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Pricing Models
To account for the limitation of the AMM mechanism, an adjusted BSM model will be used to price the option.
$d_1 = \frac{ln( \frac{S_0}{X}) + t(r-q+\frac{\sigma^2}{2})}{\sigma \sqrt{t }}$
$d_2 = d_1 - \sigma \sqrt{t}$
$C = S_0 e^{-qt}*N(d_1) - Xe^{-rt}*N(d_2)$
$P =X e^{-rt}*N(-d_2) - S_0e^{-qt}*N(-d_1)$
$S_0$
= Market Price of the Asset
$r$
= Risk Free Rate (8%-10%)
$q$
= Dividend Rate (it’s 0 for most crypto assets)
$\sigma$
= Annualized volatility of the asset
$t$
= Time to expiration in year
$X$
= Strike price
1.
2.
Base volatility used in the BSM model would be realized volatility calculated from the past 120hr hourly closed price. After transferring the hourly volatility to yearly volatility by multiplying
$\sqrt{24*365}$
, let’s say 60%, an additional 50% ramp-up will be applied to the volatility making it to 90%. The volatility multiplier is applied based on our past observation of the difference between the implied vol and realized vol of the crypto market, and such multiplier is subjected to be changed through governance mechanism.
When the option with strike price is “far” from the market price, the implied volatility increases. For example, considering the market price to be $50,000, put/call option with a strike price at$10,000/$90,000 will tend to have implied volatility higher than the put/call option with strike price$40,000/$60,000. We add this correction by including a multiplier defined as Volatility Smile Multiplier based on the absolute percentage difference between the strike price and the market price. $|Strike Price- Market Price|/Market Price*Volatility Smile Multiplier$ Assuming the Adjusted Vol calculated from the 120 hour realized close price is 90% and the market price is$50,000 If the Volatility Smile Multiplier is 2, an option with \$70,000 strike price would be applied with the volatility calculated this way:
$(|70000-50000|/50000*2+1)*90\%=162\%$
A table is made to demonstrate the following call option with different strikes would have the following volatility used in BSM to calculate their premiums.
Strike Price
Market Price
20,000
50,000
198%
30,000
50,000
162%
40,000
50,000
126%
50,000
50,000
90%
60,000
50,000
126%
70,000
50,000
162%
80,000
50,000
198%
90,000
50,000
234%