risk free rate in black-scholes?

M

MK020

In calculating option prices with black-scholes, is there any consensus on what should be used for the continuously compounded risk free rate? I was told to use the value given by the daily treasury par yield curve at 3 months. It's published daily here:

https://home.treasury.gov/resource-...yield_curve&field_tdr_date_value_month=202305

The rate listed in the link above for 5/31 was 0.0553. But this yield, I don't believe, is expressed as a continuously compounded value, so I convert 0.0553 to the continuous rate by taking natural log of (1 + 0.0553), and I get 0.0538. Is this what a bank would use as the continuously compounded risk free rate on 5/31?
 
You should use same tenor for interest rate as for your option. For most products, and under 2 year terms, LIBOR rates to closest term and adjusted from that to actual term seem to work fairly well. However, I think that may not be best rate for HTB underlying's, such as UVIX. If you subscribe to option pricing data from IVOLATILITY, they provide daily rates for the terms 1M, 2M, 3M, 6M, and 12M, that are good if converted to term of your option. (Interest rate is dependent on term, and may not be identical for all instruments, but these seem close enough for index products like SPX).
 
stepandfetchit said:
IVOLATILITY, they provide daily rates for the terms 1M, 2M, 3M, 6M, and 12M, that are good if converted to term of your option. (Interest rate is dependent on term, and may not be identical for all instruments, but these seem close enough for index products like SPX).
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Thanks, makes sense to use the same term. I'm most interested in terms of 1 to 4 weeks. Using the data from IVOLATILITY you mentioned, I could build a curve and extrapolate for values past 1 month. But any suggestions on how to best convert to a good estimate for 1 week?
 
MK020 said:
In calculating option prices with black-scholes, is there any consensus on what should be used for the continuously compounded risk free rate? I was told to use the value given by the daily treasury par yield curve at 3 months. It's published daily here:

https://home.treasury.gov/resource-...yield_curve&field_tdr_date_value_month=202305

The rate listed in the link above for 5/31 was 0.0553. But this yield, I don't believe, is expressed as a continuously compounded value, so I convert 0.0553 to the continuous rate by taking natural log of (1 + 0.0553), and I get 0.0538. Is this what a bank would use as the continuously compounded risk free rate on 5/31?
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The conversion to continous rate is correct (as such official rates usually are published not for the continous case), but you have to take the 1 year rate instead of the 3 month rate, ie. annualize if necessary.
And: with stocks, the "risk-free rate" means in fact "annual earnings yield in pct", incl. the rfr.
 
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MK020 said:
I don't think so. Risk free rate should be the amount you would pay in interest if you borrowed money to own shares of the stock during the life of the option (e.g. in delta hedging).
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Nope, another name is "drift rate" --> research this...
It (ie. "r") contains all such rates incl. the earnings yield, the risk-free-rate and the borrow rate etc., except the dividend rate ("q") as this one is a seperate param in BSM.

https://www.math.cuhk.edu.hk/~rchan/teaching/math4210/chap08.pdf
"
An important observation of the Black-Scholes equation (9) is that it does not contain the drift parameter µ of the underlying asset. [...] This means that the drift parameter µ in the stochastic differential equation for the asset can be replaced by r wherever it appears.
"
 
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FWIW: Another approach could be to extract the proper interest rate using box spreads on the targeted option chains. This would tie the interest rate to the specific security. I do not do this currently as it may be problematic with wide bid/ask spreads.

I don't require that much precision on my shorter terms, and currently use the lowest rate term I have (30 days) which I know is wrong, but the error contribution is tolerable for me.
 
Here are the current T-bills vs Market implied interest rates. We use non dividend paying high confidence tickers and show how many there are used.

Day Current T-bills Market StDev Count
30 5.55 5.34 .6 80
60 5.55 5.38 .4 121
120 5.52 5.36 .3 158
365 5.22 5.32 .3 157
730 4.46 5.02 .7 49
 
What if the stock of the company rises annually by 10%, and the risk-free rate is 5%?
What value do you in this case take for r in BSM?
IMO r should now be 15%.

Btw, some options calcuators don't allow a negative r. IMO they are broken... :-)
 
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