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Greek Lessons: Delta Explained

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Delta: A Driving Force Behind Options Pricing


  • Delta is a mea­sure of the sen­si­tiv­i­ty of price changes in the option rel­a­tive to price changes in the under­ly­ing asset.
  • The Delta of an option shifts based on changes in price of the under­ly­ing asset.
  • Delta, not to be con­fused with beta, has a non-lin­ear rela­tion­ship with the price of the under­ly­ing, which when illus­trat­ed shows how loss reduc­tion occurs as delta shifts down.
  • From a prac­ti­cal stand­point, the more the mar­ket moves down­ward, the more the port­fo­lio is pro­tect­ed as the delta shifts down.
  • The non-lin­ear shift in delta means it takes a while for down­side pro­tec­tion to ful­ly kick in.

What dri­ves the price of an option?

Option pric­ing is some­times described as “mul­ti-dimen­sion­al,” mean­ing a vari­ety of fac­tors simul­ta­ne­ous­ly influ­ence the price of an option. These fac­tors are referred to as “the Greeks” and are named using let­ters of the Greek alpha­bet- delta, theta , rho, gam­ma, and vega. Under­stand­ing all of “the Greeks” is impor­tant in under­stand­ing how the price of an option or option strat­e­gy might fluc­tu­ate. In this post we will focus on delta.

The text­book def­i­n­i­tion is this: “Delta is the ratio com­par­ing the change in the price of the under­ly­ing asset to the cor­re­spond­ing change in the price of a deriv­a­tive.”

In oth­er words, delta mea­sures the sen­si­tiv­i­ty of price changes in the option rel­a­tive to price changes in the under­ly­ing asset.

Delta vs Beta

If these def­i­n­i­tions sound famil­iar, they should. For those trained in tra­di­tion­al finance, these def­i­n­i­tions sound a lot like beta.

Before option pric­ing the­o­ry was ful­ly devel­oped, William Sharpe, Jack Treynor, and a few oth­ers devel­oped the Cap­i­tal Asset Pric­ing Mod­el and the con­cept of beta.

Beta mea­sures how sen­si­tive an asset’s price is to move­ments in the broad mar­ket.  There­fore:

  • An asset with a beta of 1.0 is expect­ed to move in lock-step with the mar­ket.
  • An asset with a beta greater than one is assumed to be aggres­sive: up more than the mar­ket when it is up, but down more when the mar­ket is down.
  • Con­verse­ly, an asset with a beta of less than one is con­sid­ered to be con­ser­v­a­tive, mov­ing less than the mar­ket when the mar­ket gyrates up and down.

The graph and table below illus­trate these three sce­nar­ios:

The graph and table below illus­trate these three sce­nar­ios:
Greek Lesson: Return Profiles of different Betas | Swan BlogTable of Returns Based on Varying Delta | Swan Blog

Take-Away #1

One key take­away from this graph is that beta is assumed to be lin­ear.

In plain Eng­lish, regard­less of where the mar­ket might be, the beta rela­tion­ship is assumed to be con­stant.

It doesn’t mat­ter if the mar­ket is down 20% or up 35%, if an asset’s beta was esti­mat­ed to be 1.15, it will be 1.15 regard­less of where the mar­ket is.

Delta is sim­i­lar to beta, but with a key dif­fer­ence.

Like beta, delta mea­sures price sen­si­tiv­i­ty rel­a­tive to changes in the inde­pen­dent vari­able. How­ev­er, the key dif­fer­ence is that delta is NOT a lin­ear rela­tion­ship.

The rela­tion­ship between an option and the under­ly­ing asset is illus­trat­ed below. The price move­ments fol­low a curved line, some­times infor­mal­ly referred to as a “hock­ey stick” pat­tern.

Price Movement of Hedged Position vs Market- Varying Delta | Swan Blog


The impli­ca­tion of this curved rela­tion­ship is that delta, unlike beta, is not a con­stant num­ber. The delta will change depend­ing upon the price of the under­ly­ing asset.

