FMP 14: Trading Strategies
Give a trader European options at every strike price and almost any payoff shape becomes reachable, provided it depends only on where the asset finishes at one future date. Such strategies fall into four groups: one option beside the asset, two or more calls, two or more puts, and mixtures of the two types. Anything built purely from calls, or purely from puts, is a spread; a mixture is a combination. Throughout, the underlying asset provides no income and every option is European.
Put-call parity, a no-arbitrage result, connects the single-option strategies. It fixes the price of a European put against that of a European call on the same asset, at one strike price and maturity.
One equation in four quantities turns around three further ways, and each version names a position.
Reading the four equations as positions
Equation (14.1) says that owning the asset and owning a put on it matures at the same value as owning a call and holding cash worth PV(K). Buying a put over an asset already held is the protective put. Equation (14.2) matches the asset plus a written call to a written put alongside cash of PV(K), and that first pairing is the covered call. Equation (14.3) inverts the protective put, so a written put against a short asset position behaves like a written call carrying a debt of PV(K), and equation (14.4) inverts the covered call the same way.
Both positions begin with an investor who already wants to own the asset. Only the worry differs.
The protective put
Construction is a long position in the asset plus a long put on it struck at K, running to the same date. Whoever sets this up is bullish, which is why the asset is held, but will not carry the whole consequence of a sharp fall.
Below the strike the put gains one for one against the asset, so the pair stop losing value and K becomes a floor. Above it the put lapses and the asset runs on without limit. With S as the purchase price and p as the premium, the largest loss is S plus p less K, profit has no ceiling, and breakeven arrives at S plus p.
The covered call
Construction is a long position in the asset plus a written call struck at K, normally out of the money. Selling it turns part of the upside into cash today, and the option premium is kept whatever follows.
Gains stop at the strike, because every further rise goes to the option buyer. Maximum profit is K less S plus the call premium c, reached at any price from K upward. The downside is barely improved: if the asset collapses to nothing the loss is S less c, which is also where breakeven sits. The covered call is payment for an opinion that the asset will not run away; the protective put is payment for insurance against the opposite.
A principal protected note, or PPN, is a security assembled around a single option so that the buyer keeps any gain on a named portfolio while none of the original money is at risk. Risk-averse investors find that attractive, and banks build it from two ordinary components.
Interest rates for three years stand at 7% with annual compounding. A bank wants to sell a three-year note of USD 10,000 returning the whole amount at maturity plus the full rise, if any, in a portfolio called Portfolio B, worth USD 10,000 today and paying no income.
What the investor gives up
Two sacrifices pay for the protection: three years of interest on USD 10,000, the whole USD 1,837.02 budget, and any income Portfolio B throws off, which goes to genuine holders rather than to the note. Together they explain why a full participation note, handing the investor 100% of the upside, needs a portfolio that generates income.
Interest rates work on the same budget. A higher rate widens the gap between the portfolio value and its present value, so the budget grows and the note becomes easier to build.
Spreads use two or more options of a single type. A bull spread suits an investor expecting the asset to rise. The trader buys a European call struck at K1 and writes one struck at K2, with K2 above K1 and both expiring together. The written call sits further out of the money, so its premium is smaller and the position costs money to open.
| Where the asset finishes | Long call at K1 | Short call at K2 | Total payoff | Profit |
|---|---|---|---|---|
| At or below K1 | 0 | 0 | 0 | minus C |
| Between K1 and K2 | S(T) less K1 | 0 | S(T) less K1 | S(T) less K1 less C |
| At or above K2 | S(T) less K1 | K2 less S(T) | K2 less K1 | K2 less K1 less C |
Source: adapted from the GARP chapter on trading strategies, profit column added.
Maximum profit is K2 less K1 less the net cost, earned anywhere from K2 upward. Maximum loss is the net cost, and breakeven falls at K1 plus that cost. Against simply buying the call struck at K1, the attraction is price: the premium received on the K2 call is a saving bought by abandoning every gain above K2. Two out-of-the-money strikes make the spread very cheap, with a slim chance both finish in the money and a large return follows, while two in-the-money strikes make it expensive, with a high probability of a modest return.
The same spread built from puts
Apply parity at each strike and subtract.
Every range pays the same amount either way, so the choice turns on which options are liquid. Note the direction of trade: a bull spread always buys the low strike and sells the high strike.
Reverse the view and the strikes swap roles. The trader buys the European put struck at K2 and writes the one struck at K1, with K2 above K1 and a shared maturity. The purchased option carries the higher strike, so again there is a net outlay.
Below K1 both puts finish in the money and the payoffs partly cancel, leaving K2 less K1. Between the strikes only the purchased put pays, giving K2 less the asset price, and above K2 neither pays. Maximum profit is K2 less K1 less the net cost, maximum loss is that cost, and breakeven sits at K2 less it.
