3. Spreads: capping the loss by capping the gain as well

Derivatives 5: Hedges and strategies

The two previous modules present the two pure positions: buying an option, expensive and with no ceiling above; selling an option, collecting little and keeping the bad part. A vertical spread is the combination of both, and its point is that each leg partly finances the other. We stay with the 200 units of the ETF at 500 dollars.

The bull call spread, with the full numbers

You buy a call with a strike of 500 at six months paying 28 dollars a unit, and at the same time you sell a call with a strike of 550 at the same expiry collecting 12. Net outlay: 16 dollars a unit, 1,600 a contract.

• Maximum loss: the 1,600 dollars. It occurs with the ETF at 500 or below.
• Break-even point: 516, which is the strike bought plus the net cost.
• Maximum profit: the width between strikes less the cost, that is 50 minus 16: 34 dollars a unit, 3,400 a contract, and it is reached with the ETF at 550 or above.

Compare it with the naked 500 call at 28. That call has unlimited upside, but its break-even sits at 528, twelve dollars higher. With the ETF at 560 at expiry, the spread makes 34 on 16 laid out, while the naked call makes 32 on 28. The spread makes more in absolute terms and far more in relative terms. The crossover comes at 562: above that price the naked call starts to make more, and from there the difference grows without a ceiling. The whole debate fits inside that figure: if you believe the ETF can go above 562, the spread is getting in your way; if not, it is financing you.

The spread applied to protection

The same mechanism works on the downside. Instead of buying the naked 450 put for 15 dollars from the first module, you buy the 475 put paying 14 and sell the 425 put collecting 6. Net cost: 8 dollars a unit, 800 a contract, 1,600 for the two. 1.6% of the portfolio against the 3% of the full policy.

What you buy with that half price:

• It starts covering sooner, at 475 instead of at 450: the excess drops from 10% to 5%.
• It runs out sooner. Below 425 the sold put cancels any additional protection. The maximum gross payout is 50 dollars a unit, 10,000 in total, and the net 8,400.
• Break-even point: 467, 6.6% below today's price.

The three scenarios, side by side

One investor buys the naked 450 put for 3,000 dollars; another buys the 475/425 spread for 1,600. Same expiry, same portfolio.

• The ETF closes at 460, 8% down. The naked put lapses worthless: minus 3,000. The spread pays 3,000 gross: plus 1,400 net. The spread wins by 4,400.
• The ETF closes at 400, 20% down. The naked put pays 10,000: plus 7,000 net. The spread reaches its cap of 10,000 gross: plus 8,400 net. The spread still wins, by 1,400.
• The ETF closes at 300, 40% down. The naked put pays 30,000: plus 27,000 net. The spread is still stuck at 8,400. The naked put wins by 18,600.

It is worth looking hard at the first scenario, because it is the one that repeats most often in practice. An 8% fall is the kind that happens almost every year, and there the full policy was no use at all while the spread made money. The reason is not that the spread is better: it is that its excess starts five points earlier. You are comparing two policies with different cover, not the same policy at two prices.

The crossover sits at 393, 21% below the starting point. And here comes the module's important warning: the spread is more efficient in the range of normal falls and it abandons you in exactly the scenario one buys insurance for. You have made the policy cheaper by selling the tail, which is the part the insurance existed to cover. That can be a perfectly reasonable decision, but it has to be taken knowing what has been sold.

Why the spread works better when implied volatility is dear

Here everything from level 3 and level 4 comes together. When implied volatility is elevated, both legs are expensive, so by selling one you recover part of the mark-up you pay on the other. In the language of sensitivities, the spread has far smaller net vega than the naked option: it matters less if you have misjudged the price of volatility. It is the idea running through the work of Natenberg and Sinclair on trading options: the result depends less on getting the direction right than on not overpaying for uncertainty.

There is a nuance that works in the protection buyer's favor. Because of the volatility smile, puts far out of the money trade with higher implied volatility than those near the money. In the 475/425 spread, the leg you sell is precisely the dearer one in volatility terms. You are buying relatively cheap volatility and selling relatively dear volatility inside the same structure.

A case worked all the way through

Suppose implied volatility shoots up and the 450 put comes to cost 24 dollars instead of 15: the full policy rises to 4,800 dollars, 4.8% of the portfolio. In that same environment the 475/425 spread might cost 11 dollars instead of 8, about 2,200. The full policy has become 60% dearer; the spread, 37%. That lower sensitivity to the price of volatility is the technical reason why spreads show up when the market is jumpy. What does not change is the arithmetic of the previous module: the more you cut the cost, the more tail you are selling.