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Greek Lessons: Gamma Explained

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The last of the “Big 5” Greeks we will be look­ing at is Gam­ma.

The Greek philoso­pher Her­a­cli­tus once stat­ed, “The only thing that is con­stant is change.” The same can be said for the­o­ret­i­cal option val­ues.

There are a mul­ti­tude of fac­tors that go into the option pric­ing mod­el such as under­ly­ing price, strike price, volatil­i­ty, days to matu­ri­ty, inter­est rates, and div­i­dends. These inputs are all used to cal­cu­late what a “fair” val­ue price of a giv­en option should be trad­ing in the mar­ket­place. If any of these inputs change, the val­ue of the option will also change.

The option pric­ing mod­el cal­cu­lates “sen­si­tiv­i­ties” to changes in pre­mi­um. In oth­er words, we can mea­sure exact­ly how much a giv­en option val­ue will change with a change in any one of the inputs.


Reviewing Delta

The most com­mon mea­sure­ment, or “Greek,” is known as Delta. In a pre­vi­ous post, we explained how Delta is the first deriv­a­tive in the option pric­ing mod­el with respect to changes in the under­ly­ing price. More specif­i­cal­ly, it mea­sures how much an option’s val­ue changes for a $1 move in the stock or index val­ue.

As a stock moves up or down, the Deltas of options in rela­tion to that stock must also change.

For exam­ple, let’s say we own the June 100 call with the stock trad­ing at $100. From our knowl­edge of Delta, we can con­clude that this option has approx­i­mate­ly a +50 Delta. If the stock moves up from $100 to $101, the call option is now in the mon­ey and the Delta moves towards 100.


The Relationship Between Gamma and Delta

Gam­ma, on the oth­er hand, mea­sures the change in Delta for a $1 move in the under­ly­ing. It is sim­ply known as the “change in change.”

Gamma Greek Symbol | Greek Lessons: Gamma ExplainedThe Greek let­ter “G,” pro­nounced ‘ɡamə’.

Gam­ma is the sec­ond deriv­a­tive in an option pric­ing for­mu­la with respect to changes in the under­ly­ing and is the only sec­ond deriv­a­tive among the “Big 5” Greeks (Delta, Gam­ma, Vega, Theta, and Rho).

Phrased dif­fer­ent­ly, Gam­ma does not mea­sure direct changes to option pre­mi­ums; instead, it mea­sures the change in Delta. By def­i­n­i­tion, Gam­ma is expressed as a per­cent­age of a $1 move in stock or index price. How­ev­er, it is much sim­pler to remem­ber that Gam­ma is just a num­ber added or sub­tract­ed to the cur­rent Delta of an option when the under­ly­ing moves $1.

Using our pre­vi­ous exam­ple, if the Delta of an option is +50 but increas­es to 60 as the stock increas­es by $1, then the Gam­ma would be 10—the dif­fer­ence between the change in Delta.

Just like Delta, Gam­ma has bound­aries. Remem­ber the Delta of an option is bound­ed by 0 to 100 cents (or -100 to 0) because as a func­tion of stock price move­ment only, an options pre­mi­um can­not move more than $1 for each $1 move in the cor­re­spond­ing under­ly­ing.

Gam­ma also has the same bound­aries because the max­i­mum amount that Delta could change is 100. In oth­er words, an option that has a Delta of zero (out of the mon­ey option about to expire) could all of a sud­den have a Delta of 100 (in the mon­ey option about to expire). This change in Delta for a $1 move in the under­ly­ing is Gam­ma.


Properties of Gamma

Put-Call Parity

Long or short options have pos­i­tive or neg­a­tive Gam­mas whether they are calls or puts. Remem­ber that option pric­ing mod­els assume Delta neu­tral­i­ty, which assumes iden­ti­cal pay-off struc­tures can be repli­cat­ed var­i­ous ways with options. It is pos­si­ble to con­struct syn­thet­ic long under­ly­ing by pur­chas­ing a call and sell­ing a put on the same under­ly­ing with the same strike and expi­ra­tion.

We know that the Delta of under­ly­ing is always 100; there­fore, the Delta of syn­thet­ic stock (long call and short put) should always be 100 as well.

In our pre­vi­ous exam­ple, the June 100 call had a Gam­ma of 10 and the Delta increased from 50 to 60 as the stock rose $1. The put will have the same Gam­ma as the call and its Delta will also move by 10. The trick with puts, how­ev­er, is that they have neg­a­tive Deltas. So, the June 100 put, which also start­ed out as an at the mon­ey option with a -50 Delta, moves from -50 to -40. This is a pos­i­tive increase in Delta. If we apply the syn­thet­ic stock check, we see that the new call Delta (+60) plus the Delta of the short put (+40) adds up to +100 Deltas. In sum­ma­ry, the Gam­mas of puts and calls on the same under­ly­ing, expi­ra­tion, and strike will equal each oth­er as a result of put-call par­i­ty.

Now let’s look at what strikes have the most Gam­ma with­in the same expi­ra­tion.

At the Money Options and Gamma

As a gen­er­al rule, the at the mon­ey strike will have the most Gam­ma with­in an expi­ra­tion. The change in Delta will be most pro­nounced as an option moves from being at the mon­ey to either in or out of the mon­ey.

By exam­in­ing an extreme exam­ple, we can imbed the con­cept of an at the mon­ey option hav­ing the most Gam­ma.

