# What is the standard risk free rate used to discount options?

Apologies if this has been asked before - but I wanted to clarify what the market standard was for discounting options i.e. what is the "risk free" rate actually used by quants and traders? I haven't really been able to find a concrete answer to this unfortunately.

I've read for instance in text books that the risk free rate suggested is the government bond rate which corresponds to your option maturity. Some books specifically suggest to use zero coupon gov bonds because there is no implicit reinvestment risk.

Practitioners seem to use a fixed rate e.g. 3/4% given rates are currently negative in some places or are too low.

I've also read that OIS rates are used, although I don't know if this reference rate is used to price all options e.g. vanilla equity options.

Any thoughts on this would be greatly appreciated.

Since pretty much all trades (at least interbank) are collateralized nowadays, you would follow the principle of CSA discounting and use the interest rate on the collateral as a discount rate. Typically it's an overnight rate, for example SONIA in GBP, EONIA in EUR, Fed Funds in USD (broad switch to SOFR hasn't yet taken place I think), so you would use OIS rates that reference those, with maturity corresponding to maturity of the option.

They are also the best available approximation to actual risk free rates, because due to the O/N tenor, both market and credit risk are minimized. Treasury yields aren't that great as a risk free rate proxy, as they suffer from many supply/demand induced effects (ex. on-the-run vs off-the-run distinction, the fact treasuries are highly sought after as collateral or safe haven asset, tax related effects) plus one could question whether they are really credit risk free, especially long-dated ones. LIBOR shouldn't be used for discounting, either, because it's just a benchmark nowadays and not investable.

In the cross currency land (FX options) this is further complicated by the fact that you need to decide on one currency as the collateral currency, then use its OIS rate and the cross currency basis to come up with the discount rate in the second currency.

Just deciding arbitrarily that actual rates are "too low" and assuming constant 4% as a discount rate is nonsensical.

• Cheers @Adam N, I’ve read around the subject a bit more since I last posted this question and I reached similar conclusions. It’s unclear to me though how the forward OIS discount curve is actually constructed. Any ideas? – AShortSqueeze Apr 24 '20 at 23:34

As a matter of fact, we know that the yield on zero coupon government bonds of all maturity change over time and that those changes aren't perfectly predictable. As this is a source of risk, it should be compensated and, therefore, the theoretical ideal would be to introduce a stochastic short rate process in your option pricing model.

Some people did just this across a nice set of models: Bakshi, Cao and Chen (1997) looked at the Black-Scholes-Merton model, the Heston model, a jump-diffusion model, as well as a jump-diffusion and stochastic volatility model. Whether you add a stochastic interest rate or not in any of the above, your pricing errors do not diminish and your hedging performance doesn't improve... The reason? Well, think about it for a minute: the small changes in interest rate that occur in a period of a few days to a few months do not command a huge premium. As long as you take into account the fact that in the real world, interest rates have a term structure, you can "cheat" your way out of modeling its behavior.

That's what you'll see academics do in research papers. They'll look for a close match in terms of maturity on a government bond and use the yield at time t on that bond to price all options that have a similar time to maturity at time t. From the stand point of your price process, they will "cheat" by working with empirical excess returns: $$$$ln(S_{t+1}) - ln S_t - r_{t,T}.$$$$

Now, there is a problem with this approach, namely that this isn't a very good approximation of the true cost of capital for trading desks and market makers. So, one thing they could do is pick another interest rate. You could opt for the LIBOR rates, US Treasury repo rates or the Sterling Overnight Index Average instead of the US Treasury bond yield curve. As far as I can tell, this seems to be the simplest way to go. You avoid the modeling problem, it's easy to implement and it takes into consideration the fact that rates always show some kind of term structure.

Using a fixed rate seems odd to me. The rates change and they have a term structure.

• So as I understand it @Stephane - using a fixed risk free rate is wrong, using a risk free treasury bond is also wrong, but using some sort of overnight funding rate (LIBOR for UK, Repo rates for US) is more correct? And the reason why this makes sense is that you have to match the delta hedge with the actual cost of the delta hedge - banks are not delta hedging at the same cost as a sovereign, yes? – AShortSqueeze Apr 20 '20 at 0:22
• A fixed rate is wrong because it ignores the term structure -- so you try to avoid it. And some people will prefer to use something else than treasury bonds for various reasons. One I mentionned is that the borrowing costs faced by traders aren't those faced by western governments. Another could be that if you go toward shorter maturities, you might fear that what you have largely reflects monetary policy. Still, the most common thing is just to use US treasury bonds if you price, say, SP500 options. – Stéphane Apr 20 '20 at 2:31
• I see what you are saying. I do have a preference for using gov bond rates given these rates seem more consistent with the risk neutral pricing framework in BSM as opposed to LIBOR rates which appear to move away from the discounting paradigm in a risk neutral/risk free world. – AShortSqueeze Apr 20 '20 at 6:59
• Also @stephane to clarify, we are clearly talking about forward rates here not historical rates per your answer ie -rt,T. For instance, if I have an SP500 call option maturing in 3 months time, the correct risk free rate is the forward risk free rate for 3 months time NOT the current 3 month risk free rate. – AShortSqueeze Apr 20 '20 at 7:07