4. Vega and rho: sensitivity to volatility and to rates

Derivatives 4: The Greeks

That leaves the two variables you cannot see on the price chart. The underlying trades and the clock ticks on in plain view of everybody; implied volatility and interest rates move under their own steam and can change the value of a position without the share doing anything at all.

Vega: what a point of volatility is worth

Vega measures the change in the premium when implied volatility rises by one percentage point, with everything else held still. It is not really a Greek letter, and in some texts it appears as kappa; the name stuck out of habit.

On our one-month 100-strike call, valued with an implied of 35%, vega is around 11 cents a share per point. That is:

• If the implied goes from 35% to 40%, the premium rises from 4.00 to around 4.60 euros.
• If it falls from 35% to 30%, the premium drops to around 3.44 euros.
• The share, meanwhile, is still at 100 and nothing else has happened.

Five points of implied are worth as much there as a move of more than a euro in the underlying. Whoever buys options has positive vega; whoever sells them, negative.

Why vega grows with the horizon and concentrates at the money

The same call, but at three months instead of one, is worth around 7.00 euros with the same implied of 35%, and its vega is around 20 cents a point, almost double.

The reason makes sense if you think about what implied volatility is: an annual rate of movement. Changing that rate affects all the days still ahead, so the more days remain, the bigger the accumulated effect. Vega grows roughly with the square root of the horizon, so multiplying the expiry by four multiplies vega by two.

• Horizon. Long expiries, high vega. The ones expiring this week have almost no vega at all.
• Distance from the strike. Vega is greatest at the money and falls away toward the extremes. A one-month call with a strike of 130 is not worth much more because the implied rises a point: it is still miles away.
• Relationship with gamma. Notice that here the horizon works the opposite way round from gamma. Short horizon, lots of gamma and little vega; long horizon, lots of vega and little gamma. They are different risks and they have to be looked at separately.

Getting the direction right and losing money

This is the case that ruins the most beginners, and it is pure vega. All the numbers are assumptions consistent with a standard model.

A company publishes results in a few days' time. Anticipation has inflated the implied of its three-month options to 40%, and a 100-strike call at that horizon costs around 8.00 euros. An investor buys it convinced the results will be good.

He is right. The share rises to 103. But, with the news out, the uncertainty dissipates and the implied deflates from 40% to 28%.

• Through delta, around 0.55, the three euros of rise contribute close to 1.65 euros in his favor.
• Through vega, close to 0.20 a point, the twelve points of implied lost subtract around 2.40 euros.
• Result: the premium falls from 8.00 to a little over 7.20 euros.

The underlying has risen 3% and the option has lost close to 9%. Nobody got the direction wrong. The other variable failed, the one that does not show up on the chart. In the jargon this is called the post-event implied vol crush, and it is so systematic that it is worth always allowing for it before buying expensive options ahead of a piece of news.

Rho: the Greek that slept for a decade

Rho measures the change in the premium when the interest rate rises by one percentage point. It is positive for calls and negative for puts.

The intuition is the one from this academy's Macro course. Buying a call is a way of controlling a hundred shares without paying out the 10,000 euros they would cost: meanwhile, that money is placed earning the risk-free rate. The more that parked money earns, the more the right to buy later rather than now is worth. With puts the mirror image applies: delaying the proceeds of a sale costs more the higher the rate, and that is why high rates make them cheaper.

The magnitudes explain why rho was ignored for years:

• On our one-month call, rho is around 4 cents for each percentage point rates rise. Next to a vega of 11 cents per point of volatility, it is noise.
• On a two-year call on the same share, with a premium of around 20 euros, rho is around 80 cents a point. There it stops being noise.

Between 2009 and 2021, with official rates pinned at zero and not moving, rho was a footnote in any practical manual. Between March and December 2022 the Federal Reserve raised its rate by some 425 basis points in nine months. Applied to that two-year call, four points times 80 cents is more than 3 euros on a premium of 20: 16% of the option's value, moved by a variable the previous generation of traders had learned not to look at.

Closing the level

Five variables, four sensitivities, and one warning that applies to all of them: each Greek measures an isolated move, and the market never moves one thing alone. A collapse in the underlying almost always arrives accompanied by a rise in the implied, and sometimes by a change in rate expectations. The Greeks are not a forecast: they are the map of where money can come in and where it can go out.