Why Path-Dependent Pricing Requires Joint Distributions
Summary
The document explains why pricing an Asian option requires modeling the joint behavior of the underlying asset across its observation dates. The payoff depends on an arithmetic average, so knowing the separate price distribution at each date is not enough: the dependence between successive prices affects the distribution of that average. A Monte Carlo simulation captures this dependence through conditional transitions along the path.
The same issue applies to a basket option, whose payoff depends on several assets at one expiry. Their individual terminal distributions do not determine the basket’s distribution without a dependence assumption; a copula is one possible way to combine marginals into a joint model. The discussion clarifies a general modeling requirement, but it does not compare local volatility with implied distribution sampling or show that one pricing approach is generally preferable. Any simulation or copula result depends on the chosen dynamics and dependence structure, which the excerpt does not specify.
Key ideas
- An Asian option payoff depends on the joint distribution of prices across its observation dates.
- Sampling each date’s marginal distribution independently does not preserve the path dependence needed for pricing.
- Monte Carlo paths represent dependence through conditional price behavior over time.
- Basket option pricing also requires a joint distribution across the component assets.
- A copula can combine asset marginals under an explicitly chosen dependence structure.
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# Why/When local volatility is preferred over implied distribution sampling?
# Why/When local volatility is preferred over implied distribution sampling?
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?
## Answer by Quantuple (score 1)
https://quant.stackexchange.com/a/34709
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.