## Abstract

In this paper, we consider generalized Asian options and propose a unified approximation method for the pricing of such options when the underlying process is a diffusion. Through numerical examples, we show that our approximation method is accurate enough to be used in practice for the pricing of *any* type of Asian options that has been treated separately in the literature. Comparisons are made with the existing methods in the literature to support the usefulness of our method.

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## Notes

- 1.
See, e.g., Fusai et al. (2008) for the usage of Asian options in the actual markets.

- 2.
For an extensive literature review, we refer to Cai and Kou (2012).

- 3.
- 4.
An extension to the stochastic volatility case is straightforward by using the method given in Funahashi (2014) and omitted.

- 5.
If a dividend rate

*d*is considered, we simply replace*r*by \(r-d\) and the same arguments can apply. - 6.
The Hermite polynomials are defined by \(h_{n}(x) = (-1)^{n} \mathrm{e}^{x^2/2} \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}} \mathrm{e}^{-x^2/2}\), \(n=1,2, \dots \), with \(h_0(x)=1\). For example, we have \(h_{1}(x)=x\), \(h_{2}(x)=x^{2} - 1\), \(h_{3}(x)=x^{3} - 3x\), etc.

- 7.
Throughout this paper, the benchmark values are computed by MC simulations with 300,000 trials and 20,000 time steps for all the cases.

- 8.
The PDFs of the averages are more concentrated and becomes more symmetric around the mean compared with the stock price \(S_T\).

- 9.
As shown in Cai and Kou (2012), the double Laplace inversion method agrees with the eigenfunction expansion method to ten decimal points.

- 10.
In practice, daily-sampled Asian options are most frequently traded. For one-year maturity options, the number of sampling is given by \(N=250\).

- 11.
The case \(\eta =1\) corresponds to the GBM case, i.e. the Black–Scholes setting.

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## Appendices

### Appendix 1: Explicit formulas of deterministic functions

For the local volatility function \(\sigma (t, S)\) given in (3.2), we define \(\sigma _0(t) =\sigma (t, F(0,t))\), \(\sigma _1(t) =\sigma _S(t, F(0,t))\), and \(\sigma _2(t) =\sigma _{SS}(t, F(0,t))\) for the sake of notational simplicity, where \(\sigma _{S}(t, S)\) and \(\sigma _{SS}(t, S)\) denote the first-order and second-order derivatives of \(\sigma (t, S)\) with respect to the price *S*, respectively. Then, according to Funahashi and Kijima (2015), the risky asset price \(S_t\) can be approximated by (3.3). The deterministic functions \(f_1(t)\), \(g_i(t)\), and \(h_{kj}(t)\) involved there are given as follows:

Next, suppose that \(X_t\) is expanded as in (4.3). Then, the functions \(q_i(t)\) involved in Lemma 4.1 (Theorem 4.1 as well) are given as follows:

### Appendix 2: Proof of Lemma 4.1

Recall that \(X_T\) in (4.3) is given as

where

and

The key point in this proof is that \(c_{1t}\) is a Wiener integral with deterministic integrand \({{\hat{f}}}_1(s,t)\), which is normally distributed with zero mean and variance \(V_T = \int _0^t {{\hat{f}}}_1^2(s,t) \mathrm{d}s\), the random variable \(c_{2t}\) is a second-order iterated stochastic integral with deterministic integrands, and the term \(c_{3t}\) is a sum of third-order iterated stochastic integrals with deterministic integrands. For such a case, we can apply the discussions given in Funahashi and Kijima (2015).

Let the characteristic function of \(X_T\) be \(\Psi (\xi ) = \mathbb {E}[ \mathrm{e}^{i \xi X_T} ] \) and expand it as

where \(R_4\) consists of the forth or higher-order multiple stochastic integrals. According to Funahashi and Kijima (2015), the term \(\mathbb {E}\left[ \mathrm{e}^{ i \xi c_{1T} } R_4 \right] \) can be regarded as zero and we obtain the following approximation:

Taking the conditional expectation on \(c_{1T}\), we then obtain

as an approximation of the characteristic function \(\Psi (\xi )\).

The conditional expectations involved in (7.1) can be evaluated explicitly by applying the formulas in Appendix C of Funahashi and Kijima (2014). Namely, we have

where the functions \(q_i(t)\), \(i=1,\dots ,5\), are given in Appendix 1.

Let us denote the density function of \(X_T\) by \(f_{X_T}(x)\). Then, from the following inversion formula, we can find an approximation of \(f_{X_T}(x)\) from the approximated characteristic function. The next inversion formula is well known. See, e.g., Funahashi and Kijima (2015).

### Lemma 7.1

Suppose that *X* follows a normal distribution with zero mean and variance \(\Sigma \). Then, for any polynomial functions *f*(*x*) and *g*(*x*), we have

where *n*(*x*; *a*, *b*) denotes the normal density function with mean *a* and variance *b*.

Now, by applying Lemma 7.1 to each term in the right-hand side of (7.1), we obtain the approximation of the density function as

Finally, by substituting (7.2)–(7.4) into (7.5), we obtain the desired result.

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### Cite this article

Funahashi, H., Kijima, M. A unified approach for the pricing of options relating to averages.
*Rev Deriv Res* **20, **203–229 (2017). https://doi.org/10.1007/s11147-017-9128-4

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### Keywords

- Generalized Asian option
- Floating strike
- Fixed strike
- Discretely sampled
- Continuously sampled
- Forward-starting
- In-progress
- Australian-Asian option

### JEL Classification

- G12
- G13
- G17