Volatility and Earning Calls

Where N is the normal distribution, T is the maturity and t is the current date (so T-t is time to maturity). Now we need to find d1 and d2 to plug them into the formula.

We then need to find every variable we need from the market except for volatility as mentioned before. While everything is quite straightforward, there something to say about the risk-free rate. Some years ago, we would have used 0% as our rate, but right now rates are higher. The 1 year treasury rate is at 5,49%.
The Black-Scholes model uses continues time, so the risk free is an instantaneous rate. We now need to convert the risk-free rate to our r which is ln( 1+risk-free rate ) = ln(1,0549) = 5,344%.
The last step is to turn everything in years, the historical option dataset i used is pricing with 365 days as time metric.

C(t)=

d1= 

d2=

To try it our first I chose Tesla’s call option ATM (on 16/07/2024). To find the implied volatility, given that it appears in both N(d1) and N(d2) we need to use an iterative process such as Newton-Raphson method. This is included in Goal Seek in Excel that makes everything simpler to do.

By doing this, we find the exact implied volatility from our historical option dataset. 

Now that we found our model working properly, we can move on with the next step.

Changes In Volatility

What I wanted to see is how volatility is priced in the market during those days close to earning calls. I chose 3 stocks: TSLA, GOOG, UNH.
From each of these stocks we select three call options, one before the earning call, one after and the one next to that. 

Just by looking at the results, we see the effects of the earning call event.

Not only that, but the implied volatility is often underestimated as we see in SVXY, in months of earning calls we can find little drops which signal unexpected hike in volatility. 

Now let’s try to see what market is expecting in terms of share price.

Implied Price Jump

We can use confidence levels to see what the market priced in for earning call days. We know that with 68% change a value will fall within one standard deviation, 95% within 2 and 99,7% within 3. Taking TSLA as an example, we can extract the implied price jump for each expiration date on different confidence levels. Here I used one call option that expires before the earning call and two that expire after.  The first step is to take implied volatility which is annual and turn it into a daily one. This is the standard deviation of returns around 0, assuming daily returns are independent from one another . We then multiply this by the number of days to expiry, which is square rooted because we are using a Brownian Motion to describe returns and if the variance of time increments is σ2T, the standard deviation must be σ√T, meaning that it doesn't linearly grow with time, but with the square root of it. The price that we get is the based on the standard deviation of return for the number of days until expiration, around 0% of return. 

Sometimes the market gets prediction correct as we can see here: 

The market priced about 15$ of movement and so happened.
Now we can make a confidence representation of the price jump expected on Tesla earning calls to the realized.
Now we can see the jump in prediction and translate it into a jump in share price.

In this case, realized volatility was higher than the implied, probably because earnings prediction were far off the actual ones.

Now, looking at the other stocks we had: 

And as shown before, the earning call volatility is underestimated, in fact there are some of the cases where this happened. 

Profiting from wrong expectation

In conclusion, what can we do to profit from mispricing during earning call times?
If, for example, we are certain that realized volatility will be higher than expected, we can perform a Long Straddle.

This involves a combination of a long call and a long put. 

To tie everything together we can see how implied volatility is linked to the price of option. If we assume we are in an efficient market, we shouldn't be able to profit (on average) from a Long Straddle, meaning that the expected value should be 0. Thus, the price of this strategy, should be coincident with how much we would profit from exercising both options. This means that if we take two ATM options, by exercising the options, we would profit exactly the change in price. The price must then be a great extimation of the movement.