VRM 16: Option Sensitivity Measures: The Greeks

Doing nothing leaves a naked position: the options are sold and nothing is held against them. The outcome is excellent so long as the stock never gets above USD 105 during the year, because the client abandons a worthless option and the trader keeps the full USD 10 million. The difficulty is that a volatility of 25% makes a substantial rise perfectly ordinary. Should the year end with the stock at USD 125, the options produce a negative payoff of USD 20 million, and crediting the premium still leaves the trader down USD 10 million. At USD 150 the net loss reaches USD 35 million, and nothing caps the damage above that.

Covering the position reverses the shape of the risk. At the moment the options are written, the trader acquires 1 million shares at a total cost of USD 100 million. Exercise then takes those shares away for USD 105 million, so the trade earns USD 5 million on the stock and USD 10 million on the premium, USD 15 million in total. What this cannot survive is a fall. A stock price of USD 75 means a loss of USD 25 million on the shares and a net loss of USD 15 million once the premium is counted. At USD 50 the net loss reaches USD 40 million.

Each strategy is comfortable in exactly the scenario that ruins the other: the naked position wants the stock to sit still or fall, the covered position wants it to rise, and nobody knows in advance which will happen.

Neither approach deserves the name of a hedge. A hedging strategy earns that name when the present value of its expected cost stays close to the Black-Scholes-Merton value of the option, USD 9.56 million here, whatever the stock price does.

The stop-loss strategy blends the naked and covered positions and switches between them according to where the stock price sits. While the option is out-of-the-money, meaning the stock price is below the strike price, the trader runs a naked position. Once it moves in-the-money, meaning the stock price is above the strike price, the trader covers by purchasing 1 million shares. Should the price later drop back below the strike the shares are sold, and a further rise through the strike brings them back again.

Suppose the shares can always be bought at exactly USD 105 as the option moves in-the-money and sold at exactly USD 105 as it moves back out. Only two endings are then possible and both look attractive. If the stock finishes below USD 105 the trader is naked, the option is abandoned, and every share bought at 105 was later sold at 105, so the profit is the full USD 10 million. If it finishes above USD 105 the trader is covered, having paid a net USD 105 million for 1 million shares and delivered them into the exercise for USD 105 million, so the profit is again USD 10 million. An option already in-the-money on day one is covered at once, and the trader keeps the gap between the price received and the initial intrinsic value.

Why the strategy does not work in practice

The flaw sits at the strike price itself. When the stock touches USD 105 nobody knows whether the next tick is up or down. A move from USD 104.9 to USD 105 triggers the purchase of 1 million shares, and the next print may take the price straight back to USD 104.9. A slip to USD 105 from USD 105.1 triggers the sale of those shares, and the price may then return to USD 105.1.

So the shares cannot be traded at the strike. They must be bought at 105 plus a small amount and sold at 105 minus the same amount. Take that amount to be 0.1: the trader buys at USD 105.1 and sells at USD 104.9, which costs USD 0.2 per share on every completed round trip. A path that never reaches USD 105, or crosses it once, costs almost nothing. A path that wanders across the level thirty times is a different matter.

Narrowing the band does not rescue the strategy, because the number of round trips rises as the band shrinks, and the smallest usable band is set by the way the stock is quoted. The all-in cost of writing an option and hedging it is therefore sometimes far more and sometimes far less than the USD 9.56 million theoretical price, and the profit is nowhere near a dependable USD 440,000.

Delta is the first and most heavily used of the Greek letters, comparing a small change in the value of a derivative with the change in the stock price that caused it, every other input frozen.

For European options on a non-dividend-paying stock, delta has a closed form that drops straight out of the Black-Scholes-Merton equations.

One convention matters on a desk. The volatility placed in these formulas is the option’s implied volatility, not a historical estimate, and that holds for every Greek letter here.

The shape of delta across stock prices

A long call option has a positive delta between zero and one, because a call gains value as the stock rises. When the call sits far out-of-the-money it is worth almost nothing, and a small move leaves it worth almost nothing, so delta sits near zero. When it is deep-in-the-money and virtually certain to be exercised, the holder is in much the same position as someone who owns the stock and owes the strike price at maturity, so the option gains nearly the full amount of any move in the share price and delta approaches one.

