Geometric Asian Put Valuation

Article Quant Q&A · Author: donpicante

Summary

The document presents an attempted comparison between a Kemna–Vorst analytic valuation and a Monte Carlo estimate for a geometric-average Asian put whose averaging starts at the initial time. It includes code for both calculations and reports that, for the stated inputs, repeated simulation estimates differ materially from the analytic result. The author asks whether the formula or simulation is specified incorrectly.

The code reveals choices that need careful alignment before the two prices can be compared, including an adjustment to volatility, the simulation time grid, whether the initial asset value is included in the geometric average, and discounting of the payoff. The document itself supplies no accepted diagnosis or corrected implementation, so it does not establish which element causes the discrepancy. The reported gap is an illustration from one parameter set, not evidence that either valuation method is generally inaccurate; convergence and matching contract definitions would need to be checked.

Key ideas

The document compares Monte Carlo pricing with an analytic formula for a geometric Asian put.
The averaging period begins at the initial valuation time, making treatment of the initial asset value relevant.
The simulation and analytic method must use consistent volatility, averaging, time-grid, and discounting conventions.
The author reports a substantial difference for one set of inputs but does not identify its cause.
The example does not establish a general failure of either pricing approach.

Tags

Options Derivatives pricing Backtesting Statistics

Full text

# asian geometric option valuation-- unable to get monte carlo simulation to converge to analytic value


# asian geometric option valuation-- unable to get monte carlo simulation to converge to analytic value












I'm trying to price asian put options in which the averaging window begins immediately (T=0). currently, I'm trying to match up geometric averaging between my Monte Carlo simulations and my attempt at replicating Kemna Vorst's 1990 analytic solution for put options.

Kemna Vorst:

```
def kemna_vorst_put_val(s0, k, r, T, sig):

    d_star = 0.5*(r-sig**2/6)*T
    d1 = (log(s0/k) + 0.5*(r + sig**2/6)*T)/(sig*sqrt(T/3))
    d2 = d1 - sig*sqrt(T/3)    
    put_val = exp(-r*T)*k*norm.cdf(-d2) - s0*exp(d_star)*norm.cdf(-d1)
    
    return put_val
```

Monte Carlo:

```

def asian_option_mc(s0, k, r, dt, sig, m, n):

 
    sig =  sig/sqrt(3)
    delta_t = dt / m  # length of time interval
    p = []
    for i in range(0, n):
        s = [s0]
        for j in range(0, m):
            s.append(s[-1] * exp((r - 0.5 * sig ** 2) * delta_t + (sig * sqrt(delta_t) * random.gauss(0, 1))))
        
        avg = scipy.gmean(s)         
        p.append(max((k - avg), 0))

    put_value = np.mean(p) * exp(-r * dt)

    return put_value
```

running `kemna_vorst_put_val(100, 95, 0.05, 1, 0.20)` gives me `0.5089522108680562`

while running

`asian_option_mc(100, 95, 0.05, 1, 0.20, 252, 100000)` gives me `0.4064389976474143`

I've run this several times and get answers that are substantially different from one another (On the order of 20-25%). Any idea what I may be doing wrong? Is there an issue with my Kemna Vorst expression? Or is there a mis-specification in my Monte Carlo simulation? I would expect them to be pretty close if I did this correctly. Thank you!

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.

Comparing Monte Carlo and Analytic Geometric Asian Put Prices

Summary

Key ideas

Tags