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1

I'm not 100% sure, please double-check. I think that both in the ITM and the OTM case (requested), a model-free answer cannot exist. In particular, the rate at which: ITM: $C(S_0) \rightarrow S_0$ as $S_0 \rightarrow \infty$ and OTM: $C(S_0) \rightarrow 0$ as $S_0 \rightarrow 0$ depends on the model-specific risk-neutral transition density $p^Q(S_T, T | ...


6

This is more of a math question than a quant question. Under Black Scholes dynamics (assuming $r=0$ for simplicity), as everyone knows we have $$C=SN(d_1)-KN(d_2)$$. In this case, we are interested in large negative $d$, since $lnS$ is large and negative. There is an asymptotic series for $N(x)$ whose first term for large negative x is $$N(x)=-\phi(x)/x$$,...


0

Put-Call parity: $C = P + (S-K),$ taking $r=0$ for simplicity but without loss of generality. Now, $\lim_{S\rightarrow 0} \,P = K$, hence $\lim_{S\rightarrow 0}\, C = 0$ Another way to see this is to note that both $N(d_1)$ and $N(d_2)$ tend to zero as $S$ tends to 0.


1

Edit: the original question didn't specify "model independence" and so the below focuses on the BS framework. Also, i focused on speed of convergence rather than order of convergence: I will try to update my answer with some thoughts on order of convergence later on. Not sure this answers your question, but the speed at which the option prices ...


4

Note that \begin{align*} U_j^{(n)} &= \frac{U_j^{(n+1)} - a_jU_{j-1}^{(n)} - c_jU_{j+1}^{(n)}}{b_j}\\ &\le \frac{\max_j|U_j^{(n+1)}| - a_j\max_j|U_j^{(n)}| - c_j\max_j|U_j^{(n)}|}{b_j}. \end{align*} Moreover, there exists $j_0$ such that \begin{align*} &\ \frac{\max_j|U_j^{(n+1)}| - a_{j_0}\max_j|U_j^{(n)}| - c_{j_0}\max_j|U_j^{(n)}|}{b_{j_0}} \\ ...


4

You are looking to solve a linear system with a band diagonal system matrix $A$ of dimensions $N \times N$ and having the property $m_1 = 2, \, m_2 = 2$, respectively the number of sub-diagonals under and over the main diagonal of $A$. $A$ is empty elsewhere: $$ a_{ij} = 0 \qquad \text{for } j > i + m_2 \quad \text{ or } \quad i > j + m_1 $$ In order ...


0

Try a discretized form of the $Put Delta*Stock increment$, since the injection of this term should produce ideally (in the limit case) a 0 variance variable, namely the risk free drift.


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