If you are like me then for most of your career in finance you’ve been in a “near-zero” world. Between global monetary easing, QE and other alphabet soup programs designed to keep interest rates down, worrying about rates seemed archaic… that was until 2021 when inflation which was obviously transitory turned out to be less transitory, and rates globally began to climb.
What’s interesting is that while most people in the space know how important interest rates are to virtually everything else in finance, few people (including myself) have really had to contend with a changing (let alone rising) rate environment.
Right now I think that this is no more evident than in the world of option pricing. I know, I know a scary topic for even the most seasoned professionals, but important nonetheless. I bring it up because lately I’ve seen a lot of market commentary on “how cheap it is to buy protection on the market right now” (from a historical perspective) and while that isn’t incorrect, the next part that sometimes follows “this implies the market isn’t worried about potential downside” is. Mainly because in a world where over the last decade rates have been low and stable, not many people have had to think about the “forgotten” Greek. I’m talking of course about Rho. As a quick refresher Rho is the change in option price to a change in interest rates… a metric that hasn’t been all that important in a world where rates have been almost certain.
But now that the environment has changed, think of this as a quick refresher on how rates impact option pricing and let’s gain a more intuitive understanding on why currently lower put prices might not necessarily mean market complacency.
Okay so simply Rho is positively correlated to Calls and negatively correlated to Puts. Interest rates go up … generally good for Calls … bad for Puts.
You can see right there why some of the market commentary is misleading… markets might not necessarily be complacent, just that in a higher rate environment Calls are going to be more expensive and Puts are going to be cheaper…
But why though?… well most people will point you to the Black-Scholes model for pricing options (let’s assume no dividends).
For Calls:
\(C = S * N(d1) - Ke^{-rT} * N(d2)\)
For puts:
\(P = Ke^{-rT} * N(-d2) - S * N(-d1)\)
If you’re not familiar with it or just haven’t looked at it in a while don’t get too hung up here on all the variables. Simply, you can see that based on where the discount factor is in these equations \(e^{-rT}\) that when \(r\) increases it will decrease the second half of the call equation (which we are subtracting) in turn increasing the call value but on the put side decrease the first half of the equation (which is getting subtracted) and in turn decreasing the put value.
technically yes \(r\) does live withing \(d1\) and \(d2\) but let’s keep it simple for now
Another interpretation of the Black-Scholes would be to state you are buying \(S * N(d1)\) worth of stock by borrowing funds \(Ke^{-rT} * N(d2)\). If the cost of borrowing has increased, then the upfront cost to purchase the call also increases.
On the put side it’s the opposite and since the rate you are earning has increased the total upfront cost decreases.
I feel like these interpretations are easy to see based on the equation but probably not very intuitive. And I feel the best way to gain intuition behind option pricing as it relates to Rho is to forget about the equations and look at a more practical model like binomial trees.
In its simplest form an option is the sum of the expected value of its future payoff. More specifically, its the present value of the expected value of its future payoff.
Here is a quick example:
We flip a coin. Heads you get $1 and tails you get $0. We both know that its a fair coin so the probability is 50/50. So simply the value of this game is $0.50.
Okay, now lets say we flip a coin today, but we keep it covered for a year. After a year we look and if it’s heads you get $1 and tails you get $0. The value now is $0.50 but discounted back 1 year. Let’s say that the risk free rate is 5% then simply
\(\$0.50/(1+5\%) = \$0.4761905\)
Basically you could take $0.47 now and invest it for a year and earn $0.5, or buy my game now for $0.47 and in a year have an expected value of earning $0.5. Since the expected value of both outcomes are the same a risk neutral investor would be indifferent to these two investments… but more on that later.
Switch it up a bit:
Now lets say there is a stock that we know for sure has a 50/50 probability of going up or down in one year. If it goes up it will be worth $2 and if it goes down it will be worth $0.5. If the risk free rate is still 5% how much should the stock be worth today?
The logic is exactly the same:
\((\$2*50\% + \$0.5*50\%)/(1+5\%) = \$1.190476\)
What if its an option:
Now let’s change it so that instead we buy an ATM option so that we get either a $1 payout if the stock goes up or $0 if it goes down. Again, knowing for sure it has a 50/50 probability.
So the option price should be:
\((\$1*50\% + \$0*50\%)/(1+5\%) = \$0.4761905\)
… okay great but I thought this was about interest rates…
Hold on.
Yea, sorry, we’re going to have to go there…
When I was writing level 2, I can’t say I fully understood this at the time and question the amount of people who actually do. But its important, especially for this.
In essence risk neutral probabilities are the implied probabilities that would make a risk neutral investor indifferent to holding a risk asset vs a risk free asset.
If that means nothing to you maybe an example would help.
Take the hypothetical stock above where in a year price either moves to $2 OR $0.5. Originally we assumed we knew the exact probabilities of this event occurring 50/50… but what if we didn’t? Well then we’d live in the real world. But here is what we likely do know… the stock’s price and the risk free rate. So lets say that the price of this asset is $1, then mathematically we can solve for what the market implied risk neutral probabilities should be:
\(\$1 = \$2*P(PositiveOutcome) + \$0.5 * P(Negative Outcome)\)
Assuming these are the only two outcomes then \(P(PO) + P(NO) = 1\) and \(P(NO) = 1 - P(PO)\)
so then
\(1 = \$2 * P(PO) + \$0.5 * [1 - P(PO)]\)
BUT this payoff is in one year so…
\(1 = [\$2 * P(PO) + \$0.5 * [1 - P(PO)]] / ( 1+ 5\%)\)
so some simple algebra:
\((1+ 5\%) = 2 * P(PO) + 0.5 - 0.5 * P(PO)\)
\((1+ 5\%) - 0.5 = 1.5 * P(PO)\)
\(P(PO) = [(1 + 5\%) - 0.5] / 1.5\)
\(P(PO) = 0.3666667\) OR ~ 37%
Intuitively this should make sense … when the probability was 50/50 price was higher ($1.190476) … so if the price is known to be lower (actually its $1) then the risk neutral probabilities should be skewed to the downside P(NO) > P(PO). Intuitively, you’d pay less for something today if the chances of that something paying off in a year were lower.