Calls do the same job through the mirror image of the earlier identity.
One rule separates the families, whichever option type is used: a bear spread always buys the high strike and sells the low strike. It inherits the same trade-off, with two out-of-the-money options giving a small chance of a large return and two in-the-money options a good chance of a small one.
Two puts on one asset share an expiry. The one struck at USD 25 trades at USD 2, the one struck at USD 30 at USD 4.50.
Three strike prices are involved, and the middle sits halfway between the outer pair. Using calls, the trader buys one call at K1, buys one at K3 above it, and writes two at the middle strike, all sharing a maturity.
Adding the legs range by range gives a triangle. Nothing is paid at or below K1, nor at or above K3. Between K1 and K2 only the first call is in the money, so the payoff is S(T) less K1, and between K2 and K3 the written calls bite and the total becomes K3 less S(T). Every one of those amounts is zero or positive, so the position can never be free. Writing the call prices as c1, c2 and c3, that constrains how prices behave across strikes.
Convexity in the strike can be proved under the Black-Scholes Merton model, and it holds under any model. Puts serve equally well: apply parity at all three strikes, add the outer equations and subtract twice the middle one.
Maximum payoff is K2 less K1, collected only at the middle strike, and maximum profit is that less the cost. Maximum loss is the cost, incurred outside the outer strikes, and the breakevens are K1 plus the cost and K3 less it. Setting K2 near the current price backs a quiet market; reversing every leg gives the short butterfly.
Three puts on one asset expire on the same date. Strikes and prices are USD 45 at USD 2, USD 50 at USD 4, and USD 55 at USD 7.
The box spread
Stack a bull spread made from calls on a bear spread made from puts, using the same two strike prices and the same maturity, and the result is a box spread. Adding the profiles removes the asset price from the answer. Below the lower strike the bull spread pays nothing while the bear spread pays the strike difference. Between the strikes they pay S(T) less K1 and K2 less S(T), summing to the same amount, and above the upper strike the roles reverse. The box delivers K2 less K1 with certainty, so maximum profit, maximum loss and breakeven all collapse into one question about price.
A certain payoff must be worth its present value, so a box spread should cost PV(K2 less K1). Cheaper and an arbitrageur buys the box, earning more than the risk-free rate; dearer and an arbitrageur shorts it, borrowing below that rate. The legs also read as forwards, since buying the call and writing the put at K1 is a long forward at delivery price K1 while the K2 pair is a short forward. All of this needs European options, because American options can be exercised early and the final payoff would then not be K2 less K1.
The calendar spread
Every position so far has used options expiring together. A calendar spread breaks that rule by writing a call maturing at time T and buying a call at the same strike price K maturing later at T*. Judge it at time T, when the written call has just died and the purchased call still has T* less T of life left. That surviving option is worth a convex function of the asset price, and the convexity shapes the payoff.
The result resembles a butterfly spread. Should the asset sit near K at time T, the written call expires nearly worthless while the longer call keeps substantial time value, and the trader gains. Move far either way and the calls converge, leaving a loss roughly equal to what the spread cost. Puts give almost the same picture, as parity at each maturity shows.
The diagonal spread
Relax both restrictions at once and the diagonal spread appears: one long option and one short option of the same type, where the strike price differs and so does the time to maturity. It crosses a bull or bear spread with a calendar spread, expressing a view on direction and timing at once.
Combinations use calls and puts together, and those below bet on the size of a move rather than its direction.
The straddle
Buy a call and buy a put sharing a strike price and a maturity, with the strike normally near where the asset trades today. Payoff at expiry is the distance between the finishing price and the strike, whichever way it falls, which draws a V. Maximum loss is the two premiums together, suffered if the asset lands on the strike. Profit is unbounded above and runs to the strike less the combined premium below, with a breakeven either side at that distance.
The strangle
Cost falls if the strikes are pulled apart. A strangle buys a put at the lower strike K1 and a call at the higher strike K2, keeping one maturity. Between them neither pays, so the profile is a flat-bottomed valley and the combined premium is lost anywhere in that range. Breakevens are K2 plus the premium and K1 less it.
An asset trades at USD 50 and a trader sets up a three-month straddle struck at USD 50. Annual volatility is 20%, with a risk-free rate of 2%. Valued by the Black-Scholes Merton model, the European call is worth USD 2.12 and the put USD 1.87.
The strip and the strap
Two variations tilt a straddle one way without abandoning the other. A strip buys two puts for every call at the same strike, so the left arm of the V rises twice as steeply, suiting a large move more likely to be downward. A strap buys two calls for every put, steepening the right arm instead. Both cost more than a straddle.