A long put option has a negative delta between minus one and zero, because a put loses value as the stock rises. Far below the strike the put is almost certain to be exercised and falls by nearly the full amount of any rise in the stock, which pushes delta towards minus one. Far above the strike it is deep-out-of-the-money and stays there after a small move, so delta approaches zero.

So d1 works out at 0.0898 and delta at 0.5358, meaning a rise of USD 1 in the stock price lifts the value of one call option by roughly USD 0.5358.

Return to the trader who has sold options on 1 million shares. Multiplying the position by the delta of a single option gives the delta of the whole book, and a short position carries the opposite sign: -1,000,000 multiplied by 0.5358 gives -535,800. Buying 535,800 shares brings the total to zero.

Test the hedge with a small move. A rise of USD 0.2 in the stock price lifts each option by about USD 0.1072 (= 0.2 x 0.5358), so the short option position loses 1,000,000 multiplied by 0.1072, or 107,200, while the shares gain 535,800 multiplied by 0.2, which is 107,160. These are the same quantity computed twice, and the USD 40 between them is purely the rounding of delta to four decimal places. The gain on the shares cancels the loss on the options.

The awkward part is that delta does not stay at 0.5358. It moves with the stock price and with time, so the hedge cannot be set up and left alone. The position is adjusted every day, often more frequently, and those adjustments are called rebalancing.

Two days of rebalancing

Suppose the stock price rises to USD 100.4 after one day, leaving 251 days of a 252 day year. Recomputing d1 gives a delta of 0.5418, so the option position now stands at -541,800, being -1,000,000 multiplied by 0.5418. The hedge calls for 541,800 shares and the trader buys an additional 6,000 shares (= 541,800 – 535,800).

Suppose the stock then falls to USD 99.9 by the end of the following day, leaving 250 days. Delta drops to 0.5334, the required holding falls to 533,400 shares, and the trader sells 8,400 shares.

That pattern continues until expiry, and as maturity approaches delta resolves towards one extreme or the other. With the stock at USD 110 and two business days remaining, delta is 0.9827 and the trader is nearly fully covered. With the stock at USD 100 and only two business days left, delta is 0.0152 and the position is nearly naked. The uncomfortable case is a stock sitting almost exactly at the strike at maturity, because delta is then close to 0.5: the trader is under-hedged if the option closes in-the-money and over-hedged if it closes out-of-the-money.

Delta hedging is not free. Every rebalancing trade buys shares just after the price has risen and sells just after it has fallen, a buy-high sell-low pattern bound to cost money. In exchange the position is hedged day by day, and under the Black-Scholes-Merton assumptions the discounted total of those costs comes out close to the theoretical price of the option. Unlike the stop-loss rule, the position is never fully naked and never fully covered but always partially covered, and the fraction covered is delta, a rough measure of the probability that the option finishes in-the-money.

Every calculation so far has held volatility fixed. Real volatility moves, and that is a risk in its own right. Take the same trader on the first day of the option’s life, with the stock still at USD 100 but the volatility rising from 25% to 28%. The model now values each option at USD 10.75, so the block sold for USD 10 million is worth USD 10.75 million, which is USD 1.19 million more than the day before, the difference between 10.75 and 9.56. The stock has not budged and the trader has lost money.

Vega is the Greek letter for that exposure, comparing the change in the value of a derivative with a small change in volatility. The volatility in question is the implied volatility.

Long positions in calls and in puts alike carry a positive vega, because more volatility widens the distribution of outcomes and raises the chance of a valuable exercise. Vega is largest when an option is close-to-the-money and shrinks to nothing at both extremes, where exercise is close to settled whatever volatility does and a change in it has very little to work on.

There is something odd about extracting a volatility sensitivity from a model that assumes volatility is constant. Models in which volatility follows its own random process produce vega estimates close to the Black-Scholes-Merton figure, so the practical objection is small. With interest rates held constant, an option price depends on just the asset price and the implied volatility, so a trader hedged against both has very little risk left.

Gamma measures how quickly delta itself changes as the stock price moves, which makes it a statement about the shape of the option price curve rather than its slope.

A delta hedge treats the link running from stock price to option price as a straight line when it is really curved, and gamma is the size of the error that assumption introduces.

