Why Long-Dated At-the-Money Calls and Puts Can Have Equal Prices

# Put-Call Parity for Long Time Frames


# Put-Call Parity for Long Time Frames












Suppose we're dealing with European call options and put options on stocks (say Berkshire Hathaway, that pays no dividends and is unlikely to for the foreseeable future), and assume that the current interest rate environment continues (i.e. we can assume ~0% risk-free rates).

Then for a call and put whose strike price is the current stock price $S_0$, the put-call parity relationship implies that $C = P$ (the price of the call option is equal to the price of the put option).

Now assume we are dealing with super long-dated LEAPS, (say 10 years before expiry). What I don't understand is, on the one hand, if we examine longer and longer dated options, the price of the call option must increase, (to account for general asset price inflation over long time horizons, otherwise the call option would be too cheap). On the other hand, the price of the call option cannot increase too much, because this would imply that $P$ would also increase, but then the risk/reward for selling a put would dramatically improve. (Imagine BRK B shares at \$250, and a 10 year call and put option selling for \$200).

Is this not a contradiction? This option identity does not make sense to me when looking out into the future sufficiently long.

## Answer by fes (score 1)

https://quant.stackexchange.com/a/63655

Nice question. The short answer is of course that if $C>P$ you could make riskless profits by buying puts and stocks and writing calls.

But the prices of long dated securities can indeed look counterintuitive. For example Warren Buffett has said that the Black-Scholes model gives strangely high prices for very long dated puts (like 100 year maturity). It somehow seems like it doesn't take into account the positive drift in stock prices that makes it highly probable that stock prices after 100 years are above current prices. This criticism is different from fat tails that here would imply an even higher put price.

If $M$ is the stochastic discount factor, your version of put call parity implies

$$\mathbb{E}[M\max(S-S_0,0)]=\mathbb{E}[M\max(S_0-S,0)]$$

How can this be true given over large horizons when stock prices have a positive drift implying that it is much more likely for the call to land in the money?

There are two reasons. First even though the call has a higher chance of landing in the money, there is a small chance that the put will pay very well. Moreover, payoffs in the extreme states in which the put pays well are more valuable.

Ian Martin wrote a paper (https://personal.lse.ac.uk/martiniw/long%20run.pdf) where he explains these issues more formally. He shows that payoffs in extreme states drive the pricing of assets with very long horizons.