A unified approach for the pricing of options relating to averages

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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Fig. 1

Notes

  1. 1.

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

  2. 2.

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

  3. 3.

    A symmetry result between the floating-strike case and the fixed-strike case is first obtained in Henderson and Wojakowski (2002). This symmetry has been extended to various setting as a duality by Vecer and Xu (2004) and further by Eberlein et al. (2008).

  4. 4.

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

  5. 5.

    If a dividend rate d is considered, we simply replace r by \(r-d\) and the same arguments can apply.

  6. 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. 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. 8.

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

  9. 9.

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

  10. 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. 11.

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

References

  1. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.

    Article  Google Scholar 

  2. Cai, N., & Kou, S. (2012). Pricing Asian options under a hyper-exponential jump diffusion model. Operations Research, 60, 64–77.

    Article  Google Scholar 

  3. Cai, N., Li, C., & Shi, C. (2014). Closed-form expansions of discretely monitored Asian options in diffusion models. Mathematics of Operations Research, 39, 789–822.

    Article  Google Scholar 

  4. Chang, C. C., & Tsao, C. Y. (2011). Efficient and accurate quadratic approximation methods for pricing Asian strike options. Quantitative Finance, 11, 729–748.

    Article  Google Scholar 

  5. Dewynne, J. N., & Shaw, W. T. (2008). Differential equations and asymptotic solutions for arithmetic Asian options: Black-Scholes formulae for Asian rate calls. European Journal of Applied Mathematics, 19, 353–391.

    Article  Google Scholar 

  6. Dewynne, J. N., & Wilmott, P. (1993). Partial to the exotic. Risk, 6, 38–46.

    Google Scholar 

  7. Dufresne, D. (2000). Laguerre series for Asian and other options. Mathematical Finance, 10, 407–428.

    Article  Google Scholar 

  8. Eberlein, E., Papapantoleon, A., & Shiryaev, A. N. (2008). On the duality principle in option pricing: semimartingale setting. Finance and Stochastics, 12, 265–292.

    Article  Google Scholar 

  9. Fouque, J. P., & Han, C. H. (2003). Pricing Asian options with stochastic volatility. Quantitative Finance, 3, 352–362.

    Article  Google Scholar 

  10. Funahashi, H. (2014). A chaos expansion approach under hybrid volatiltiy models. Quantitative Finance, 14, 1923–1936.

    Article  Google Scholar 

  11. Funahashi, H., & Kijima, M. (2014). An extension of the chaos expansion approximation for the pricing of exotic basket options. Applied Mathematical Finance, 21, 109–139.

    Article  Google Scholar 

  12. Funahashi, H., & Kijima, M. (2015). A chaos expansion approach for the pricing of contingent claims. Journal of Computational Finance, 18, 27–58.

    Article  Google Scholar 

  13. Fusai, G., Marena, M., & Roncoroni, A. (2008). Analytical pricing of discretely monitored Asian-style options: Theory and application to commodity markets. Journal of Banking and Finance, 32, 2033–2045.

    Article  Google Scholar 

  14. Henderson, V., Hobson, D., Shaw, W., & Wojakowski, R. (2007). Bounds for in-progress floating-strike Asian options using symmetry. Annals of Operations Research, 151, 81–98.

    Article  Google Scholar 

  15. Henderson, V., & Wojakowski, R. (2002). On the equivalence of floating- and fixed-strike Asian options. Journal of Applied Probability, 39, 391–394.

    Article  Google Scholar 

  16. Ju, N. (2002). Pricing Asian and basket options via Taylor expansion. Journal of Computational Finance, 5, 79–103.

    Article  Google Scholar 

  17. Kemna, A. G. Z., & Vorst, A. C. F. (1990). A pricing method for options based on average asset values. Journal of Banking and Finance, 14, 113–129.

    Article  Google Scholar 

  18. Lapeyre, B., & Temam, E. (2001). Competitive Monte Carlo methods for pricing Asian options. Journal of Computational Finance, 5, 39–57.

    Article  Google Scholar 

  19. Linetsky, V. (2004). Spectral expansions for Asian (average price) options. Operations Research, 52, 856–867.

    Article  Google Scholar 

  20. Moreno, M., & J.F. Navas (2002). Australian Asian options. IE Working Paper, Instituto de Empresa, Spain.

  21. Shiraya, K., & Takahashi, A. (2011). A moment matching approach to the valuation of a volume weighted average price option. Journal of Futures Market, 10, 407–439.

