The classic definition of leverage for options, frequently called 'lambda' ( sometimes effective gearing or also leverage factor), is defined as Delta times Stock price/Option price (Δ*s/p_call). It is the percentage change in an option price given a percentage change in an underlying price. A leverage of 4 would mean that the option price increases by 4% if spot moves 1% up. Soc Gen offers a quick explanation of gearing and effective gearing for options (a vanilla warrant is really just an option, ignoring details like dilution etc.).
Since most tools provide strike (and IVOL), you can simply compute the ratio yourself and see which one is closest to 4%.
The paper is about Bitcoin options, which I admittedly know nothing about. I presume it will be similar to FX and you can price it with Garman Kohlhagen - unless the option is on a BTC future, in which case one would use Black. Either way, there might be specifics about numeraire, delta convention (FX can be premium included, excluded, spot or forward delta for example - see for example here or here). I think generally, the presented method seems to be fairly simple where the exact details should not matter too much.
I'll use Julia to show this in simple code (correct pricing is a bit more nuanced as can be seen here).
# define packages and cdf
using Distributions, DataFrames, Dates
N(x) = cdf(Normal(0,1),x)
# generic call pricer
function GK(S, K,t,rd,rf,σ)
d1 = ( log(S/K) + ( rd -rf + 0.5*σ^2)*t ) / (σ*sqrt(t))
d2 = d1 - σ*sqrt(t)
c = S*exp(-rf*t)*N(d1)-K*exp(-rd*t)*N(d2)
delta = exp(-rf*t)*N(d1)
return c, delta, delta*(s/c)
end
# get dates
d = Date(2023,02,24)-Date(2023,01,11)
# define inputs
s,k,rd,rf,vol,t = 17561.49, 14000, 0.04, 0.02, 0.609, d.value/365
# get results
res = GK(s,k,t,rd,rf,vol)
DataFrame(Call = res[1], Delta = res[2], Leverage = res[3])
I solved for a strike that results in a leverage of roughly ~4 beforehand, to see what BTC option from Delta Exchange matches.
My toy example results in a call price of $3819, and an effective leverage of 4.048. The price is within $20 of the exchange which is OK, given I did not care about exact daycount etc.
What is left now is to show that this is indeed a leverage of 4. All we need to do is to bump the underlying BTC price by one percent (s*1.01), run the option pricer and compute the difference in the option price.
P.S. with regards to the other suggestion, I do not think this is used. The price of the option alone has very little to do with leverage. Using the current option chain for SPY, you would get for spot of 395.52 that you need to aim for a price of ~32.96.
Looking at Yahoo finance we get the following data for the option that meets this criteria best (I cross checked on Nasdaq, and it is identical, but I would need to compute IVOL myself because delta is also not displayed).
I'll use the FX model because it is essentially the same, just another interest rate instead of a dividend. The results:
The price of the option matches again (within bid ask spread as IV seems to be mid here, last price is outdated). However, leverage is close to 9.
