Consider the backward induction relation, i.e.
\[
\widetilde{v}(t, x)=\left(1-p_{N}^{*}ight) \widetilde{v}\left(t+1, x\left(1+a_{N}ight)ight)+p_{N}^{*} \widetilde{v}\left(t+1, x\left(1+b_{N}ight)ight),
\]
using the renormalizations \(r_{N}:=r T / N\) and
\[
a_{N}:=\left(1+r_{N}ight)(1-|\sigma| \sqrt{T / N})-1, \quad b_{N}:=\left(1+r_{N}ight)(1+|\sigma| \sqrt{T / N})-1
\]
of Section 3.6, \(N \geqslant 1\), with
\[
p_{N}^{*}=\frac{r_{N}-a_{N}}{b_{N}-a_{N}} \quad \text { and } \quad p_{N}^{*}=\frac{b_{N}-r_{N}}{b_{N}-a_{N}}
\]
a) Show that the Black-Scholes PDE of Proposition 6.1 can be recovered from the induction relation when the number \(N\) of time steps tends to infinity.
b) Show that the expression of the Delta \(\xi_{t}=\frac{\partial g_{\mathrm{c}}}{\partial x}\left(t, S_{t}ight)\) can be similarly recovered from the finite difference relation (3.19), i.e.
\[
\xi_{t}^{(1)}\left(S_{t-1}ight)=\frac{v\left(t,\left(1+b_{N}ight) S_{t-1}ight)-v\left(t,\left(1+a_{N}ight) S_{t-1}ight)}{S_{t-1}\left(b_{N}-a_{N}ight)}
\]
as \(N\) tends to infinity.

Data from Proposition 6.1

Transcribed Image Text:

Call-put parity. We have the relation