Asian option

An Asian option is a special form of an exotic option whose payout profile when exercised depends on the difference between the exercise price and an average value of past prices of the underlying asset .

backgrounds

The designation as an Asian option is not linked to Asia via the place of exercise or the place of origin . Presumably the name came about because this type of warrant was first traded by the Tokyo office of the Bankers Trust.

In terms of exercise type, Asian options can be of either American or European type .

The main feature of Asian options is that, on the exercise date, the value of the option is not determined by the current price of the underlying, but by the average of the prices of certain past days specified in the contractual conditions.

Asian options are suitable e.g. B. to hedge against exchange rate risks if a product is sold at a certain future point in time, but the production and manufacturing costs are incurred up to this point in time and are therefore subject to a continuous exchange rate risk. When hedging commodity price risks using options, this type of option is often chosen - especially by customers - because the underlying asset to be hedged is purchased continuously.

Types of Asian options

There are several types of Asian options, which differ from one another in the way they are averaged:

Asian arithmetic options

Given a base value (such as a share or an index ) with the price . An arithmetic Asian call option on the underlying with a term T> 0 is a contract that gives the option buyer the right to have the seller (also called writer) pay out a certain amount at time T. This amount is the difference between the arithmetic mean of the price on certain days in the period [0, T] and the agreed base price X. is defined as ${\displaystyle (S_{t}),\;t\geq 0}$ ${\displaystyle {\bar {S}}}$ ${\displaystyle {\bar {S}}}$

${\displaystyle {\bar {S}}={\frac {1}{n}}\sum _{i=1}^{n}S_{t_{i}}}$,

where the times are the days on which the price of the base value is determined for averaging. This can be, for example, the last 10 business days before the exercise date or the ultimos of the last 3 months of the term. ${\displaystyle t_{1},t_{2},\ldots t_{n}}$

As is usual with options, the option buyer will normally only exercise his right to withdrawal if there is a positive amount. ${\displaystyle {\bar {S}}-X}$

An arithmetic Asian put option works in the same way, only here is the payout amount , ie the exercise is cheap if the average value of the courses is below the exercise price X. ${\displaystyle X-{\bar {S}}}$

Geometric Asian options

The geometric Asian option works accordingly, but instead of the arithmetic, the geometric mean is used to calculate the average:

${\displaystyle {\hat {S}}=\left(\prod _{i=1}^{n}S_{t_{i}}\right)^{1 \over n}}$

A fundamental relationship between arithmetic and geometric options can be seen if the value of the base value is logarithmized , i.e. calculated. Then namely applies ${\displaystyle L_{t}:=\ln(S_{t})\ }$

${\displaystyle {\hat {S}}=e^{ln({\hat {S}})}=\exp \left({\frac {1}{n}}\sum _{i=1}^{n}\ln \left(S_{t_{i}}\right)\right)=e^{\bar {L}}}$.

A geometric Asian call option with an underlying value X = 0 pays out exactly the exponential of a corresponding arithmetic option that refers to the underlying value instead . This property can be beneficial because, in many financial models, the logarithm of the course is easier to handle than the course itself. ${\displaystyle S}$ ${\displaystyle L}$ ${\displaystyle S}$

The geometric mean represents an approximation of the integral . With the help of Jensen's inequality it can be shown that it always holds. Accordingly, the premium for a geometric call option (put option) must always be lower (higher) than the premium for an arithmetic call option (put option) of the same term. ${\displaystyle \exp \left({\frac {1}{T}}\int _{0}^{T}\ln(S_{t})\mathrm {d} t\right)}$ ${\displaystyle {\hat {S}}\leq {\bar {S}}}$

Evaluation of Asian options

In financial mathematics one is always interested in calculating the expected value of the payout in terms of a certain probability measure . This is often difficult, if not impossible , with path-dependent options, i.e. those in which the payout does not only depend on the closing price (this is the case, for example, with normal buy and sell options ). This applies to Asian options to a greater extent, as these, for example with a three-year term and daily observation, can mean the calculation of an average of over a thousand interdependent random variables . In such cases, a Monte Carlo simulation is usually the only way to estimate the expected value.

One way out of this can often be the strategy of looking at the actual integral instead of the mean value actually paid out. In some capital market models, S is modeled by special stochastic processes whose integrals are known at least in terms of their distribution . This is particularly often the case with the geometric Asian option, as it is mostly assumed (for example in Lévy models ) , where X is a "simple" process. In the best-known capital market model, the Black-Scholes model , this is, for example, a Brownian movement . ${\displaystyle S_{t}=e^{X_{t}}}$

Individual evidence

1. ↑ Archive link ( Memento of the original from April 16, 2010 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. ISDA FAQ list , question 34