The Black–Scholes formulas for the prices of European calls and puts on a non-dividend paying stock are:
C = SN(d1) – Xe-rT N(d2)
P = Xe-rT N(-d2)- SN(-d1)
Where d1 = [ ln s/x +(r+δ2/2)T ]/ δ √T
And d2 = d1 –δ√T
• The Black Scholes equation is done in continuous time. This requires continuous compounding. The “ r ” that in this is ln(1+r). E.g. if the interest rate per annum is 12%, you need to use ln1.12 or 0.1133, which is continuously compounded equivalent of 12% per annum.
• N () is the cumulative normal distribution. N(d1 ) is called the delta of the option which is a measure of change in option in option price with respect to change in the price of the underlying asset.
• δ a measure of volatility is the annualized standard deviation of continuously compounded returns on the underlying. When daily sigma are given, they need to be converted into annualized sigma.
• Σ annual = Σ daily * √number of trading days per year. On a average there are 250 trading days in a year.
• X is the exercise price, S is the spot price and T the time to expiration.
C = SN(d1) – Xe-rT N(d2)
P = Xe-rT N(-d2)- SN(-d1)
Where d1 = [ ln s/x +(r+δ2/2)T ]/ δ √T
And d2 = d1 –δ√T
• The Black Scholes equation is done in continuous time. This requires continuous compounding. The “ r ” that in this is ln(1+r). E.g. if the interest rate per annum is 12%, you need to use ln1.12 or 0.1133, which is continuously compounded equivalent of 12% per annum.
• N () is the cumulative normal distribution. N(d1 ) is called the delta of the option which is a measure of change in option in option price with respect to change in the price of the underlying asset.
• δ a measure of volatility is the annualized standard deviation of continuously compounded returns on the underlying. When daily sigma are given, they need to be converted into annualized sigma.
• Σ annual = Σ daily * √number of trading days per year. On a average there are 250 trading days in a year.
• X is the exercise price, S is the spot price and T the time to expiration.