# Setting the r in put-call parity?

Put-call parity is given by $C + Ke^{-r(T-t)} = P + S$.

The variables $C$, $P$ and $S$ are directly observable in the market place. $T-t$ follows by the contract specification.

The variable $r$ is the risk-free interest rate -- the theoretical rate of return of an investment with zero risk.

In theory that's all very simple. But in practice there is no one objective risk-free interest rate.

So in reality, how would you go about setting $r$? Why?

• everyone here seems to be assuming that the stock does not pay dividends! – NIKO Jun 21 '18 at 16:12

## 6 Answers

If your trades are collateralized/margined, you should use the rate paid on your collateral/margin.

This is not a trivial question. Here's a relevant excerpt (an appetizer, really) from Hull's book (7th Edition, P. 75):

It is natural to assume that the rates on Treasury bills and Treasury bonds are the correct benchmark risk-free rates for derivative traders working for financial institutions. In fact, these derivative traders usually use LIBOR rates as short-terrn risk-free rates. This is because they regard LIBOR as their opportunity cost of capital (see Section 4.1). Traders argue that Treasury rates are too low to be used as risk-free rates because:

1. Treasury bills and Treasury bonds must be purchased by financial institutions to fulfill a variety of regulatory requirements. This increases demand for these Treasury instruments driving the price up and the yield down.
2. The amount of capital a bank is required to hold to support an investment in Treasury bills and bonds is substantially smaller than the capital required to support a similar investment in other instruments with very low risk.
3. In the United States, Treasury instruments are given a favorable tax treatment compared with most other fixed-income investments because they are not taxed at the state level.

LIBOR is approximately equal to the short-term borrowing rate of a AA-rated company. It is therefore not a perfect proxy for the risk-free rate. There is a small chance that a AA borrower will default within the life of a LIBOR loan. Nevertheless, traders feel it is the best proxy for them to use. LIBOR rates are quoted out to 12 months. As we shall see in Chapter 7, the Eurodollar futures market and the swap market are used to extend the trader's proxy for the risk-free rate beyond 12 months.

• Things have changed in the rates world since Hull wrote that. Traders typically use an OIS rate as the risk free rate, not LIBOR. As quant_dev suggests below, the correct rate to use in the put-call parity formula (or indeed for discounting any future cash flow) is the rate you pay/receive on collateral from your counterparties. This is typically the relevant currency's OIS rate (eg EONIA for EUR) – ldnquant May 14 '11 at 18:43
• Too add to complexity, it would also depend on what you are posting or taking in as collateral. The whole "rates and govies is now a risky business" thing is forcing all longer-dated OTC trades to move to forward premium. – Strange Nov 24 '12 at 0:48
• LIBOR rates are also prone to manipulation, since they are declared, not traded rates. – quant_dev Mar 23 '16 at 12:24

I think you might use the relevant OIS-rate like EONIA or Fed Fund Rate, at least this is the current fad when discounting interest rate swaps.

If you are trying to arbitrage the put-call parity, then use your collateral interest rate for the options side, and your cost of funds on the stock side of the equation. Yes, that's right, 2 different interest rates. Also, don't forget to incorporate bid-ask spreads.

If you are trying to turn a put into a call for your own book, you don't actually need this computation, since the total delta is 1.0.

If you are trying to infer volatilities, then you should use the customary interest rates and borrow costs of the market makers.

Look at it the way you would have to realize it, whatever your position in the market is, as the price of a synthetic bond that pays $K$ at time $T$:

\begin{array}{c}Ke^{-r(T-t)} & = & S & + & P & -& C \\ (\text{bid}) &=& (\text{bid}) &+& (\text{bid}) &-& (\text{ask})\\ (\text{ask}) &=& (\text{ask}) &+& (\text{ask}) &-& (\text{bid}) \end{array}

There are other ways to construct a synthetic bond: \begin{array}{c}(K_2-K_1)e^{-r(T-t)} & = & C_1 & - & C_2 & + & P_2 & -& P_1\\ (\text{bid}) &=& (\text{bid}) &-& (\text{ask}) &+& (\text{bid})&-& (\text{ask})\\ (\text{ask}) &=& (\text{ask}) &-& (\text{bid}) &+& (\text{ask})&-& (\text{bid}) \end{array}

That's a lot of spread when it's all said and done, and the "risk-free rate," however you define it, will be right in the middle of that spread. There is likely only an arbitrage here if you're a bookie and make a market in all these options.

I think you should put YOUR attainable interest rate. Because it is your view of how much the forward is worth. So, on the offer the rate at which you are indifferent is computed with the interest rate at which you borrow. And if you go short, the interest at which you can put your have your money fructified.

No what rate can your money be fructified at ? Well you are a department inside of a bank. So that rate should be in between the state funding's rate, if your bank as a whole was cash neutral to begin with, and you actually have some money to place, and the marginal cost of borrowing that you save to your bank if the bank had to go in the market.

But trust the cashiers to always tell your the wrong story..