Rho measures the sensitivity of an option's price to changes in interest rates. It represents the change in the option's price per one percentage point change in interest rates. Call options generally have positive rho, meaning their price increases with higher interest rates, while put options have negative rho and their price decreases with higher rates. Rho is usually a small number and has less influence on option prices than other Greeks like delta and gamma. The effect of interest rates is also greater for options with longer times to expiration, as it represents the cost of delaying payment.

1. Lecture No 69
Option Greek -
Rho
Rohan Sharma (Coach)

2. What is Rho ?
Rho measures an option’s sensitivity to changes in interest rates –
how much option premium will change if the risk-free interest rate
increases by one percentage point.
Rho is the ratio of option price change (in dollars) to interest rate
change (in percentage points).
Therefore, its units are dollars per percentage point, although in
practice units are rarely mentioned, as with the other Greeks.
No theoretical limit on the values rho can reach. It can be positive
or negative, but it is usually a very small number.

3. Example - Rho
• Consider a call option on a stock with 3 months left to expiration,
which is currently trading at $2.35 (option premium).
• Its rho is 0.15 and the 3-month risk-free interest rate is 3%.
• The option’s rho indicates that if the interest rate increases by one
percentage point to 4%, the option premium should rise by $0.15
to $2.50.
• Conversely, if the interest rate declines by one percentage point to
2%, the option premium should decrease by $0.15 to $2.20.
• All this assumes that the other factors (the underlying stock’s
price, implied volatility, and time to expiration) remain the same.

4. Rho Values
In most cases, interest rates have far smaller effect on
option prices than underlying price or volatility.
Interest rate changes tend to be smaller and take longer
time (a one percentage point change in the short-term risk-
free interest rate is always a huge event in the markets, while
the same change in implied volatility often happens in
minutes or seconds on many underlying's).

5. Rho Values
Rho often receives less attention than the other main Greeks –
delta, gamma, theta, and vega – and it is less understood by the
typical option trader, also because it is harder to make universal
rules about effects of different factors (underlying price, time or
volatility) on rho values.
These effects depend on option type, underlying type, and the
settlement procedures of both. For example, rho of stock options
behaves differently from futures options rho or FX options rho
(currency options are in fact affected by two interest rates –
domestic and foreign – and have two rhos).

6. Call Option Rho
• The main benefit of holding a call option is the optionality,
right but not obligation to buy the underlying.
•If the stock goes up, you make money, but if it goes down,
you don’t need to exercise the option and you are protected.
You have a choice.
•That said, a call option has another benefit that is sometimes
forgotten: it allows you to control the underlying (at least its
upside) without paying for it.
•It delays payment and improves cash flow.

7. Call Option Rho
 The main benefit of holding a call option is the optionality, right
but not obligation to buy the underlying.
If the stock goes up, you make money, but if it goes down, you don’t
need to exercise the option and you are protected. You have a
choice.
That said, a call option has another benefit that is sometimes
forgotten: it allows you to control the underlying (at least its upside)
without paying for it.
It delays payment and improves cash flow.
Call options are more valuable with higher interest rates and have
positive rho.

8. Example - Call Option Rho
• For example, consider a stock trading at $50 per share. You can buy 100
shares in the stock market and pay $5,000 immediately.
• Alternatively, you can buy a 3-month, $50 strike call option for $2 and
pay only $200 (for one contract of 100 shares). In both cases, you control
100 shares of the stock and make money if the stock goes up.
• However, in the first case you pay $5,000 now, while in the second case
you only pay $200 now (the option premium) and $5,000 (the option’s
strike price) later, if you exercise.
• The option effectively delays your payment for 3 months (from now to
the option’s expiration). It is like a 3-month loan of $5,000.
• Therefore, the option’s time value must reflect not only the optionality,
but also the value of the loan

9. Put Option Rho
• Let’s say you already own 100 shares of the same stock. You are
worried the stock will fall. You can sell the stock in the stock market
and get $5,000 in cash.
• Alternatively, you can buy a 3-month, $50 strike put option. If the stock
does fall, you can exercise the put later and get $5,000.
• Without the put, you receive $5,000 now (and you can put it in a bank
for 3 months and earn interest); with the put you get it in 3 months.
• The higher the interest rate, the more attractive it is to sell the stock
and get the money earlier, rather than buy the put and get the money
later.
• Put options are less valuable when interest rates are higher. They
have negative rho.

10. Rho and Time to Expiration
 From the above examples it should be obvious that the
effect of interest rates on options is greater with longer time
to expiration.
In general, rho approaches zero as an option gets closer to
expiration.

11. Rho of Futures Options
 Underlying security settles like a stock – you must pay the full price in cash
when buying, and you receive the full price in cash when selling. Not all
underlying's settle like that.
When you buy (go long) a futures contract, you don’t pay anything (you only
deposit a margin) and only your profits or losses are marked to market and
settled daily.
Therefore, the above logic of calls having positive rho and puts having
negative rho is valid for stock options or currency options, but not for futures
options or currency futures options.
In general, futures options tend to be less sensitive to interest rates than
stock or currency options, as the cash flow advantage of calls and
disadvantage of puts explained above does not apply.

12. Rho of Currency Options
 With foreign currency options, two interest rates are involved. For
example, consider an option on euros, traded on a US exchange in
dollars.
This option’s price will be affected by both the US (domestic)
interest rate and the Eurozone (foreign) interest rate.
Rho measures the effect of the domestic rate, which is similar to
stock options.
Generally, higher domestic interest rate makes foreign currency calls
more valuable (positive rho) and puts less valuable (negative rho),
because the domestic rate is the cost of financing.

13. Rho of Currency Options
 The effect of the foreign interest rate is measured by rho2
(sometimes called phi).
It is like the effect dividend yield has on stock options.
When you hold the underlying (a stock / euros), you earn it.
When you hold a call option instead, you don’t.
As a result, the higher the EUR interest rate, the less attractive call
options are as an alternative to holding the euros directly.
It is the opposite with puts.
An increase in foreign interest rate makes call options on that
currency less valuable and put options more valuable.

14. Effect of Interest Rates on Underlying
 Understand that rho (and rho2) only measures the effect of
interest rate on option price if the underlying price and all
the other factors remain the same.
 In reality, interest rate changes often move the underlying
price as well – particularly for currency options (interest
rates are the main driver of exchange rates), options on fixed
income instruments (bond prices are closely related to
yields), but also some stock options (e.g. bank stocks may
strongly react on interest rate moves).

15. Effect of Interest Rates on Underlying
Therefore, interest rate changes can affect option prices via
two channels: directly and indirectly via moving underlying
price.
Rho only measures the direct effect, but not the indirect
one.
The effect of underlying price changes on option premium is
measured by delta.
For some underlying's, such as bonds or currencies, the
indirect effect is often much stronger than the direct one.

16. Conclusion
Rho measures how option premium will change if the risk-free
interest rate increases by one percentage point.
Call options on most underlying's have positive rho; put options
have negative rho.
Rho is generally greater (in absolute terms) with more time to
expiration.
For many underlying's like currencies or bonds, interest rates may
also affect underlying price, and thereby option prices.
This indirect effect, though often greater than the direct effect, is
not measured by rho.