Arbitrage free volatility surface and negative implied dividend yield

[brownianpotion]

I am currently trying to tidy up the implied volatility for options on the SPX and I am finding a negative yield for the SPX.

The current spot is 3969.61,
the strike is 3970,
the risk-free rate is 0.047631 (calculated from zero coupon),
the time to expiration is 0.016438 (expiration on 17th January 2023),
the call price is 50.3,
and the put price is 45.35.
The implied dividend yield is -0.03421.
Dividend futures yield is currently 0.017=~1.7%.

Does this mean that the implied borrow rate for the SPX is around 5% or have I made a mistake?

 

[Robert Morse]

For spot, are you using the cash index or the corresponding future for that expiration

 

[brownianpotion]

For spot, are you using the cash index or the corresponding future for that expiration

the cash not the forward. I'm using the formula in Hull for the implied dividend

implied dividend= -1/T * ln(c-p+Strike*e^(-rfr*T) / S(o))

 

[newwurldmn]

Likely a mistake.

what’s the vol you are getting ?

However this short term you might have a div yield close to zero as the real arbitrage free will be the actual divs paid. Over a short period it may not be 1.7percent. The borrow and risk free rate is a yield as you pay that every day but divs are actually discrete.

 

[stepandfetchit]

I am currently trying to tidy up the implied volatility for options on the SPX and I am finding a negative yield for the SPX.

The current spot is 3969.61,
the strike is 3970,
the risk-free rate is 0.047631 (calculated from zero coupon),
the time to expiration is 0.016438 (expiration on 17th January 2023),
the call price is 50.3,
and the put price is 45.35.
The implied dividend yield is -0.03421.
Dividend futures yield is currently 0.017=~1.7%.

Does this mean that the implied borrow rate for the SPX is around 5% or have I made a mistake?

Can you provide the time of that sample? Something seems off with your data! The time to expiration (and value of spot) seems to infer price from close yesterday, but the price of the PUT and CALL don't agree, so unsure what the reference point in time really was! -- Below I post what I find for the close yesterday for those two options. -- Warning: These shorter terms (dte's under 14 days) should be taken with grain of salt. -- I have not pursued making these precise and have not addressed better interest rates for <30 day options. Hopefully they will be in the ballpark.

 

[stepandfetchit]

 

[stepandfetchit]

 

[brownianpotion]

Can you provide the time of that sample? Something seems off with your data! The time to expiration (and value of spot) seems to infer price from close yesterday, but the price of the PUT and CALL don't agree, so unsure what the reference point in time really was! -- Below I post what I find for the close yesterday for those two options. -- Warning: These shorter terms (dte's under 14 days) should be taken with grain of salt. -- I have not pursued making these precise and have not addressed better interest rates for <30 day options. Hopefully they will be in the ballpark.

Cheers! I've attached the data I obtained from yfinance eod on the 11th when the market was closed. I used the bid/ask mid for calls and puts (the numbers read last Price, bid, ask). I can see how you arrived at your forward here from the pictures you've attached using the bid/ask mid for the calls and puts. Is that the bid/ask from the 11th eod?

How do you determine the risk-free rate? I've utilised a term structure model to fit the zero coupon bonds data from the Treasury to the time left to expiry.

What is value U in your picture and how is it calculated?

Your implied dividend + borrow cost seems to have come out to ~3.06% here, in order for you to arrive at the forward using S*exp^(rfr-borrow-dividend).

 

[brownianpotion]

Likely a mistake.

what’s the vol you are getting ?

However this short term you might have a div yield close to zero as the real arbitrage free will be the actual divs paid. Over a short period it may not be 1.7percent. The borrow and risk free rate is a yield as you pay that every day but divs are actually discrete.

strike iv delta gamma rho theta vega
3975 0.2352 0.5067 0.0033 0.3228 -4.6071 2.0301
3970 0.2361 0.5233 0.0033 0.3332 -4.6244 2.0269

Fascinating fact about the arbitrageurs, I am utilizing FactSet's method for volatility surface modelling as a guide and found it quite surprising that they employ continuous implied dividends for the index and discrete for equities (https://insight.factset.com/hubfs/White Papers/Implied_Volatility_Surface_WP.pdf)

What would be the most suitable approach for obtaining the dividend and borrow that fits the market implied?

 

[brownianpotion]

Something that just occurred to me: I should be able to take the ln() of the forward formula forward= spot*e^(rfr-div-borrow rate), use the option implied forward and then solve for the aggregate of implied borrow and dividend rate