Asian Options: Pricing Based on Averages

Asian options are probably the most practical exotic derivatives. In crude oil markets, they are not even considered exotic. They are the vanilla. Chapter 25 applies the framework from Chapter 24 to options whose payoff depends on an average price.

Why Averages?

Cheaper. The volatility of an average is less than the volatility of the underlying. Averaging smooths out swings. Lower vol means lower premiums.

Manipulation-resistant. One day’s price barely moves the average. Important for thinly traded commodities.

Matches business exposure. A company buying oil weekly cares about the average price over six months, not one date.

Crash-proof(ish). One bad day barely dents the accumulated average.

Asian options were first priced in 1987 by David Spaughton and Mark Standish at Bankers Trust in Tokyo, working on crude oil. The name stuck for geographic reasons only.

The Two Flavors: Strike and Rate

Every Asian option uses an average. The question is: what role does the average play in the payoff?

Average strike options replace the strike price with the average. An average strike call pays:

$$\max(S(T) - A, 0)$$

You get the difference between the final stock price and the average. If the stock ended higher than its average, you profit. An average strike put pays:

$$\max(A - S(T), 0)$$

Average rate options (also called average price) replace the stock price with the average. An average rate call pays:

$$\max(A - E, 0)$$

where E is a fixed strike. You get paid if the average exceeds the strike. An average rate put pays:

$$\max(E - A, 0)$$

From a pricing perspective, the strike/rate distinction matters. Average strike options are generally easier to handle numerically because of certain symmetry properties (similarity reductions) that reduce the dimensionality.

Types of Averaging

Two choices define the average:

Arithmetic vs. Geometric

The arithmetic average: $A = \frac{1}{n}\sum_{i=1}^{n} S(t_i)$

The geometric average: $A = \exp\left(\frac{1}{n}\sum_{i=1}^{n} \ln S(t_i)\right)$

The geometric average is always $\le$ the arithmetic (AM-GM inequality) and is mathematically easier because the geometric average of lognormal variables stays lognormal. Arithmetic does not, making closed-form pricing harder.

Continuous vs. Discrete

Continuous sampling uses every instant (average becomes an integral). Discrete sampling uses specific data points like Friday closing prices. Real contracts are discrete. The difference matters when samples are far apart.

The PDE Approach

For the arithmetic case, the pricing PDE is:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} + S\frac{\partial V}{\partial I} - rV = 0$$

For geometric, replace S with $\ln(S)$ in the path-dependent term.

For discrete sampling, solve Black-Scholes between dates, apply jump conditions at each sample. Arithmetic jump condition:

$$V(S, A_{i-1}, t_i^-) = V\left(S, \frac{(i-1)A_{i-1} + S}{i}, t_i^+\right)$$

The Similarity Reduction

A nice trick for average strike options. The payoff $\max(S(T) - I/T, 0)$ scales linearly with S. Wilmott introduces $R = I/S$ and writes $V(S, I, t) = S \cdot W(R, t)$ where W satisfies:

$$\frac{\partial W}{\partial t} + \frac{1}{2}\sigma^2 R^2 \frac{\partial^2 W}{\partial R^2} + (1 - rR)\frac{\partial W}{\partial R} = 0$$

Three dimensions down to two. Works for average strike but not average rate. There is also a put-call parity for average strike:

$$V_C - V_P = S - \frac{I}{T}\left(\frac{1 - e^{-r(T-t)}}{r(T-t)}\right)$$

The Asian Tail

Sometimes averaging covers only the final portion of the option’s life. This Asian tail protects against sudden moves right before expiry. Some pension schemes use this. The path-dependent PDE term only appears during the averaging window.

Closed-Form Solutions

Exact formulas exist only for geometric averages (because geometric average of lognormal stays lognormal). The formula looks like Black-Scholes but with $\sigma/\sqrt{3}$ replacing $\sigma$. The $\sqrt{3}$ comes from the variance of the integral of Brownian motion.

For arithmetic averages, no exact formulas. Approximations include:

• Turnbull and Wakeman (1991): moment-matching to lognormal. Not great for discrete sampling
• Curran (1992): geometric conditioning. More accurate
• Thompson (2000): upper and lower bounds via double integrals

Volatility Term Structure

For geometric Asians, the effective volatility is time-weighted:

$$\sigma_{\text{eff}}^2 = \frac{3}{T^3}\int_0^T (T-t)^2 \sigma^2(t) , dt$$

Early volatility matters more than late volatility. Wilmott cites Haug, Haug and Margrabe (2003): the error from ignoring term-structure effects is comparable to using an approximate formula. Both matter. An exact formula with flat vol may not beat an approximate formula with the right term structure.

Multi-Asset Asians

The anteater option depends on the average ratio of two underlyings. Four dimensions: $S_1$, $S_2$, I, t. At the edge of PDE feasibility. Beyond this, use Monte Carlo.

The Takeaway

Asian options are the workhorse of exotic derivatives, especially in commodities and FX. The key points:

• Average strike vs. average rate is the first classification
• Arithmetic vs. geometric is the second
• Continuous vs. discrete sampling is the third
• The PDE approach works well, especially with similarity reductions for average strike options
• Geometric averages have closed-form solutions; arithmetic averages need approximations or numerical methods
• The volatility term structure matters as much as the pricing model
• For low dimensions, PDEs beat Monte Carlo. For high dimensions, Monte Carlo wins

The framework from Chapter 24 makes pricing Asian options systematic. You define the averaging rule, write the state variable, and the math follows.

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