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.

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(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

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

(7.5)

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