Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 28J / d

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- Solution d

Solution

Write

ϕ = Φ^{'}

for the standard normal density. Differentiating the Black-Scholes formula and using

S_{0} ϕ (d_{1}) = K e^{- r T} ϕ (d_{2})

gives

\frac{\partial C}{\partial σ} = S_{0} ϕ (d_{1}) (\frac{\partial d _{1}}{\partial σ} - \frac{\partial d _{2}}{\partial σ}) = S_{0} T ϕ (d_{1}) > 0 .

The last equality follows by differentiating

d_{1} - d_{2} = σ T

. This derivative is the option's vega.

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