Updating a Discrete Arithmetic Asian Option Value at an Intermediate Time

Summary

The document asks how to value a discretely sampled arithmetic Asian option at an intermediate time using Monte Carlo simulation. It gives a simplified setup with three observation dates, zero interest rate, zero strike, and a single simulated path. At the initial date, the proposed payoff uses the average of the asset values at all three dates, floored at zero.

At the middle date, the question proposes retaining the already observed initial and intermediate prices and simulating only the remaining terminal price. This highlights the path-dependent nature of an arithmetic average: realized observations are fixed, while future observations remain uncertain. However, the document contains no answer confirming the proposed procedure, and a single simulated path is not enough to estimate a Monte Carlo price. It also does not specify a pricing measure, discounting beyond the simplifying assumption, or how to average results across simulations.

Key ideas

• An arithmetic Asian option payoff depends on the asset prices observed across its sampling dates.
• At an intermediate date, past observed prices are known inputs to the eventual average.
• Future observations can be simulated conditional on the information available at the pricing date.
• A single simulated path does not provide a Monte Carlo estimate of option value.
• The document poses the procedure but provides no answer or pricing-measure details.

Tags

# Pricing Asian option at discrete times


# Pricing Asian option at discrete times












I hope you can help me again regarding pricing an arithmetic Asian option.

Assume we have a time grid $(0=t_0,t_1,t_2=T)$ and we buy an Asian option at time 0 and the maturity is at T. Now we would like to price the option at each time with a Monte-Carlo simulation. Now assume the risk-free rate and the strike $K=0$ and we just do one simulation (a simplification for this example) .

At $ t=t_0$: Clear. First of all simulate the future price at $t_1$ and $ t_2$. Then $$ \text{Asian} =\max\left(0,\frac{S(t_0)+S(t_1)+S(t_2)}{3}\right)$$.

At $ t=t_1$: Here is now my problem. We still have the information at time $t_0$. I think just simulate a new price at $t_2$ and hence $$ \text{Asian} =\max\left(0,\frac{S(t_0)+S(t_1)+S_{\text{new}}(t_2)}{3}\right)$$ $S(t_0),S(t_1)$ are from $t_0$.

Is that correct? Thank you very much!!!!

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