Consider the backward induction relation, i.e. [ widetilde{v}(t, x)=left(1-p_{N}^{*}ight) widetilde{v}left(t+1, xleft(1+a_{N}ight)ight)+p_{N}^{*} widetilde{v}left(t+1, xleft(1+b_{N}ight)ight), ] using the renormalizations

Question:

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

image text in transcribed

image text in transcribed

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Question Posted: