Let's say we have an option whose payoff is path dependent (let's say it's asian option with observations every month). Then why these are usually priced with local vol instead of sampling from implied distributions corresponding to consecutive expiries spaced out monthly?

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    $\begingroup$ How do you account for the correlation between the returns in M months and those of M+1 months in your approach? I mean sampling from two marginals is not the same as sampling from the joint (you lose the path dependence). $\endgroup$ – Quantuple Jun 14 '17 at 17:22
  • $\begingroup$ could you please tell me more about this? $\endgroup$ – CC89 Jun 14 '17 at 17:57
  • $\begingroup$ Let $X_1$ denote the 1Y stock price, $X_2$ denote the 1Y+1M stock price. Sampling from $p(x_1)$ and then $p(x_2)$ is not the same as sampling from $p(x_1,x_2)$. Except if you sample from $p(x_1)$ and then $p(x_2 \vert x_1)$. $\endgroup$ – Quantuple Jun 14 '17 at 18:19
  • $\begingroup$ Glad to hear. Now, this approach could make sense for e.g. European basket options, see quant.stackexchange.com/questions/34558/… $\endgroup$ – Quantuple Jun 15 '17 at 8:04

Consider an Asian option on a traded asset $S$ with payout $$ (A - K)^+ = \left( \frac{1}{N} \sum_{i=1}^N S(t_i) - K \right)^+ $$ To sample the arithmetic mean $A$ you need to sample from the joint distribution of $S(t_1),\dots,S(t_N)$. Thing is, individually sampling from the corresponding marginals will not the same as sampling from the joint because $$ p(s_1, \dots, s_N) \ne p(s_1) \dots p(s_N) $$ since $$ p(s_1, \dots, s_N) = p(s_1) p(s_2 \vert s_1) \dots p(s_N \vert s_1, \dots, s_{N-1} ) $$ a conditional behaviour (path-dependence) which you implicitly construct when running a Monte Carlo simulation.

The same goes for pricing a European basket option of expiry $T$ with payout $$ (B - K)^+ = \left( \sum_{i=1}^N w_i S_i(T) - K \right)^+ $$ in the sense that in order to sample the terminal basket value $B$ you need to sample from the joint distribution of the terminal individual asset prices $S_1(T),\dots,S_N(T)$. Sampling from the individual marginals is not enough, but you could come up with a copula to construct the joint distribution from the marginals (the copula function reflecting the choice of dependence structure you would like to impose between the individual terminal asset prices).

  • $\begingroup$ Agreed for the underlying price simulation $S(t_i)$. However, I think the question was more about the implied volatility. $\endgroup$ – JejeBelfort Jun 15 '17 at 9:02
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    $\begingroup$ @JejeBelfort: nowhere does the OP mentions implied volatility. Instead he is interested in pricing using local vol + Monte Carlo vs. pricing by sampling from the implied marginals (i.e. distributions inferred from the full vol smile) $\endgroup$ – Quantuple Jun 15 '17 at 9:38
  • $\begingroup$ After reading the question carefully, it seems that you are correct. However the latter option is far less common than the first... $\endgroup$ – JejeBelfort Jun 15 '17 at 9:55
  • $\begingroup$ I've seen it applied by some banks for basket options. $\endgroup$ – Quantuple Jun 15 '17 at 13:14
  • $\begingroup$ Another approach similar to the latter that works well for Asian options is to just moment match on the first two moments. It's super easy and gives nice accurate prices. $\endgroup$ – will Jun 17 '17 at 16:13

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