This is a question frequently asked during technical interviews. I will try to answer it through this short article giving examples and the logic behind them. Here are the main points I deal with here:
- Reminder: Gamma and Theta
- Black Scholes vs Real World
- Vertical Spread
- Conclusion – Smile skew
Gamma is defined as the second derivative of the option price with respect to the underlying. Gamma for a vanilla call and put is given by:
with the normal density function. A long position on an option makes you Gamma positive. Gamma tells us how our delta will change for a 1$ move in the underlying.
Theta (silent killer)
Theta is defined as the first derivatives of the option price with respect to the time. Theta for a vanilla call is given by:
The formula for the put option can be derived with the call-put parity. Theta is the decay of the extrinsic value (time value) of an option, all else equal. One has a positive Theta when selling an option and a negative Theta when buying it.
Black & Scholes
Within the B&S world, volatility is flat. Options on the same underlying will have the same volatility whatever the strike and the maturity chosen. In this case, it’s impossible to find a combination leading to a positive gamma and theta (it can be proven from the Black&Scholes PDE that Gamma and Theta are opposite signs).
In reality, we all know volatility for all options isn’t flat. We talk about implied volatility and it depends on the strike and the maturity of your options. It’s deduced from the market price of each option and therefore by supply and demand. This screenshot shows a set of implied volatilities for different strikes and maturities:
Notice the pattern here: OTM (Out The Money) puts have higher implied volatility than OTM calls (spot is 24.74€).
The smile changes everything for our research. Actually, Greeks are sensitive to volatility. Let’s illustrate this with a little example I made on Excel:
- Call S=45 K=50 T=1Y vs
We can see here that every greeks changed. Now let’s focus on Theta: with a higher volatility and all else equal, we finally get a higher Theta in absolute value.
Now let’s plot the gamma for different spot value:
We can notice that Gamma is max for ATM options and decreases for ITM and OTM options.
Here we are, we have everything we need to make a Theta and Gamma positive combination! Being Theta and Gamma positive is possible when shorting the high volatility (OTM option) and going long the low volatility (near ATM).
Bull Call Spread
A bull call spread is a combination in which you go long a call and short a call with . Let’s see what happen when inserting a big smile into the call spread pricer:
We did it, we are Gamma and Theta positive.
Insight: the volatility gap between the two calls is so big that the OTM call Theta is higher than the ATM call Theta.
Bear Put Spread
Naturally, we can also find a Bear Put Spread that leads to the same kond of result. All we need is a smile, sell the OTM put and buy ATM put:
We are Gamma and Theta positive, the logic behind is exactly the same.
I tried to find another trick playing with a calendar spread but I did not find a plausible combination that leads to a positive Gamma and Theta. In fact, it works in inserting a really dropping implied volatility for the longer maturity but it joins the same logic as we saw just before.
Consider 2 options near the money with same strike, one long-term and one short-term. You sell the short-term and buy the long-term. The short-term option has both a higher Theta and Gamma than the long-term and then it leads to a positive Theta and negative Gamma. Indeed, for ATM options, Theta and Gamma increase when expiration is coming.
However, if you include a lower volatility (short-term uncertainty about a company?) for the long-term option its Gamma will increase and may exceed the Gamma of the short-term option. Example :
Theta and Gamma positive combinations exist in the real world. However, it’s not that easy to find for some reasons:
First, we need a large implied volatility gap between our two options:
- In the vertical spread, so that the OTM Theta is higher in absolute value than the ATM Theta while not going too far away out the money.
- In the horizontal spread, so that short-term option has less Gamma than the long-term option.
Second, in the equity derivatives market, OTM calls have lower implied volatility OTM puts. We usually find skew in the smile, like this one:
Reason for this skew is the fear of the downside. Investors are long stock in general and perceive the risk of a crash much greater than the risk of the soaring price. Thus they buy OTM puts as a protection.
In this market, it will be easier to find a Put Spread leading to a positive Gamma and Theta.
In commodities, we see the converse. OTM alls have higher implied volatilities than OTM puts. Companies run their business with commodities, their margin disappears if prices go up, they fear the upside. Then they buy OTM calls a protection.
In this market, it will be easier to find a call spread leading to a positive Gamma and Theta.