# 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.

• "to account for general asset price inflation" that's where the discount factor (via na interest rate) comes into play - short term rates are close to zero, but 10-year rates (expected inflation) are not. So that expected inflation would be accounted for in the 10-year interest rate. Apr 28, 2021 at 19:05
• When I say asset price inflation, I do not mean inflation in the CPI sense or the GDP deflator sense. I mean that stocks are valued with an implicit discount rate, and all the studies show that even in the most risk-on times, the equity risk premium is above 3-4%. So I don't think it's necessarily enough to say it's accounted for in the 10 year interest rate. Apr 28, 2021 at 19:18

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.

• Thanks for the answer and the link. Apr 28, 2021 at 19:27
• While I understand that there is a small chance of disaster, the payoff does not seem commensurate with the probability. For example, if we assume that 10 year calls are priced such that if the average return of the underlying is above r=6% over the next 10 years, then you are in the money, then the cost of a call option with \$250 strike is approximately \$198. But if a very unlikely event (alien attack) occurs that destroys Berkshire, you would have made (only) 25% off your options. Apr 28, 2021 at 19:39
• Doing exercises such as this for r=5%, r=4%, etc. also yields pretty poor returns for the put options, but we're in a territory where 10 years in, given a good 10 year stretch, we are far ahead of where the call was priced, with non-negligible probability. Apr 28, 2021 at 19:41
• There is something about merging the two schools of thought (asset prices as reflecting intrinsic value of a business) and (asset prices modelled as a time series using some probability distribution) which doesn't seem to mix very well. Apr 28, 2021 at 19:43
• @Snowball In your case I think the question is ultimately more about why drift doesn't directly affect arbitrage free pricing and how that can create situations which are seemingly counterintuitive.
– fes
Apr 28, 2021 at 20:16