How to use it
- Enter the spot price and the strike price in CAD.
- Enter the number of days to expiry.
- Enter an annual volatility, as a percentage.
- Enter a risk-free interest rate, as a percentage.
- Read the call and put values, deltas and breakevens, then change the volatility to see how sensitive they are.
What the model calculates
Black–Scholes values a European option, which can be exercised only at expiry. It assumes the underlying price follows a lognormal random walk with constant volatility, that you can borrow and lend at a constant risk-free rate, and that you can trade continuously with no fees. Under those assumptions an option can be replicated with the underlying asset and cash, which fixes its price.
The calculator counts calendar days over a 365-day year and uses a standard approximation of the normal distribution that is accurate to about seven decimal places.
Delta is the model's estimate of how much the option's value changes for a small move in the spot price. A call delta of 0.359 means the call would gain roughly $0.36 for each $1 rise in spot, all else equal; a put delta of −0.641 means the put would lose roughly $0.64. Delta itself changes as the price moves and as expiry approaches.
Worked example
Illustrative inputs: spot $3,000, strike $3,300, 30 days to expiry, 70% volatility, a 3% rate.
N(d2) for the call is 28.67%: the model's risk-neutral probability that the call finishes in the money. It is not a forecast of real-world odds. The call minus the put equals spot minus the discounted strike (−$291.87). That relationship, put–call parity, is a useful check on any pair of option quotes.
The breakevens ignore trading fees and the bid–ask spread, which in thin option markets can be large relative to the option's value.
| Output | Call | Put |
|---|---|---|
| Theoretical value | $131.84 | $423.71 |
| Delta | 0.359 | −0.641 |
| Breakeven at expiry | $3,431.84 | $2,876.29 |
Volatility drives the price
Volatility is the one input you cannot observe directly, and it dominates the result. With the same inputs at 40% volatility, the call is worth $42.55 instead of $131.84. At the money (strike $3,000, 70% volatility), the call is $243.20 and the put $235.81.
Markets quote an implied volatility: the figure that makes the model match the traded price. Implied volatility usually differs by strike and expiry, which is itself evidence that the model's assumptions do not hold.
Why crypto breaks the assumptions
Because of the violations below, model prices for crypto options are most useful for comparing quotes and understanding sensitivities, not for deciding that an option is cheap or expensive. Bought options can expire worthless, and sold options can lose far more than the premium received.
- Volatility is not constant: it clusters, jumps during sell-offs and shifts with leverage across the market.
- Returns have fat tails and sudden jumps, so large moves happen more often than a lognormal model expects, and options far from the money tend to trade above model values.
- Trading is not frictionless: spreads, fees, gas and gaps in liquidity make continuous hedging impossible.
- There is no single risk-free rate in crypto; on-chain lending rates are neither risk-free nor constant.
- Settlement adds risk: an on-chain option depends on smart contracts and price oracles (see oracle risk), and an off-chain one on the venue that holds your funds.
Canadian context
Platforms that offer crypto products to Canadians, including platforms located outside Canada, are expected to be registered with Canadian securities regulators. Check any platform's status before trading, and see regulation.
Frequent option trading may weigh toward business income under CRA's factors, which include how often you transact and your knowledge of the markets; see capital vs business income.