Asian Options (Path-Dependent)

Asian Options

Asian options are path-dependent derivatives where the payoff depends on the average price of the underlying over the life of the option, rather than just the final price.

Why Asian Options?

• Reduced manipulation risk: harder to manipulate the average than a single price at expiry
• Smoothed exposure: useful when a company's cash flows accumulate over time (e.g., exporters)
• Lower cost: average price reduces volatility → lower option price than vanilla

Arithmetic Average Asian Call

The payoff of an arithmetic average Asian call is:

$\text{Payoff} = \max\left(\bar{S} - K, 0\right)$

where $\bar{S} = \frac{1}{n} \sum_{i=1}^{n} S_{t_i}$ is the arithmetic average of stock prices at observation times.

Monte Carlo Algorithm

For each simulated path:

1. Simulate n_steps stock prices along the path
2. Compute the arithmetic average of all prices
3. Calculate the payoff: max(avg - K, 0)

Then discount the average payoff: $C_{\text{Asian}} \approx e^{-rT} \cdot \text{mean payoff}$

Comparison to Vanilla

Asian options are always cheaper than (or equal to) vanilla calls with the same strike, because the average price is less volatile than the terminal price.