Suppose the stock position is adjusted once a day to keep delta at zero. The risk left overnight depends on how far the stock could travel before the next adjustment, and on how much curvature sits in the price relationship. Small moves barely matter, because a curve looks straight over a short span. Larger moves between rebalancing dates are where a delta-neutral portfolio gets hurt, and gamma says how badly.

A short option position with a gamma of -0.1 that sees the stock price rise by USD 2 can expect to lose 0.2 in value. Delta and gamma do for an option what duration and convexity do for an interest rate exposure: the slope, then the correction for the slope changing.

A long call option has positive gamma, and so does a long put option. Like vega, gamma peaks near the strike and falls towards zero once an option is far in-the-money or far out-of-the-money. The two part company on time: vega grows as the time to maturity lengthens, while gamma shrinks.

Delta can be moved by trading the underlying asset. Gamma and vega cannot, because a spot position has zero gamma and zero vega, so buying or selling shares leaves both untouched. They can only be changed with another derivative on the same asset, and that trade brings its own delta, which then has to be neutralised.

The sequence is always the same: choose a traded option, work out how many of them set the unwanted Greek letter to zero, then trade the underlying asset to bring delta back. Zeroing two Greek letters at once takes two different options, since one instrument cannot satisfy two conditions independently.

Source: portfolio and option sensitivities as given in the chapter; the final column is derived in Example 5 below.

Delta is brought back to zero at the end of every trading day. Gamma and vega are not, and cost is the reason: the underlying asset trades in size at a competitive price, while the options needed to shift gamma or vega are expensive when they can be found at all. Traders watch for opportunities to lay them off rather than rebalancing mechanically.

Time helps a little. Both gamma and vega are largest near the strike, and an option written there often drifts far in-the-money or far out-of-the-money as the asset moves, which makes both risks fade on their own. Options that stay near the strike are where the largest gamma and vega risks sit.

Theta is the rate at which an option loses or gains value purely because time is passing, with everything else unchanged, and time is measured in years.

For the running example, d1 carried to five decimal places is 0.08984 and d2 is -0.16016. Substituting the stock price of USD 100, the strike of USD 105, the risk-free rate of 4%, the volatility of 25% and one year to maturity gives a theta of -6.73, so a long position loses value at USD 6.73 per year if nothing else moves. Desks quote theta by the day instead, and with 252 trading days that is USD 0.0267 (= 6.73/252).

Theta is normally negative for a long option position, since an option is worth less the less time remains for the asset to move. It also sits apart from the other Greek letters: nobody knows what the stock price or the volatility will do, but there is no uncertainty about time passing, so nothing needs hedging against the calendar.

Traders monitor it anyway, because of a relationship the Black-Scholes-Merton analysis delivers.

Set delta to zero, as a hedged book does, and the middle term on the left disappears. Theta and gamma are then tied together with opposite signs: a strongly negative theta comes with a strongly positive gamma, and the reverse. On a delta-neutral portfolio, theta is a readable proxy for the gamma risk being carried.

Rho

Rho measures sensitivity to the level of interest rates, with rates expressed as decimals.

In the running example the call option has a rho of 44, so an increase of 10 basis points, which is 0.1% or 0.001 as a decimal, would raise the option price by USD 0.044 (= 0.001 x 44). For most option books, interest rate uncertainty is a much smaller worry than uncertainty about the asset price or its volatility. Options written on interest rates themselves are the exception.

Any Greek letter for a portfolio of derivatives on the same asset is the weighted sum across the components, short positions entering as negative quantities. Take a book made up of three holdings, each written on the same stock, with the per-option sensitivities and the resulting contributions set out below.

Source: per-option sensitivities from the chapter; the three contribution columns and the totals are calculated here.

Watch the signs on the third row. Those put options carry a negative delta and the position in them is short, so the two minus signs combine into a positive delta contribution of 16,200 while the vega and gamma contributions stay negative.

The limits a trader works to

Delta, vega and gamma limits are set in different units, which is easy to trip over. A delta limit might be USD 100,000, expressed as an equivalent position in the underlying asset; if that asset trades at USD 20, the limit on delta as calculated here is 5,000 (= 100,000/20). A vega limit might be USD 200,000 per 1%, so a one point move in volatility must not change the book by more than that. A gamma limit might be 500 deltas, so a USD 1 move in the asset price must not change delta by more than 500.