    Article  Google Scholar 

  22. Takahashi, A. (1999). An asymptotic expansion approach to pricing financial contingent claims. Asia-Pacific Financial Market, 6, 115–151.

    Article  Google Scholar 

  23. Takahashi, A., & Takehara, K. (2007). Pricing currency options with a market model of interest rates under jump-diffusion stochastic volatility processes of spot exchange rates. Asia-Pacific Financial Market, 14, 69–121.

    Article  Google Scholar 

  24. Vecer, J. (2001). A new PDE approach for pricing arithmetic average Asian options. Journal of Computational Finance, 4, 105–113.

    Article  Google Scholar 

  25. Vecer, J., & Xu, M. (2004). Pricing Asian options in a semimartingale model. Quantitative Finance, 4, 170–175.

    Article  Google Scholar 

  26. Zhang, J. (2000). Arithmetic Asian options with continuous sampling. Journal of Computational Finance, 2, 59–79.

    Google Scholar 

Download references

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Correspondence to Hideharu Funahashi.

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:

$$\begin{aligned} f_1(s)= & {} \sigma _0(s) + \left\{ F(0,s) \sigma _1(s) + \frac{1}{2} F^2(0,s) \sigma _2(s) \right\} \left( \int _{0}^{s} \sigma ^{2}_{0}(u) \mathrm{d}u \right) , \\ g_{1}(s)= & {} \sigma _{0}(s) + F(0,s) \sigma _1(s), \\ g_{2}(s)= & {} \sigma _0(s) ,\\ h_{11}(s)= & {} \sigma _{0}(s) + 3 F(0,s) \sigma _1(s) + F^2(0,s) \sigma _2(s),\\ h_{21}(s)= & {} \sigma _{0}(s) ,\\ h_{31}(s)= & {} \sigma _{0}(s) ,\\ h_{12}(s)= & {} \sigma _{0}(s) + F(0, s) \sigma _1(s), \\ h_{22}(s)= & {} F(0, s) \sigma _1(s) , \\ h_{32}(s)= & {} \sigma _{0}(s) . \end{aligned}$$

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:

$$\begin{aligned} q_{1}(t)= & {} \int _{0}^{t} {\hat{f}}_1(s,t) {\hat{g}}_1(s,t) \left( \int _{0}^{s} g_2(u) {\hat{f}}_1(u,t) \mathrm{d}u \right) \mathrm{d}s, \\ q_{2}(t)= & {} \int _{0}^{t} {\hat{f}}_1(s,t) {\hat{h}}_{11}(s,t) \left( \int _{0}^{s} {\hat{f}}_1(s,t) h_{21}(u) \left( \int _{0}^{u} {\hat{f}}_1(r,t) h_{31}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \int _{0}^{t} {\hat{f}}_1(s,t) {\hat{h}}_{12}(s,t) \left( \int _{0}^{s} {\hat{f}}_1(u,t) h_{22}(u) \left( \int _{0}^{u} {\hat{f}}_1(r,t) h_{32}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s , \\ q_{3}(t)= & {} q^2_{1}(t) , \\ q_{4}(t)= & {} 2 \int _{0}^{t} {\hat{f}}_1(s,t) {\hat{g}}_1(s,t) \left( \int _{0}^{s} {\hat{f}}_1(u,t) {\hat{g}}_1(u,t) \left( \int _{0}^{u} g_2^2(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \int _{0}^{t} {\hat{f}}_1(s,t) {\hat{g}}_1(s,t) \left( \int _{0}^{s} {\hat{g}}_1(u,t) g_2(u) \left( \int _{0}^{u} {\hat{f}}_1(r,t) g_2(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \int _{0}^{t} {\hat{g}}_1^2(s,t) \left( \int _{0}^{s} {\hat{f}}_1(u,t) g_2(u) \mathrm{d}u \right) ^{2} \mathrm{d}s, \\ q_{5}(t)= & {} \int _{0}^{t} {\hat{g}}_1^2(s,t) \left( \int _{0}^{s} g_2^2(u) \mathrm{d}u \right) \mathrm{d}s. \end{aligned}$$

