Asian option — details
About this app
Monte-Carlo pricing with a quantifiable error: the running price estimate with its ±1.96·SE band cinches in at 1/√M; the European estimate from the same paths validates against the analytic Black-Scholes reference. “+1000 paths” animated, “+10000” fast. Debunks “simulation is guessing”.
Pricing by simulation
There is no closed-form price for an arithmetic-average Asian option. Instead you use Monte Carlo: simulate many price paths, compute each payoff, average them and discount with e^(−rT). The error can be quantified. With M paths and sample standard deviation s, the standard error is SE = s/√M, and a 95% confidence band spans ±1.96·SE. Four times as many paths halve the band.
What you see
The app plots the running price estimate of the Asian option against the number of paths, surrounded by its ±1.96·SE band. The same paths also price a European option, whose estimate settles onto the dashed analytical Black–Scholes line. If the method is right there, the Asian estimate can be trusted. The values panel lists M, both Monte Carlo prices, the Black–Scholes price, the Asian-to-European ratio and the standard error.
Things to try
Press “+1000 paths” a few times and watch the band tighten around the price; “+10000” runs ahead quickly. Check that after four times as many paths the band is roughly half as wide. Change the option type, the strike K or the volatility σ, and the estimator resets and builds up again. Compare the Asian-to-European ratio at high and low volatility.
Frequently asked questions
- How do you price an Asian option?
- For the arithmetic average there is no closed-form formula, so Monte Carlo simulation is the usual tool. You simulate risk-neutral price paths, compute the payoff on each path’s average and discount the mean. For a geometric average, a closed-form solution does exist.
- How accurate is Monte Carlo option pricing?
- The standard error of the estimate is s/√M, where s is the spread of the payoffs and M the number of paths. A 95% confidence interval covers about ±1.96 standard errors around the estimate. Accuracy improves with the square root of the number of paths.
- Why does the error only halve when you quadruple the paths?
- Because the standard error is proportional to 1/√M. Multiplying M by four doubles √M and halves the error.
- Why does the app also price a European option?
- It uses the same simulated paths as the Asian estimate but can be checked exactly against the Black–Scholes formula. When it lands on the analytical line, that validates the simulation.
Subject: Financial derivatives
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