Arithmetic Asian Option (Turnbull-Wakeman)
Computes the price of an arithmetic-average Asian call with the Turnbull-Wakeman approximation. Unlike the geometric average, the arithmetic average has no exact closed form, so Turnbull and Wakeman match the first two moments of the average's distribution and apply a Black-Scholes with adjusted volatility and carry. It's the market-standard approximation for arithmetic Asians, common in commodities and FX. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.
Result
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Arithmetic Asian Option (Turnbull-Wakeman)
Computes the price of an arithmetic-average Asian call with the Turnbull-Wakeman approximation. Unlike the geometric average, the arithmetic average has no exact closed form, so Turnbull and Wakeman match the first two moments of the average's distribution and apply a Black-Scholes with adjusted volatility and carry. It's the market-standard approximation for arithmetic Asians, common in commodities and FX. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.
The Asian the market actually uses
Asian options pay based on the average price, which makes them ideal for commodities and FX, where what matters is the average price over the period. The problem is that the arithmetic average, the one everyone actually uses, has no exact closed form: the sum of lognormal variables isn't lognormal.
Turnbull and Wakeman got around this with a moment trick. They compute the exact mean and variance of the arithmetic average's distribution and pretend it's lognormal with those two moments, then apply a Black-Scholes with adjusted volatility and carry. The approximation is surprisingly good and became the market standard for arithmetic Asians.
Enter the spot price, the strike, the interest rate, the cost of carry, the volatility and the term, with the averaging covering the option's whole life. The tool returns the arithmetic Asian call premium. Since the average smooths out volatility, the premium is much lower than an equivalent plain option's.
Related Tools
Geometric Asian Call (Kemna-Vorst)
Computes the price of a geometric-average Asian call option with the Kemna-Vorst closed form. Asian options pay based on the average price over the period, which reduces the impact of expiry manipulation and makes the premium cheaper. The geometric-average version has an exact solution: it's a Black-Scholes with volatility adjusted to ฯ/โ3 and an adapted cost of carry. Enter the spot price, the strike, the interest rate, the term and the volatility.
Basket Option (Levy Approximation)
Computes the price of a call on a basket of two assets with the Levy approximation. The weighted sum of two lognormal assets isn't lognormal, so there's no exact formula; Levy matches the basket's mean and variance to an equivalent lognormal and applies a Black-Scholes. It's the practical way to price options on indices and portfolios, where the correlation between assets is decisive. Enter the two prices, the weights, the strike, the volatilities, the correlation, the rate and the term.
Spread Option (Kirk Approximation)
Computes the price of a spread option with Kirk's approximation: a call on the difference between two assets, S1 minus S2, with a strike. Spreads are everywhere in commodities (oil crack spread, power spark spread) and no exact formula exists, so Kirk proposed a clever approximation that reduces the problem to a Black-Scholes with an effective volatility combining the two vols and the correlation. Enter the two forward prices, the strike, the volatilities, the correlation, the rate and the term.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.