# Holding all other parameters constant, why is theta higher for ATM calls than for ATM puts?

Compare theta for ATM puts and ATM calls with all other parameters (e.g. volatility, strike price, price of underlying, risk-free rate, etc.).

• I don't know enough of the theory to put in a formal answer, but I think this has to do with different volatility skews between the puts and calls. And this is not always the case, although mostly correct for stocks, you may find some options on some commodities that exhibit the opposite. Also something to do, for securities, to the fact that moves on the downside tend to be more violent/pronounced than on the upside, and this gets reflected in the the IV of each option which will drive the price. – Pascal Belloncle Apr 23 at 3:32

Assuming 0 dividends, the underlying is not hard to borrow, and the option is European:

If the risk free rate is 0, the theta decay for strike and maturity matched options on the same underlying are equal. If the risk free rate is negative, the put will have a higher theta decay.

The intuition for this is that a long call option position borrows money to buy stock. There is implicit interest expense.

A put is effectively short the stock and lending cash. This is implicit interest income.

Consider a long call option and short put option position with the same strike and expiration. It replicates a futures contract. Hopefully, this example will aid the intuition about why the call is a form of borrowing cash and the put is a form of lending cash.

I'm not well versed in the Greeks since I only use delta. I'll take a stab at it but take this answer with a grain of salt.

Theta has different formulas for both call and put options. Whether it's merely a change of signs to account for direction or whether it relates to the distribution curve, I couldn't tell you. It's above my pay grade. Perhaps a calculus quant will drop by and offer that explanation.

What I can tell you is the pricing relationship of puts to calls when at-the-money:

• P + I = C - D

Let's assume that there's no dividend. Then:

• P + I = C

Due to the carry cost, the call's premium will exceed the put's premium by the amount of the carry cost or "I" .

Since theta is the amount that an option's value will decline every day until expiration and since the call premium is higher then the call's theta must be higher.

Theta is the phenomenon that if the underlying's price remains stable, the option's value goes down. Stocks usually go up, and options are priced with that in mind. So if the stock does not, in fact, go up, then it's performing less well as expected, which helps puts and hurts calls.

Another way of looking at it: when you say "ATM", that means that the strike price is equal to the current underlying price. But the expected price at expiration is higher than the current price, so a call that is currently ATM is, in some sense, underwater (in that the option will be valuable only if the stock underperforms expectations). And theta is higher for underwater options.

Charles Fox talks about borrowing money, which is yet another way of looking at this principle.