need not all be from the same asset class.
So, for instance, an option may have a payout which is linked to the performance of the S&P index, the price of Brent crude and the dollar-yen exchange rate.
There is an additional special case where payments from a single asset option are made in a different currency to the denominating currency of the underlying asset.
These are generally called currency translated , quantized or just quanto options.
The values of many of these options depend upon correlation and as such are described here.
Quantization can also be applied to most types of derivative instrument
(see the entries for currency protected swap, quanto FRA and the descriptions of cross-currency swaptions and quanto caps under the entries for swaption and cap).
Options on all asset classes can be quantized.

Asset-linked foreign exchange option
A quanto option on a foreign asset in which the exchange rate used on maturity is the greater of zero, and the exchange rate at maturity less a preset strike.
So the payout for a FTSE-100 equity-linked foreign exchange option would be:
$[FTSE-100 x max($/GBPmaturity - $/GBPstrike], 0)] where the maximum function determines the exchange rate to be used.
This strategy combines a currency option with an equity forward to create a variable quantity forex option.
It gives the holder of the option the ability to gain from any dollar strengthening but places a floor (the strike) on the exchange rate component of the investment since if at maturity sterling has weakened from the strike level then the option payout is zero.
An alternative way to look at this is that it is a foreign exchange option which is quantoed into FTSE units.

Basket option
An option that pays out on the basis of the aggregate value of a specified 'basket' of financial assets rather than on the value of the individual assets.
The premium of the option will reflect the correlations of the basket components: if they are negatively correlated, then moves in the value of one component will be neutralized by opposite movements of another.
Unless all the components are perfectly correlated, the option will be cheaper than a series of individual options on each of the assets in the basket.
Basket options are used in all asset markets and can be cross-asset i.e. contain more than one asset type and can be constructed with digital payoff characteristics.
For example:
Better off (worse off)
An option that pays the holder a return based on the price level/percentage price change achieved by the better (worse) performing of two or more underlying assets.
So its payout is: Max(asset1, asset2, asset3...assetn) where asset i is either the price level or percentage change in asset i.
For example, an investor could buy a better-of option on two equity indices and receive the returns from the better-performing of the two.
If all the underlying assets fall in value, the holder must pay the performance of the asset with the smallest decrease in value.

Composite option
An option where the strike is denominated in the payout currency of the option and fixed at inception.
The asset price is then translated at spot at expiry of the option.
So in the example described if the FTSE option is struck at the money spot (i.e. the strike is 4800*1.6 = $7680 per FTSE unit) then the payout in the two cases is max(5000*1.5 - 7680,0) = max(7500 - 7680,0) = $0 and max(5000*1.7 - 7680,0) = max(8500-7680,0) = $820.
In this case the investor is still exposed to currency movements but the hedger has significant correlation risk.
He will be funding his position in dollars and every hedge re-balance will involve a sale or purchase of sterling and a sale or purchase of the sterling asset.
Thus his cost of hedging will be affected by the frequency with which the exchange rate and the FTSE index move.
Currency basket option
An option that gives the holder the right to exchange a portfolio of predetermined amounts of currency for a fixed amount of a base currency.
Example
A US multinational is expecting flows of Euros, Canadian dollars, Japanese Yen and Swiss Francs in three months and has budgeted rates for all flows.
Unwilling to lose upside potential by using forwards, the company must choose between buying currency puts for each currency struck at the budgeted rate or buy a basket option struck at the total US dollar value of all the currency amounts at the budgeted rates.
If on the option expiry date, the dollar value of the currencies at the final spot rates is less than the basket strike, the corporation will exercise the option.
If the total dollar value of the portfolio is higher than the basket put strike then the option expires worthless and the currencies are converted at spot.
Currency-translated/Quanto options
Also known as quantos or quantized options, these options payout in a currency different from the natural denominating currency of the asset.
For instance an option on the Nikkei index that pays out in dollars or an option on dollar-mark that pays out in Italian lire.
There are five main types.
To illustrate them we will take the example of a dollar-based investor who wishes to buy a call option on the FTSE-100 but receive the proceeds in dollars.
Let us suppose that at inception of the trade the FTSE-100 is trading at 4800 and the dollar-sterling exchange rate is 1.60.
At expiry let us suppose that the FTSE-100 is trading at 5000 and the dollar-sterling exchange rate is either 1.50 or 1.70.

