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## New answers tagged numerical-methods

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$Stends to 0. 1 Edit: the original question didn't specify "model independence" and so the below focuses on the BS framework. Also, i focused on speed of convergence rather than order of convergence: I will try to update my answer with some thoughts on order of convergence later on. Not sure this answers your question, but the speed at which the option prices ... 4 Note that \begin{align*} U_j^{(n)} &= \frac{U_j^{(n+1)} - a_jU_{j-1}^{(n)} - c_jU_{j+1}^{(n)}}{b_j}\\ &\le \frac{\max_j|U_j^{(n+1)}| - a_j\max_j|U_j^{(n)}| - c_j\max_j|U_j^{(n)}|}{b_j}. \end{align*} Moreover, there existsj_0such that \begin{align*} &\ \frac{\max_j|U_j^{(n+1)}| - a_{j_0}\max_j|U_j^{(n)}| - c_{j_0}\max_j|U_j^{(n)}|}{b_{j_0}} \\ ... 4 You are looking to solve a linear system with a band diagonal system matrixA$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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