Appendix 2: Proof of Lemma 4.1

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

$$\begin{aligned} X_T = \sum _{i=1}^3 c_{iT} , \end{aligned}$$

where

$$\begin{aligned} c_{1t} = \int _0^t {\hat{f}}_1(s,t) \mathrm{d}W_s , \qquad c_{2t} = \int _0^t {\hat{g}}_1(s,t) \left( \int _{0}^{s} g_2(u) \mathrm{d}W_u \right) \mathrm{d}W_s , \end{aligned}$$

and

$$\begin{aligned} c_{3t} = \sum _{j=1}^2 \int _{0}^{t} {\hat{h}}_{1j}(s,t) \left( \int _{0}^{s} h_{2j}(u) \left( \int _{0}^{u} h_{3j}(r) \mathrm{d}W_r \right) \mathrm{d}W_u \right) \mathrm{d}W_s . \end{aligned}$$

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

$$\begin{aligned} \Psi (\xi )= & {} \mathbb {E}\left[ \mathrm{e}^{ \left\{ i \xi \left( c_{1T} + c_{2T} + c_{3T} \right) \right\} } \right] \\= & {} \mathbb {E}\left[ \mathrm{e}^{ i \xi c_{1T} } \left( 1 + i \xi c_{2T} + i \xi c_{3T} - \frac{1}{2} \xi ^{2} c_{2T} ^{2} + R_4 \right) \right] , \end{aligned}$$

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:

$$\begin{aligned} \Psi (\xi ) \ \approx \ \mathbb {E}\left[ \mathrm{e}^{ i \xi c_{1T} } \left( 1 + i \xi c_{2T} + i \xi c_{3T} - \frac{1}{2} \xi ^{2} c_{2T} ^{2} \right) \right] . \end{aligned}$$

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

$$\begin{aligned} \Psi (\xi )\approx & {} \mathbb {E}[ \mathrm{e}^{ i \xi c_{1T}} ] + i \xi \mathbb {E}\left[ \mathrm{e}^{ i \xi c_{1T}} \mathbb {E}[ c_{2T}\, |\, c_{1T}(t) ] \right] \\&\quad +\, i \xi \mathbb {E}\left[ e^{ i \xi c_{1T}} \mathbb {E}[ c_{3T} \, |\, c_{1T} ] \right] - \frac{1}{2} \xi ^{2} \mathbb {E}\left[ e^{ i \xi c_{1T}} \mathbb {E}[ c_{2T}^{2}\, |\, c_{1T}] \right] , \nonumber \end{aligned}$$
(7.1)

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

$$\begin{aligned} \mathbb {E}[ c_{2T} | c_{1T} = x ]= & {} q_{1}(t) \left( \frac{x^{2}}{V_T^{2}}- \frac{1}{V_T} \right) , \end{aligned}$$
(7.2)
$$\begin{aligned} \mathbb {E}[ c_{3T} | c_{1T} = x ]= & {} q_{2}(t) \left( \frac{x^{3}}{V_T^{3}}- \frac{3x}{V_T^{2}} \right) , \end{aligned}$$
(7.3)
$$\begin{aligned} \mathbb {E}[ c^2_{2T} | c_{1T} = x ]= & {} q_{3}(t) \left( \frac{x^{4}}{V_T^{4}} - \frac{6x^{2}}{V_T^{3}} + \frac{3}{V_T^{2}} \right) + q_{4}(t) \left( \frac{x^{2}}{V_T^{2}}- \frac{1}{V_T} \right) + q_{5}(t) , \nonumber \\ \end{aligned}$$
(7.4)

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

$$\begin{aligned} \frac{1}{2 \pi } \int _{\mathcal {R}} \mathrm{e}^{-iky} g(-ik) \mathbb {E}\big [f(X) \mathrm{e}^{ikX} \big ] \mathrm{d}k = g\left( \frac{\partial }{\partial y}\right) f(y) n(y;0,\Sigma ) , \end{aligned}$$

where n(xab) 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

$$\begin{aligned} f_{X_T}(x)\approx & {} n\left( x; 0, V_T \right) - \frac{\partial }{\partial {x}} \left\{ \mathbb {E}[ c_{2T} | c_{1T} = x ] n\left( x; 0, V_T \right) \right\} \nonumber \\&-\, \frac{\partial }{\partial {x}} \left\{ \mathbb {E}[ c_{3T} | c_{1T} = x ] n\left( x; 0, V_T \right) \right\} \nonumber \\&+ \frac{1}{2} \frac{ \partial ^{2}}{\partial x^2} \left\{ \mathbb {E}[ c_{2T}^{2} | c_{1T} = x ] n\left( x; 0, V_T \right) \right\} . \end{aligned}$$
(7.5)

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

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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