Flexi-quanto/flexo
An option where the payout of an option is translated at spot from the denominating currency of the underlying asset into the denominating currency of the option.
So for instance in the example described if the FTSE option is struck at the money spot then the payout to the investor in the two cases is (5000-4800)*1.50 = $300 and (5000-4800)*1.70 = $340.
In this case the investor is fully exposed to currency movements and the hedger has no correlation risk since he funds his position in sterling, transacts his delta hedges in a sterling asset and simple does a spot foreign exchange transaction at expiry.

Hybrid barrier option
An option on one asset knocked-in or out by movements in another.
These are also called two-factor or outside barrier options and also dual trigger options. The commonest types are hybrid barrier caps and floors linking exchange rates (example 1), commodity prices (example 2) or equity indexes with interest rates (example 3) though hybridization has also been applied to other types of derivative. For example semi-fixed swaps (and other resettable or contingent structures) have been constructed with the rate reset trigger dependent on the price of oil (a swap plus a binary oil option) and other assets.
Example 1
A Japanese exporter with large floating-rate debt outstandings might be very profitable when dollar-yen exceeds 105 but below 95 cash flow becomes critical and he requires interest rate protection. A normal five-year 7% cap might cost 364 bp. Instead they buy a five-year 7% cap which knocks-in when the dollar-yen rate hits 95. This reduces the cost of the cap by 160 bp.
Example 2
Or take a gas producer whose profits usually rise with rising gas prices but fall with rising interest rates.
The company fears the combination of rising rates and falling prices.
It could buy a standard interest rate cap, but is unwilling to pay so much premium as it is only the combination of factors that pose a threat.
Instead it can buy an interest rate cap that is knocked-out if the gas price exceeds a specified barrier in any quarter.
The strike and knock-out levels are set at the company's combination pain or breakeven threshold.
The company pays floating interest rates only when it has profits with which to pay.
Example 3
A UK-based company might wish to buy interest rate cover for some debt.
However, it is contemplating floating off a large subsidiary in the next two years in which event it will not require the cap.
Instead of buying a three-year cap at a cost of 339 bp, they buy a knock-out cap that knocks-out when the FTSE midcap index rises by 15%
This cap costs 140 bp less.
As well as the lower premium, the cap will disappear at exactly the right time: when the company will be able to float its company or sell it at an attractive price and pay down its liabilities. The knock-out can either be permanent, as in these examples, or the cap can be structured so that it is only knocked-out for the period in which the knock-out trigger is breached. If the underlying moves back through the knock-out trigger, then the cap is reactivated, making it resemble a range transaction. Interest rate basket option An option on a basket of interest rates designed to reduce overall interest costs across a number of different markets.
Example
A borrower may believe that his European interest rate bill would rise more than is implied in the market but is unwilling to fix in case his view is incorrect. Instead of purchasing a series of options on the individual markets, he wants a basket because the mix of EU and non-EU currencies exhibit some negative correlations that will reduce the premium cost. He could buy a one year 8% strike basket option denominated in his base currency with the underlying the average two-year swap rate in the chosen currencies. If the average rate rose above 8% this hedger would be protected. The sensitivity of the basket would be similar to that of a basket of payer swaptions. (An investor would buy the product if he wanted a customized, balanced exposure to a region and is prepared to accept a degree of upside limit.)

Joint quanto option
An option where the payout is translated from the denominating currency of the underlying asset into the denominating currency at a rate which is at least as good as a predetermined exchange rate.
So in the example described if the FTSE option and fixed dollar/sterling exchange rate are at the money spot then the payout in the two cases is max(5000-4800,0)*max(1.6,1.5) = $320 and max(5000-4800,0)*max(1.6,1.7) = $340.
In this case the investor is fully protected from adverse movements in the foreign exchange rate and participates fully in favourable movements.
The hedger clearly has significant correlation risk.
In fact he has two rainbow options since he has a contingent flexo option and a contingent quanto option where the contingency is based on the exchange rate: he will have the flexo option only if the exchange rate is above the predetermined foreign exchange rate and he will have the quanto option only if the exchange rate is below the predetermined foreign exchange rate.