Size matters here. Two traders, one running ten options on an asset and one running 1,000, each restore delta with a single trade at the end of the day. The larger book spreads the bid-ask spread on that trade across 1,000 positions and the smaller across ten, which may not earn enough to cover it. Economies of scale of that kind are much of why derivatives dealing sits with a handful of large institutions.

Beyond the five main letters, desks sometimes track three more: vanna, which links delta to volatility, charm, which links delta to elapsed time, and vomma, which links vega to implied volatility. All fall out of the same Taylor series expansion.

Every Greek letter looks at a small move in one variable over a short interval. Traders also run sensitivity analysis over longer horizons with several variables moving together, asking what a month of a sharply higher or lower asset price, combined with a change in volatility, would do to the book.

When the underlying asset provides a dividend yield at rate q, each formula picks up a discount factor and d1 changes.

The value of q depends on what the option is written on.

Source: substitutions for q given in the chapter, with the futures exception alongside.

Reading the Greek letters off a binomial tree

American options have no closed-form Greek letters, so they are taken numerically from the valuation tree. Take a four-step tree built for an American call option on a currency, struck at 0.8000, with the exchange rate currently 0.7800 and one year to maturity. Volatility is 12%, the domestic risk-free rate is 2%, and the foreign risk-free rate is 6%.

Delta comes from the two nodes at the end of the first step, as the change in the option price over the change in the exchange rate: (0.0394 – 0.0051)/(0.8282 – 0.7346) = 0.37. Gamma needs two deltas, both from the end of the second step. The upper pair gives (0.0794 – 0.0127)/(0.8794 – 0.7800) = 0.6710 at an average rate of 0.8297, and the lower pair gives (0.0127 – 0)/(0.7800 – 0.6918) = 0.1440 at an average rate of 0.7359. Gamma is the change in delta over the change in the rate: (0.6710 – 0.1440)/(0.8297 – 0.7359) = 5.6.

Theta compares the initial node with the middle node two steps later, six months on. The option price falls by 0.0061 (= 0.0188 – 0.0127) over that half year, so theta is -0.012 per year. Vega needs a second tree: rebuilding this one with a volatility of 12.1% rather than 12%, keeping the number of steps the same, raises the option value by 0.00026 to 0.01904, which puts vega at 0.0026 per 1% of volatility. Rho is found the same way by nudging the interest rate.

Delta hedging can be run in reverse. The day-to-day rebalancing described earlier effectively manufactures a long position in the call option that was sold, and an asset manager who wants a put option on a portfolio but cannot buy one can use the same machinery deliberately. Rather than taking a position that offsets the delta of an option already held, the manager takes one that matches the delta of the option being created.

Suppose a portfolio is worth USD 100 million and the manager wants protection against its value falling below USD 90 million over the next six months. Volatility on the portfolio is 25%, the dividend yield is 2%, and the risk-free rate is 3%. So the option to be created is a put with K = 90 million against S0 = 100 million, with r = 0.03, q = 0.02, a volatility of 0.25 and T = 0.5.

That gives d1 = 0.7127. The delta of the put option is e raised to the power of minus qT, multiplied by N(d1) minus one, which comes to -0.236. To match it the manager sells 23.6% of the portfolio and holds the proceeds in cash. Delta changes as markets move and time passes, and the manager keeps the amount sold equal to the current delta. Hold that discipline and something very close to the desired put option has been created out of the portfolio itself.

The cost is built into the mechanics. Matching a put delta means selling into a falling market and buying back into a rising one, the pattern that makes delta hedging expensive. A manager whose portfolio tracks an index has an easier route: leave the portfolio alone and buy options on index futures on an exchange. If that options market is not deep enough, the required options can themselves be created synthetically by trading the index futures.

Portfolio insurance was popular into the late 1980s, and its weakness was exposed on Monday, October 19, 1987, commonly called Black Monday, when the Dow Jones Industrial Average fell by more than 20% in a single day. As prices dropped, portfolio insurers sent sell orders to the exchange, those orders pushed the market down further, which called for more selling, and the volume overwhelmed the exchanges so that trades could not be handled promptly. The strategy protected far less than promised, and it has been much less popular ever since.