Calculate the theta, gamma, vega and rho for European call and put options.
Note: I have assumed a non-dividend stock.
Θ = ∂Π/∂t
Θc = - ½S0N'(d1)σ/√T - rKe-rTN(d2)
Θp = - ½S0N'(d1)σ/√T + rKe-rTN(-d2)
Γ = ∂2Π/∂S2
Γc = Γp = N'(d1)/S0σ√T
v = ∂Π/∂σ
vc = vp = S0N'(d1)√T
ρ = ∂Π/∂r
ρc = KTN(d2)e-rT
ρp = -KTN(-d2)e-rT
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2 comments:
Could you pleas show, how by using your transformation you can go from the heat equation back to Black-Scholes.
That's an interesting question. I've never done that. I *believe* that every step in my BSE->HE proof is reversible (i.e., if one simply starts at the bottom and works backward the proof holds together), but this would need verification, as there may be assumptions for which the converse assumption is not valid.
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