# Which risk free rate is assumed by market when pricing american options?

I'm just started with finance, so maybe my question is dumb or answered elsewhere. Please guide me to relevant materials.

According to put-call parity more time to expiration means more difference between Put and Call prices Call - Put = Spot - Strike*e^(-r*T) My understanding this is to avoid arbitrage between Stock plus Put vs Call plus Deposit. The arbitrage is avoided by embedding deposit returns into Call price.

Now looking at real prices I do not see large difference between Put and Call options prices even for options which have about a year till expiration which suggest near zero risk-free rate. For example, today data from google:

Stock | Expiration   | Spot   | Strike | Put Bid | Call Bid |
AAPL  | Jan 15, 2016 | 109.41 | 110    | 14.95   | 13.40    |
SBUX  | Jan 15, 2016 | 80.43  | 82.50  | 9.20    | 6.55     |


I calculate risk-free rate, assuming T ~ 1, as r = -ln((Put + Spot - Call)/Strike)

In both cases (AAPL, SBUX) risk free rate is slightly less than 0. By looking at this two questions arise:

1. Does my calculations correct?
2. If market assume zero risk free rate does this means call are underpriced? One can still get risk free rate by investing into bonds or saving account. In this case Call plus Deposit will earn more than Stock plus Put since Call price does not have risk-free rate embedded in it.
• Put-call parity is valid only for European options because they are exercised only at maturity. For American options it is not valid. – Arrigo Dec 18 '14 at 13:02
• Good point Arrigo, thank you! If put call parity does not hold can you suggest a way to calculate risk free rate assumed by market when option prices are defined? – averbin Dec 18 '14 at 13:32

First of all, if you are new in quantitative finance, I suggest to read the Hull'book, that's the basic for who wants to get topic fundamentals.

Your evaluation is correct if you assume that linear relationship, but on real prices anything is linear; so, it depends on whath you're looking for: if you have to conclude a project work at your university, it is fine, otherwise it is not.

As regards what you need for about risk-free rate estimation, each option trader has different opinions about the question you raised. For instance, Hull himself suggests to use a fixed risk-free rate equal to 3% in the examples you'll read on the pdf file.

In my humble opinion, you should use the return of the less risky government bond of the area you're studying, as the US T-Bill for North America option market or the German Bund return for the Euro option market.

Moreover, there're a lot of model that deal this topic with and that estimate the proper risk free-rate. If you need particularly something for like that, I suggest to look for papers on SSRN or Google Scholar

• could you please clarify one more thing. If I see that r ~ 0, then I better buy Call and put money into savings account rather than buy stock and put. No matter what former strategy will yield more that latter one, isn't it? – averbin Dec 21 '14 at 1:59

For American options there is no parity rule, as I stated in the comments. However, there is the following disequality:

$$S_0 - D - K \leq C - P \leq S_0 - K e^{-rT}$$

where $C$ and $P$ are prices of American call and put respectively, $S_0$ is the spot price today, $K$ is the strike price, $D$ is present value of the cash dividend (not as percentage), $r$ risk-free rate and $T$ the maturity (this is covered in problem 10.19 of Hull's book). This helps you find a lower bound for $r$, nothing more unfortunately.

The risk-free rate used in the valuation of options must be the rate at which banks fund the cash needed to create a dynamic hedging portfolio that will replicate the final payoff at expiry. Dealers borrow and lend at a rate close to LIBOR, which is the funding rate for large commercial banks. The LIBOR swap curve is therefore the rate to be used when pricing options. It is therefore quite wrong to use a Government bond yield curve.

• Good point. Since especially the 2008 crisis, there is some default risk included in the LIBOR rate. It is better to use rates from Overnight Indexed Swaps.fincad.com/resources/resource-library/wiki/… – mth_mad Jul 24 '16 at 18:29
• Yes that is true. I did not want to get into OIS but wanted to make clear that previous answers referencing government curves were wrong. I will extend my response. – Dom Jul 26 '16 at 8:47