# Implying risk-free rates using Put/Call parity

I recently purchased SPX options data from the CBOE. Normally, if the data is OK and the Put-Call parity holds, one should expect to correctly imply ZC (Zero Coupon bond) prices and forwards by performing a linear regression on all available collar prices (long put and short call of the same strike), given a fixed maturity $T$.

This works indeed very well, until we come close to maturity. More specifically, the linear regression gives wrong results for collars on their last trading day. For example, if I perform linear regression on all the collars available at date 2005-02-17 (Thursday) which expire on 2005-02-19, I get an annualized risk-free rate of 84.85% (or a ZC being worth ~99.5297%), which is clearly wrong.

What I find weird is that this behavior happens only one day before the 3rd friday. Is there something I am missing about how the prices behave when close to expiration ?

EDIT :

I'm adding here a plot of how of the term structure of the implied risk-free rate (that I have smoothed using a Nelson-Siegel fit but the behavior is present regardless of whether I smooth the curves or not) evolves as we approach the third Thursday 2005-02-17 :

When we are a little far from that date, we get more coherent term structures :

• Are these options on spx futures? Thanks
– dm63
Oct 29 '17 at 22:21
• The underlying is the S&P 500 Index and the settlement is in cash.
– BS.
Oct 29 '17 at 22:23
• To me the problem lies in the fact that the closer you get to the expiration date, the less a pure diffusion model makes sense and the more a jump diffusion model does. Oct 30 '17 at 10:59
• I admit I have no clue as to why this should be related to any model whatsoever, since replicating a synthetic ZC bond (corresponding to the risk-free rate) using for example a collar spread (short collar of strike $K_1$ and long collar of strike $K_2$ with $K_2>K_1$) is normally model independent.
– BS.
Oct 30 '17 at 22:04
• See this post: quant.stackexchange.com/questions/1807/… Oct 31 '17 at 18:14

Once upon a time, all option contracts ceased trading on the third Friday of every month. There was no after hours trading for the underlying. When the exchanges closed, everything was done. This is no longer true. Contracts do not exclusively cease trading on the third Friday, although some still do. Likewise, the underlying can continue trading after the options cease trading.

This change creates liquidity mismatches. In fact, an option can close out of the money at the end of options trading, but land in the money by the end of the aftermarket trading of the underlying. From what I have inferred from your posts the cost of liquidity is not being modeled separately from the discount factor.

In addition, some derivative contracts are used as proxies for other assets. This also creates liquidity issues.

You should consider formally modeling liquidity. Bonds will do similar things as they get close to maturity. The market maker has to make back their cost of capital regardless of how long until the closing date. These are not "jumps" so much as a failure to control for the width of the bid-ask spread. The spread often widens towards maturity to prevent a savvy investor from dumping bad inventory on the market maker at a favorable price.

Look at:

A. Abbott, Valuation Handbook, ch. Measures of Discount for Lack of Marketability and Liquidity, pp. 474–507. Hoboken, NJ: Wiley Finance, 2009.

I have used it to model liquidity.

Your model does not consider the costs to the market maker and so is behaving badly on a date that is the common nexus of a variety of contracts. This isn't a "rigor" issue so much as a misspecification issue. This isn't the only type of contract that has this behavior.

• It is indeed the fact that I didn't account for liquidity, and especially how the bid-ask behaves when approaching maturity. Thank you for the reference, I'll spend some time on it to correctly take into account liquidity.
– BS.
Dec 1 '17 at 13:07

Elaborating on my comment: consider a 100 point in the money collar, one day before expiration. You are effectively claiming the price of this is 99.5. But if you call the pit and get a two way price of 99-100 there is nothing to do.