I struggled with this concept for a while, not fully understanding what it meant. But it’s an important topic in finance because it allows us to use one of the few known variables (or at least few agreed upon variables) which is the risk free rate and few observable variables: underlying price, to then price risk assets like derivatives.
First, what is a risk neutral investor again?
They are an investor who is indifferent to the expected value of outcomes.
Remember the coin toss game above… that $0.5 guaranteed now, vs a game where you get $1 for heads or $0 for tails , are equal to this investor as they both have an expected value of $0.5.
But why does it matter that the expected values are the same?!
Well because if these events actually happen in the future, it means I can use the risk free rate for both.
Why?
Well the $0.5 I know is guaranteed, so the risk free rate is appropriate. BUT since I also know that (to the investor) the $0.5 risk free outcome and the game of the heads/tail are equal (since EV is the same), I can also use the risk free rate to price the present value of the game.
This is incredibly useful because in the real world we seldom know the likelihood of any outcome… but we do have markets for pricing. That means we can use the price to determined what probabilities are implied that would make a risk neutral investor indifferent to investing in a risk free asset vs buying this risk asset. And when we do that we can then use those weights with the risk free rate to price our derivatives.
The other solution would be…not to do this… but then we’d need to make an assumption on the actual probability of the events occurring and find a non risk-free discount rate to account for the fact that there is now risk.
Here is a quick example
Let’s go back to the stock above trading at $1 and in a year will either be worth $2 or $0.5.
I decide the P(Positive Outcome) should be 70%.
Which means the expected value in one year is:
\(\$2 * 70\% + \$0.5 * [1 - 70\%] = \$1.55\)
If I discount this back at the risk free rate I would get a value of \(\$1.55/(1+5\%) = \$1.47619\)
But I know for a fact that the price is $1 … so what’s changed? Well the outcome has more risk than the risk free rate implies so needs a higher discount rate. Since its only one period this would simply mean that the implied discount rate is 55%.
At the end of the day, both methods get to a price of $1, but look at how much more uncertainty we’ve added to the model. For starters the weights are completely random and we’ve effectively removed interest rates from the model because we’re now solving for the implied discount rate.
Using the risk neutral approach, we can take two know variables: the risk free rate and underlying price and solve for the implied probability. We can then use these three variables to solve for our derivative price.
… I thought this was about interest rates and options …
I’m getting there.
I know I seem hung up on risk neutral probabilities but its because, if you haven’t figured it out yet, it is at the heart of how rates relate to option pricing.
Let’s revisit the example above, the stock is trading at $1, payoff is still either $2 or $0.50 but now interest rates go up to 10%… the stock price though doesn’t move… what does this mean?
Lets go back for a second.. we know that
\(Price = [\$2 * 37\% + \$0.5 * (1 - 37\%)] / ( 1+ 5\%)\) will equal 1 … but now
\(Price = [\$2 * 37\% + \$0.5 * (1 - 37\%)] / ( 1+ 10\%) = \$0.95454545\) …
What happened?
Our denominator increased so obviously we have a lower value if we keep all other variables the same… but remember we know looking in the market that price is still equal to $1. Well the only other variable that can change is P(Positive Outcome)… this probability needs to go up … it has to because we need a larger expected value to discount back at the larger risk free rate… and if we do re-solve for P(Positive Outcome) when rates increase to 10% we now get P(Positive Outcome) = 40%.
\(Price = [\$2 * 40\% + \$0.5 * (1 - 40\%)] / ( 1+ 10\%)= 1\)
If at this point you’re thinking… “wait but then we are using a price to solve for a price”… well yes… but different prices. We are using the stock price to determine the risk neutral probabilities in the stock… then we use the same risk neutral probabilities to find the option price … basically the two need to be internally consistent… which makes sense because one is just a derivative of the other.
So now if we sub the expected stock price with the expected payoff of the option, take the risk neutral probabilities from the underlying and discount at the risk free rate we can find a fair price for the option that is consistent with the current underlying price and risk free rate.
You can also see that when rates go up, the risk neutral probability of the price going up, increases. The key assumption of course is: everything else is equal. Simply the risk neutral probability of positive outcome is positively correlated to rates. This is also fairly intuitive. Most people know that generally rates and price have inverse relationships (think of bonds), so for stocks if rates go up but all other variables including price remain stable, then obviously the implied probability of a higher payoff must have gone up. This is advantageous to Calls where the payoff is related to prices going up… so if the probability of this happening has increased… the value of the call should increase. On the other hand as rates went up the risk neutral probability of the price going down must decrease. This is not favorable for Puts where the payoff and value are related to prices moving down.
To tie it all back, this interpretation also shows up in the Black-Scholes model as well. Where? Remember? \(d1\) and implicitly then in \(d2\). Where you can interpret it as the probability of being ITM.
Hopefully this provides some context into option pricing as it relates to interest rates. As we move into what is potentially a new regime of higher secular inflation, having a better understanding of how rates are priced into all financial assets will be increasingly important. Gone are the days of passively setting the risk free rate… Rho is back.