Quantitative finance notes
Bachelier model
for european options on stock
\( c= S N(d_1) - K e^{-rT}N(d_1)+\sigma \sqrt{T}n(d_1) \)
\( p= K e^{-rT}N(-d_1)-S N(-d_1)+\sigma \sqrt{T}n(d_1) \)
where
- \(c\) = call price
- \(p\) = put price
- \(S\) = stock price
- \(K\) = strike price
- \(T\) = time to expiration (year fraction)
- \(r\) = risk-free interest rate %
- \(\sigma\) = annualized volatility %
- \(N(x)\) = Cumulative Normal Distribution function
- \(n(x)\) = Normal Density function
- \(d_1=\frac{S - K}{\sigma \sqrt{T}} \)