Let’s say an at the mon­ey option is sec­onds from expir­ing. The option pric­ing mod­el will still show the option as hav­ing a 50 Delta. If the under­ly­ing moves up and the option expires, a call option will now have a 100 Delta and the cor­re­spond­ing put will have a 0 Delta. This is quite a large move in the Delta and thus shows how at the mon­ey options have the most Gam­ma.

For anoth­er way to under­stand what Gam­ma is like, let us turn to an anal­o­gy.

Say, for exam­ple, you have two bas­ket­ball teams, maybe North Car­oli­na and Vil­lano­va. When are changes to the score most impor­tant? Obvi­ous­ly, in a close game with only a few sec­onds left on the clock. If the score is Vil­lano­va 74, North Car­oli­na 71 with only 6.7 sec­onds left, ANY change in the vari­ables mat­ter, but a made three point­er will change the com­plex­ion of the game entire­ly. This is the “high Gam­ma” part of the game. If the score is tight, that is like hav­ing a near- or at-the-mon­ey option where small moves in price will deter­mine whether that option is a win­ner (in the mon­ey) or los­er (out of the mon­ey).

Out of the Money Options and Gamma

What about strikes that are fur­ther away from the under­ly­ing price? Well, as expi­ra­tion approach­es, out of the mon­ey options are approach­ing 0 Delta and in the mon­ey options are approach­ing 100 Delta, so their Delta will not move as much as the Delta of an at the mon­ey option. Thus, when Gam­ma is small in absolute terms, Delta changes slow­ly with changes in the under­ly­ing asset price. How­ev­er, if Gam­ma is large in absolute terms, Delta is high­ly sen­si­tive to the changes in the under­ly­ing asset price.

Using our pre­vi­ous exam­ple of a bas­ket­ball game, imag­ine a sit­u­a­tion where we are only five min­utes into the first half of the game or maybe a game that is a blowout. A buck­et is still worth two or three points, but its impact on the final out­come is much less impor­tant. Changes to the score aren’t as impor­tant in those sit­u­a­tions. Along those same lines, if there is a lot of time left to expi­ra­tion, or if the option is deep in or out of the mon­ey, Gam­ma is low­er.

Time and Gamma

With respect to time, short­er term options will have more Gam­ma than longer term options. Refer­ring to our expi­ra­tion exam­ple, an option with more time will not have such a pro­nounced Delta change as one that is expir­ing soon­er. The chart below illus­trates both Gam­mas with respect to time and strike loca­tion. Options with more time until expi­ra­tion have smoother Gam­mas, with the at the mon­ey strike (at the mon­ey = 100) still hav­ing the most Gam­ma with­in a giv­en expi­ra­tion. One can clear­ly see how the Gam­ma increas­es great­ly as expi­ra­tion approach­es, again, with the at the mon­ey strike still hav­ing the most Gam­ma than sur­round­ing strikes.

Gamma, Expiration, Strike Prices | Greek Lessons: Gamma Explained



Even though Gam­ma does not direct­ly mea­sure changes in option val­ues for changes in under­ly­ing prices, it is still an impor­tant risk mea­sure­ment.

It sig­ni­fies the poten­tial changes in option pre­mi­um as mea­sured by Delta. Many option pre­mi­um strate­gies are short term in nature, and as a result, these strate­gies take on a tremen­dous amount of Gam­ma risk.

Pre­mi­um sell­ing strate­gies that appear Delta-neu­tral may have hid­den risks such as Deltas mov­ing around as options approach expi­ra­tion or the poten­tial for cer­tain types of spreads to pro­duce an inor­di­nate num­ber of Deltas under cer­tain mar­ket con­di­tions.

One of the ben­e­fits of short pre­mi­um strate­gies is the col­lec­tion of Theta, or time decay; how­ev­er, Gam­ma can quick­ly wipe out any Theta ben­e­fit if the under­ly­ing begins to move towards the short strike and the options approach expi­ra­tion. A pre­mi­um sell­ing pro­gram must not only be con­cerned about max­i­miz­ing Theta, but min­i­miz­ing the effects of Gam­ma and the result­ing Deltas on a port­fo­lio.

A port­fo­lio man­ag­er can nev­er just focus on one type of risk but must jug­gle all of them to pro­duce the most suit­able risk-reward pro­file. Only a sound man­age­ment process void of “blind­ers” can con­cur­rent­ly nav­i­gate all the risks while adding incre­men­tal returns to a port­fo­lio in all mar­ket con­di­tions.

Check out the rest of the Greek Lessons blog series:

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About the author: 

Chris HausmanChris Haus­man, CMT®, Direc­tor of Risk Man­age­ment and Chief Tech­ni­cal Strate­gist,  focus­es on risk assess­ment and man­age­ment for the Defined Risk Strat­e­gy invest­ments and posi­tions. He mon­i­tors risk across all of Swan’s port­fo­lios and pre­pares stress tests, risk assess­ment reports and con­tributes to strate­gic deci­sion mak­ing for the invest­ment man­age­ment team, as well as serv­ing as an addi­tion­al lay­er of over­sight for the trad­ing team. As a Char­tered Mar­ket Tech­ni­cian, he also acts as Chief Tech­ni­cal Strate­gist at Swan Glob­al Invest­ments.




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By |2018-10-02T11:16:51+00:00July 6th, 2017|Blog|Comments Off on Greek Lessons: Gamma Explained