Sensitivities and Greeks - Computational Methods for Quantitative Finance - page 153

Fig. 11.1 Convergence rates
of Greeks for a European put
in the Black-Scholes ( top )
and variance gamma model
( bottom )

11.3.1 One-Dimensional Models

We consider two models, the Black-Scholes model and the variance gamma model.
For both models, we consider a European put with strike K

= 100, maturity T =

1 . 0 and interest rate r = 0 . 01, and we calculate the Greeks Delta, = ∂ s V , and

Gamma, Γ = ∂ ss V . For the Black-Scholes model, we additionally compute the

Ve g a , V = ∂ σ V . We choose for both models the parameter σ = 0 . 3 and additionally

for the variance gamma model ν = 0 . 2. The convergence rates on G 0 =

(K/ 2 , 3 / 2 K) are shown in Fig. 11.1 . As predicted in Theorems 11.2.4 and 11.2.7 ,
all Greeks converge with the optimal rate as the price V itself.

0 . 04, θ =−