1. How much of the portfolio can hang on one option

Derivatives 9: Derivatives in a real portfolio

The previous eight levels have described how a derivative behaves on the inside. This last one changes scale: it looks at the portfolio containing it. And its first question is the simplest in the course and the worst answered: how much of your wealth is really tied to the option you have just bought.

The premium is the price, not the size

When you buy 100 shares at 40 euros there is no ambiguity: you have put in 4,000 euros and that is your exposure. When you buy an options contract on those shares and pay 180 euros of premium, the statement says 180 euros, and instinct concludes that the position is a 180-euro one.

It is not. A standard contract on shares refers to 100 of them. That contract, which has cost you 180 euros, is hooked to the behavior of 4,000 euros of shares. That figure is called the notional controlled, and it comes out of a multiplication with no mystery to it: price of the underlying times contract multiplier times number of contracts.

The magnitudes change with the underlying: ten contracts on a 40-euro share control 40,000 euros, and one contract on an index at 5,000 points with a multiplier of 100 controls 500,000. The premium is in both cases a tiny fraction of those figures, and that disproportion explains at once why derivatives exist and why they do damage.

Delta translates contracts into shares

The notional controlled is the ceiling, not today's exposure. For that you need the delta of level 4, which measures how much the option moves for each euro of the underlying. The translation is this: equivalent exposure in shares equals delta times multiplier times number of contracts times price of the underlying.

An example. You hold ten calls on a share trading at 40 euros, with a delta of 0.35. The sum gives 0.35 times 100 times 10 times 40, that is, 14,000 euros. If the total premium paid was 1,800 euros and your portfolio is 100,000, the position does not weigh 1.8%: today it behaves as if you had 14% of the portfolio in that share.

The important word is today. Delta is not constant: it is a function of price, and its rate of change is the gamma of level 4. If the share rises to 46 euros, delta may go to 0.65 and the equivalent exposure to some 30,000 without your having placed an order; if it falls to 34, it caves in and is left at a few thousand. The exposure of an options position is not a number: it is a number that moves on its own, and it moves faster the closer expiry is.

The three figures to write down before trading

• The premium. What leaves the account today. It is the one almost everybody looks at and the least informative.
• Today's equivalent exposure. Delta times multiplier times contracts times price. It is the one you compare with the portfolio.
• The notional controlled. What the position becomes if delta approaches one. It is the scenario you have to be able to bear.

Whoever writes down only the first believes they hold a small position. Whoever writes down all three knows what they hold.

When it is true that you are only risking the premium

It is true, literally, in a long call and in a long put: you pay, and if it goes badly the option expires worthless. And in any strictly long position with a net debit, such as the long spread of level 5, where the cap is the difference in premiums.

It stops being true the moment a sold leg appears that is not entirely offset by a bought one of the same size and expiry. Level 8 laid out the catalog:

• The ratio, where you sell more contracts than you buy: the unoffset part is a naked position.
• The calendar, where the leg bought expires after the one sold, so that in between the worst case depends on a volatility you do not control.
• The short put presented as a way of buying cheaply: the obligation is for the full notional, not for the premium.

One qualification for the honest case: the cap is per trade, not per strategy. Repeating the purchase twelve months in a row allows you to lose twelve premiums.

How much it can weigh: where that question gets answered

This course gives you the translation; size is decided elsewhere. How much a position can weigh, how it interacts with the others and what maximum loss a given wealth tolerates are the subject matter of this academy's Risk and portfolio management course, on this same intermediate track. That is where volatility-based sizing, correlation and risk of ruin live.

What does belong here is the conversion rule: those rules apply to the equivalent exposure and to the notional controlled, never to the premium. A 5% per position limit does not mean a premium below 5%, it means equivalent exposure below 5%, with the added check that the notional does not turn that 5% into 40% when the market moves.

Worked scenario: the portfolio that thought it had 1.5% in options

A portfolio of 200,000 euros. The investor buys 20 calls on a share trading at 50 euros, strike 55, three months to expiry, premium 1.50 per share. Outlay: 20 times 100 times 1.50, that is, 3,000 euros, 1.5% of the portfolio. That is where what he thinks he holds ends.

The other two figures of the module say otherwise. With a delta of 0.30, the equivalent exposure is 0.30 times 100 times 20 times 50, that is 30,000 euros: 15% of the portfolio. And the notional controlled is 20 times 100 times 50, that is, 100,000 euros, half of his wealth.

Three weeks later the share rises to 58 and delta goes to 0.62. The equivalent exposure is now some 71,900 euros, 36% of the portfolio. He has placed no order: that concentration was built for him by the instrument. In the opposite direction, with the share at 44 delta caves in to around 0.06 and exposure is left at some 5,300 euros.

And the finishing touch. If, to cheapen the purchase, he had also sold 20 puts at a strike of 45, the line about only risking the premium would be false. With the share at 44, that sold leg is the obligation to buy 2,000 shares at 45: 90,000 euros, 45% of the portfolio, committed for a premium that was there to save him a few hundred.