Max/min option
An option that pays the holder a return based on the asset that performs best against its strike out of a basket of assets each with its own pre-set strike price.
Unlike the better (worse) of option, the strikes are above zero and the payout cannot be negative.
So a Max call option pays out: Max(max(asset1-strike1, asset2-strike2,... assetN-strikeN), 0) and a Min call pays out: max(min(asset1-strike1,asset2-strike2,... assetN-strikeN),0).
Outperformance option
An option that pays the holder the difference in the performance of two assets - that is: max(asset2-asset1, 0)
An investor with a cash position in the FTSE-100 but worried that the S&P500 might outperform it could buy a rainbow outperformance option that paid a return based on the difference if positive between the two indices' performance.
So if the S&P500 did outperform the FTSE-100 by 8%, then the holder would receive a payout based on that 8% difference.
If the S&P500 did not outperform, then there would be no payout.
Also known as difference options.

Quanto option
An option where the payout is translated from the denominating currency of the underlying asset into the denominating currency of the option at a predetermined exchange rate.
So in the example described if the FTSE option and fixed dollar/sterling exchange rate are struck at the money spot then the payout in the both cases is max(5000 - 4800,0)*1.6 = $320.
In this case the investor is fully protected from movements in the foreign exchange rate and the hedger has correlation risk.
He will have a funding position in both dollars and sterling always maintaining the net value of sterling assets and liabilities at 0.
This means that correlation impacts him as the size of his FTSE position varies linearly with the dollar/sterling exchange rate.

Rainbow option
An option whose payout is based on the relationship between multiple assets as opposed to the price or performance of a single asset.
A rainbow option whose payout depends on two assets is said to be a two-colour rainbow, on three assets a three-colour rainbow and so on.
There are five basic types of rainbow option.

Spread option
An option on the spread between two asset prices or indices.
They differ from outperformance options as they are struck not at zero but at some level of the spread.
So the payout is: max(asset2-asset1-strike, 0).
Example 1
A yield curve option is an option on the spread between interest rates at two different points on the same yield curve.
They are usually struck on the yield of a longer maturity bond/index less the yield of a shorter maturity bond/index.
So an at-the-money call on the US dollar two-year CMS versus the 10-year CMS would have a strike equal to the implied spread between the expected levels of the two rates on the exercise date.
Yield curve calls profit if the spread widens (yield curve steepens), puts if it narrows (yield curve flattens).
They allow investors to take a view on the shape of the yield curve without taking a directional view on the underlying bond market.
A yield curve option costs less than the series of calls and puts on the underlying securities/indices used to construct the yield curves because it only pays off on the change in spread whereas one of a pair of separate options might be in-the-money as the result of a parallel shift in the yield curve.
For the option writer a key consideration, as in all multi-factor options, is the correlation between the two points on the yield curve.
An investor holding a three-year floating rate asset yielding Libor believes that the spread between the three-year swap rate and six-month Libor will be higher than implied by the forward curve.
So he buys a digital cap (i.e. series of digital calls) on the three year swap rate minus six month Libor spread struck at 60bp with an immediate payoff of 120bp as soon as the spread hits the strike.
The premium is 65bp a year semi-annually.
If the spread is lower than 60bp the investor receives Libor less 65bp.
If the spread is higher he receives Libor plus 55bp. He could bet on a narrowing of the spread by buying a digital floor.
Example 2
A crack-spread option is an option on the spread between the price of crude oil and one or more of its refined products.
Simple versions are traded on Nymex - one on the spread between heating oil and crude oil prices and one on the spread between crude and unleaded gasoline.
They are quoted in terms of the price of one barrel of the refined product less the price of one barrel of crude.


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