  • At a cer­tain point in the curve, the delta will be 0.5, mean­ing that a 1.0% price move up or down in the under­ly­ing asset would result in a cor­re­spond­ing 0.5% move in the option’s price.
  • At the far end of the curve, a delta of 1.0 would indi­cate a 1.0% price move in the under­ly­ing asset’s price should cause a 1.0% move in option price.
  • On the oth­er extreme end of the curve, a delta of 0.0 means a 1.0% price move in the under­ly­ing option price won’t have any impact on the price of the option.

But most impor­tant is the fact that, the delta of an option changes or shifts; it is not con­stant.

Take-Away #2

From a prac­ti­cal stand­point, the more the mar­ket moves down­ward, the more the port­fo­lio is pro­tect­ed as the delta shifts down. This means it takes a while for down­side pro­tec­tion to ful­ly kick in.

In a future blog post we will dis­cuss the rea­sons why delta is not con­stant. It has to do with a con­cept known as the “mon­ey­ness” of an option. In order to round out this blog post, we will instead focus on how delta impacts the per­for­mance of a hedged strat­e­gy.


Impact of Delta on Performance of Hedged Equity Strategy

Let’s assume a hedged strat­e­gy starts off with a delta of around 0.50.

Sce­nario 1:

From that ini­tial point, if the mar­ket were to go down 10%, the strat­e­gy would like­ly go down 5%. If the mar­ket con­tin­ued down, the delta would now be dif­fer­ent, more in the neigh­bor­hood of 0.25. If the mar­ket sold off anoth­er 10% the strat­e­gy should be down an addi­tion­al 2.5% or so. Once mar­ket has sold off from the hedge point by 20% or more, that is where we enter the flat part of the curve where the delta is zero.

Sce­nario 2:

Con­verse­ly, if the mar­ket were to go up 10% from the hedge point, the strat­e­gy would go up in the neigh­bor­hood of 5%. If the mar­ket con­tin­ued upwards, the strat­e­gy would like­ly cap­ture three-quar­ters of the next 10%, as the delta would move toward 0.75.  As the delta reach­es 1.0, the strat­e­gy should then see dol­lar-for-dol­lar gains past that point.


It is impor­tant to note this sim­ple exam­ple and sce­nar­ios are meant to be a gen­er­al frame­work on under­stand­ing delta and price rela­tion­ship between options and the under­ly­ing asset.

The exam­ple excludes some of the oth­er fac­tors that will undoubt­ed­ly impact the total per­for­mance of Swan’s Defined Risk Strat­e­gy. These changes include, but are not lim­it­ed to, the impact of the mar­ket-neu­tral, pre­mi­um-col­lec­tion income trades, which is a sig­nif­i­cant com­po­nent of the over­all DRS returns. Also, per­for­mance dis­per­sions can occur due to the equal­ly-weight­ed sec­tor approach used rel­a­tive to the cap-weight­ed S&P 500. Final­ly, as we men­tioned at the out­set, option pric­ing is mul­ti-dimen­sion­al, and will be influ­enced by the oth­er Greeks like gam­ma, vega, theta, and rho. We will take a look at these oth­er fac­tors in future blog posts.



Marc Odo, Marc Odo, CFA®, CAIA®, CIPM®, CFP®, Director of Investment Solutions - Swan Global InvestmentsAbout the author: Marc Odo, CFA®, CAIA®, CIPM®, CFP®, Direc­tor of Invest­ment Solu­tions, is respon­si­ble for help­ing clients and prospects gain a detailed under­stand­ing of Swan’s Defined Risk Strat­e­gy, includ­ing how it fits into an over­all invest­ment strat­e­gy. For­mer­ly Marc was the Direc­tor of Research for 11 years at Zephyr Asso­ciates.



Impor­tant Notes and Dis­clo­sures:

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By |2018-10-02T12:16:18+00:00May 5th, 2016|Blog|Comments Off on Greek Lessons: Delta Explained