Asian option

In the 1980s Mark Standish was with the London-based Bankers Trust working on fixed income derivatives and proprietary arbitrage trading. David Spaughton worked as systems analyst in the financial markets with Bankers Trust since 1984 when the Bank of England first gave licences for banks to do foreign exchange options in the London market. In 1987 Standish and Spaughton were in Tokyo on business when "they developed the first commercially used pricing formula for options linked to the average price of crude oil." They called this exotic option the Asian option because they were in Asia.[3][4][5][6]

There are numerous permutations of Asian option; the most basic are listed below:

• Fixed strike (also known as an average rate) Asian call payout

${\displaystyle C(T)={\text{max}}\left(A(0,T)-K,0\right),}$

where A denotes the average price for the period [0, T], and K is the strike price. The equivalent put option is given by

${\displaystyle P(T)={\text{max}}\left(K-A(0,T),0\right).}$

• The floating strike (or floating rate) Asian call option has the payout

${\displaystyle C(T)={\text{max}}\left(S(T)-kA(0,T),0\right),}$

where S(T) is the price at maturity and k is a weighting, usually 1 so often omitted from descriptions. The equivalent put option payoff is given by

${\displaystyle P(T)={\text{max}}\left(kA(0,T)-S(T),0\right).}$

The Average ${\displaystyle A}$ may be obtained in many ways. Conventionally, this means an arithmetic average. In the continuous case, this is obtained by

${\displaystyle A(0,T)={\frac {1}{T}}\int _{0}^{T}S(t)dt.}$

For the case of discrete monitoring (with monitoring at the times ${\displaystyle 0=t_{0},t_{1},t_{2},\dots ,t_{n}=T}$ and ${\displaystyle t_{i}=i\cdot {\frac {T}{n}}}$) we have the average given by

${\displaystyle A(0,T)={\frac {1}{n}}\sum _{i=1}^{n}S(t_{i}).}$

There also exist Asian options with geometric average; in the continuous case, this is given by

${\displaystyle A(0,T)=\exp \left({\frac {1}{T}}\int _{0}^{T}\ln(S(t))dt\right).}$

A discussion of the problem of pricing Asian options with Monte Carlo methods is given in a paper by Kemna and Vorst.[7]

In the path integral approach to option pricing,[8] the problem for geometric average can be solved via the Effective Classical potential [9] of Feynman and Kleinert.[10]

Rogers and Shi solve the pricing problem with a PDE approach.[11]

A Variance Gamma model can be efficiently implemented when pricing Asian style options. Then, using the Bondesson series representation to generate the variance gamma process can increase the computational performance of the Asian option pricer.[12]

Within Lévy models, the pricing problem for geometric Asian options can still be solved.[13] For the arithmetic Asian option in Lévy models, one can rely on numerical methods[13] or on analytic bounds.[14]

European Asian call and put options with geometric averaging

We are able to derive a closed-form solution for the geometric Asian option; when used in conjunction with control variates in Monte Carlo simulations, the formula is useful for deriving fair values for the arithmetic Asian option.

Define the continuous-time geometric mean ${\displaystyle G_{T}}$ as:

$${\displaystyle G_{T}=\exp \left[{1 \over {T}}\int _{0}^{T}\log S(t)dt\right]}$$

where the underlying ${\displaystyle S(t)}$ follows a standard geometric Brownian motion. It is straightforward from here to calculate that:

$${\displaystyle G_{T}=S_{0}e^{{1 \over {2}}\left(r-{1 \over {2}}\sigma ^{2}\right)T}e^{{\sigma \over {T}}\int _{0}^{T}(T-t)dW_{t}}}$$

To derive the stochastic integral, which was originally ${\textstyle {\sigma \over {T}}\int _{0}^{T}W_{t}dt}$, note that:

$${\displaystyle d[(T-t)W_{t}]=(T-t)dW_{t}-W_{t}dt}$$

This may be confirmed by Itô's lemma. Integrating this expression and using the fact that ${\displaystyle W_{0}=0}$, we find that the integrals are equivalent - this will be useful later on in the derivation. Using martingale pricing, the value of the European Asian call with geometric averaging ${\displaystyle C_{G}}$ is given by:

$${\displaystyle C_{G}=e^{-rT}\mathbb {E} \left[(G_{T}-K)_{+}\right]={e^{-rT} \over {\sqrt {2\pi }}}\int _{\ell }^{\infty }\left(G_{T}-K\right)e^{-x^{2}/2}dx}$$

