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Greek Lessons: Rho Explained

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Rho: Interest Rates’ Impact on Option Pricing

An option’s sen­si­tiv­i­ty to changes in inter­est rates is known as “rho,” which is one of the Greeks used to deter­mine how an option’s val­ue can change.

Rho | Swan Blog

The Greek let­ter “P”, pro­nounced ‘rōw’.

One of the more hot­ly debat­ed top­ics is exact­ly when and how much the Fed­er­al Reserve will be rais­ing inter­est rates in the near future.  It final­ly appears a cred­i­ble plan of sys­tem­at­ic inter­est rate rais­es is in place.

How­ev­er, mar­ket con­di­tions can still have an impact on whether or not the Fed­er­al Reserve ulti­mate­ly acts or remains “data depen­dent.”  After many years of essen­tial­ly zero rates, it was inevitable that rates should final­ly move high­er.

This is a good oppor­tu­ni­ty to review how option prices are affect­ed by changes in inter­est rates (Rho) and what that means.

Federal Reserve Interest Rate Probability

Federal Rate Probability - Greek Lessons Rho | Swan Blog

Source: Bloomberg

In gen­er­al, of the five basic inputs used to price options (Delta, Theta, Vega, Gam­ma, and Rho), inter­est rates are of the least con­cern.  This is because most option strate­gies are short-term in nature, and changes in inter­est rates will have very lit­tle impact.

In con­trast, option strate­gies that have longer matu­ri­ties, such as Long-Term Equi­ty Antic­i­pa­tion Secu­ri­ties (LEAPS), can have a sig­nif­i­cant impact on option pric­ing. Once again, Rho risk gets lit­tle respect since most traders employ short term strate­gies; thus there is lit­tle impact on option prices with changes in inter­est rates.  How­ev­er, there can be mate­r­i­al changes to pre­mi­ums with longer-dat­ed options, so Rho risk should not be ignored.

There are cash flows involved in trad­ing options that can result in either cred­its or deb­its.  These cash flows will either earn inter­est or be charged a bor­row­ing fee. There­fore, inter­est rates must be con­sid­ered as one of the basic inputs that go into the gen­er­al­ly accept­ed Black-Scholes option pric­ing mod­el.

Inter­est rates affect the for­ward price of a giv­en under­ly­ing.

For our exam­ple, we will use stock-type set­tle­ment.

  • If inter­est rates increase, that increas­es the for­ward price of the under­ly­ing, and call prices will rise and put prices will decline.
  • If inter­est rates decrease, that decreas­es the for­ward price of the under­ly­ing and now put prices will rise and call prices will decline.

Option pric­ing mod­els assume delta neu­tral­i­ty. 

Let’s explain this by using put-call par­i­ty, which assumes iden­ti­cal pay-off struc­tures can be repli­cat­ed var­i­ous ways with options.

Under the assump­tion of par­i­ty, there are no arbi­trage oppor­tu­ni­ties.

  1. A long stock posi­tion can be hedged by short­ing a call and pur­chas­ing a put.
  2. Con­verse­ly, a short stock posi­tion can be hedged by pur­chas­ing a call and sell­ing a put.

If inter­est rates decline, the hold­er of long stock will be pay­ing less in bor­row­ing costs and should be will­ing to sell the call at a cheap­er price and pur­chase the put at a high­er price to hedge the posi­tion.

If short stock is held and inter­est rates rise, then the short stock rebate col­lect­ed will increase and the short stock hold­er should be will­ing to pur­chase the call at a high­er price and sell the put at a low­er price to hedge the posi­tion.

In oth­er words, if one does not account for Rho, a clever trad­er would be able to earn a risk­less, arbi­trage prof­it.

Typ­i­cal­ly, the risk-free rate is the rate used in the option pric­ing mod­el to derive the­o­ret­i­cal prices.  Yields on gov­ern­ment secu­ri­ties with matu­ri­ties equiv­a­lent to the expiry of the option would be the appro­pri­ate input into the mod­el.  In real­i­ty, how­ev­er, we can­not lend and bor­row at the same gov­ern­ment rate so a more real­is­tic, or “trad­able,” rate may be the yield on Eurodol­lar futures, for exam­ple.

Com­pli­cat­ing mat­ters even more, most traders must bor­row at a high­er rate and lend at a low­er rate, so their mod­els might even incor­po­rate mul­ti­ple rates like required when eval­u­at­ing options on for­eign cur­ren­cies.

In short, Rho is pos­i­tive for calls and neg­a­tive for puts.

As dis­cussed, the Rho is much greater for longer term options.  Rho increas­es as calls go fur­ther in the mon­ey and decreas­es (gets more neg­a­tive) as puts go fur­ther in the mon­ey.

Illustrating the Impact of Rho

Below are two graphs demon­strat­ing the Rho char­ac­ter­is­tics for calls and puts.  In the call exam­ple, the stock price is at $2120.  Strikes range from 1000 up to 3500 and the time until expi­ra­tion ranges from 0 out to 720 days.  The graph clear­ly shows that deep­er in the mon­ey calls (strikes less than the cur­rent stock price and with more time to matu­ri­ty) have the most sen­si­tiv­i­ty to changes in inter­est rates.

Call Rho Stock 2120 - Greek Lessons Rho | Swan Blog

Source: Swan Glob­al Invest­ments

The same can be said for puts, except now high­er strikes with more time until expi­ra­tion will have the most sen­si­tiv­i­ty to changes in inter­est rates.

Put Rho Stock 2120 - Greek Lessons Rho | Swan Blog

Source: Swan Glob­al Invest­ments

Those who uti­lize option strate­gies must always be cog­nizant of all the poten­tial risks. No option in any strat­e­gy is “per­fect.”  There is always a trade-off between the poten­tial fac­tors that can alter an option’s val­ue.  Suf­fice it to say, even though options with longer-term matu­ri­ties do have more expo­sure to changes in inter­est rates, this risk is off-set by low­er decay costs and more sta­bil­i­ty with respect to volatil­i­ty than short­er-term options.  Ris­ing inter­est rates actu­al­ly decreas­es the cost of put cal­en­dar spreads, so it is pos­si­ble that loss­es suf­fered as a result of a ris­ing inter­est rate envi­ron­ment in the short-term, may be off­set in the future when it is time to roll puts by pur­chas­ing a “cheap­er” spread.

Over the last few years with short-term rates held close to zero, Rho risk has essen­tial­ly been a non-issue.  This will like­ly change mov­ing for­ward as we are about to begin a new inter­est rate cycle that many have not wit­nessed or sim­ply for­got what it felt like.

Con­se­quent­ly, adher­ing to sound risk man­age­ment prin­ci­ples and prop­er assess­ment of all risks will help nav­i­gate you in mer­ri­ly achiev­ing your finan­cial goals.

Check out the rest of the Greek Lessons blog series:

Feel free to review more infor­ma­tion on the Defined Risk Strat­e­gy per­for­mance, or its com­po­nents, or call 970.382.8901.

For more infor­ma­tion on the Defined Risk Strat­e­gy per­for­mance, call 970.382.8901.


Chris HausmanAbout the author:
Chris 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 |2018-10-02T11:17:10+00:00May 2nd, 2017|Blog|Comments Off on Greek Lessons: Rho Explained

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