In order to find ${\displaystyle \ell }$, we must find ${\displaystyle x}$ such that:

$${\displaystyle G_{T}\geq K\implies S_{0}e^{{1 \over {2}}\left(r-{1 \over {2}}\sigma ^{2}\right)T}e^{{\sigma \over {T}}\int _{0}^{T}(T-t)dW_{t}}\geq K}$$

After some algebra, we find that:

$${\displaystyle {\sigma \over {T}}\int _{0}^{T}(T-t)dW_{t}\geq \log {K \over {S_{0}}}-{1 \over {2}}\left(r-{1 \over {2}}\sigma ^{2}\right)T}$$

At this point the stochastic integral is the sticking point for finding a solution to this problem. However, it is easy to verify that the integral is normally distributed as:

$${\displaystyle {\sigma \over {T}}\int _{0}^{T}(T-t)dW_{t}\sim {\mathcal {N}}\left(0,\sigma ^{2}{T \over {3}}\right)}$$

This is equivalent to saying that ${\textstyle {\sigma \over {T}}\int _{0}^{T}(T-t)dW_{t}=\sigma {\sqrt {T \over {3}}}x}$ with ${\textstyle x\sim {\mathcal {N}}(0,1)}$. Therefore, we have that:

$${\displaystyle x\geq {\log {K \over {S_{0}}}-{1 \over {2}}\left(r-{1 \over {2}}\sigma ^{2}\right)T \over {\sigma {\sqrt {T/3}}}}\equiv \ell }$$

Now it is possible the calculate the value of the European Asian call with geometric averaging! At this point, it is useful to define:

$${\displaystyle b={1 \over {2}}\left(r-{1 \over {2}}\sigma _{G}^{2}\right),\;\sigma _{G}={\sigma \over {\sqrt {3}}},\;d_{1}={\log {S_{0} \over {K}}+\left(b+{1 \over {2}}\sigma _{G}^{2}\right)T \over {\sigma _{G}{\sqrt {T}}}},\;d_{2}=d_{1}-\sigma _{G}{\sqrt {T}}}$$

Going through the same process as is done with the Black-Scholes model, we are able to find that:

$${\displaystyle C_{G}=S_{0}e^{(b-r)T}\Phi (d_{1})-Ke^{-rT}\Phi (d_{2})}$$

In fact, going through the same arguments for the European Asian put with geometric averaging ${\textstyle P_{G}}$, we find that:

$${\displaystyle P_{G}=Ke^{-rT}\Phi (-d_{2})-S_{0}e^{(b-r)T}\Phi (-d_{1})}$$

This implies that there exists a version of put-call parity for European Asian options with geometric averaging:

$${\displaystyle C_{G}-P_{G}=S_{0}e^{(b-r)T}-Ke^{-rT}}$$

There are some variations that are sold in the over-the-counter market. For example, BNP Paribas introduced a variation, termed conditional Asian option, where the average underlying price is based on observations of prices over a pre-specified threshold. A conditional Asian put option has the payoff

${\displaystyle \max \left(K-{\frac {\int _{0}^{T}S(t)I_{\{S(t)>b\}}dt}{\int _{0}^{T}I_{\{S(t)>b\}}dt}},0\right),}$

where ${\displaystyle b>0}$ is the threshold and ${\displaystyle I_{A}}$ is an indicator function which equals ${\displaystyle 1}$ if ${\displaystyle A}$ is true and equals zero otherwise. Such an option offers a cheaper alternative than the classic Asian put option, as the limitation on the range of observations reduces the volatility of average price. It is typically sold at the money and last for up to five years. The pricing of conditional Asian option is discussed by Feng and Volkmer.[15]