2.1. Pricing model

In this paper, we price the timer option under a multiscale stochastic volatility model for the stock price. This class of models has been studied previously [Reference Fouque, Papanicolaou, Sircar and Solna12], and it has been developed as an effective framework in which the principal components of derivative prices can be efficiently captured. In our pricing model, most of the notation adopted in this paper conventionally follows that defined in [Reference Fouque, Lorig and Sircar8, Reference Fouque, Papanicolaou, Sircar and Solna12] and extensively used in the literature whenever this class of models is under consideration. Choosing a risk-neutral measure $\mathbb {P}^{\star }$ and a standard three-dimensional $\mathbb {P}^{\star }$ -Brownian motion $W_{t}^{\star }$ [Reference Durrett5], the evolution of the underlying asset price S satisfies the following stochastic differential equations (SDEs):

(2.1) $$ \begin{align} \left\{ \begin{aligned} dS_{t} &=r S_{t}\,dt +f(Y_{t},Z_{t})S_{t}\,dW_{t}^{(0)\star }\\ dY_{t} &=\bigg(\frac{1}{\epsilon}\alpha(Y_t)- \frac{1}{\sqrt{\epsilon}}\Lambda(Y_{t},Z_{t})\beta(Y_t)\bigg)\,dt +\frac{1}{\sqrt{\epsilon}} \beta(Y_t)\,dW_{t}^{(1)\star}\\ dZ_{t} &=(\delta c(Z_{t})-\sqrt{\delta}\Gamma(Y_{t},Z_{t})g(Z_{t}))\,dt +\sqrt{\delta}g(Z_{t})\,dW_{t}^{(2)\star }, \end{aligned} \right. \end{align} $$

where $W_t^{\star } =(W_t^{(0)\star },W_t^{(1)\star },W_t^{(2)\star })$ is a standard three-dimensional Brownian motion with correlation structure

$$ \begin{align*} \begin{aligned} \left(\begin{matrix} W_{t}^{(0)\star}\\ W_{t}^{(1)\star}\\ W_{t}^{(2)\star} \end{matrix} \right)= \left( \begin{matrix} 1&0&0&\\ \rho_{1}&\sqrt{1-\rho_{1}^2}&0\\ \rho_{2}&\tilde{\rho}_{12}&\sqrt{1-\rho_{2}^2-\tilde{\rho}_{12}^2} \end{matrix} \right)W_{t}. \end{aligned} \end{align*} $$

Here $W_{t}$ is a standard three-dimensional Brownian motion, and the coefficients $\rho _{1}$ , $\rho _{2}$ and $\tilde {\rho }_{12}$ satisfy

$$ \begin{align*}|\rho_{1}|<1, \quad \rho_{2}^2+\tilde{\rho}_{12}^2<1\quad \text{ and } \quad \rho_{12}=\rho_{1}\rho_{2}+\tilde{\rho}_{12}\sqrt{1-\rho_{1}^2}.\end{align*} $$

The risk-free interest rate $r $ is a positive constant. The combined market prices of volatility risk are given by

$$ \begin{align*} \Lambda(y,z) &= \sqrt{1-\rho_{1}^2}\frac{(\mu - r)}{f(y,z)} + \gamma(y,z)\rho_{1}, \\ \Gamma(y,z) &= \sqrt{1-\rho_{2}^2-\tilde{\rho}_{12}^2}\frac{(\mu - r)}{f(y,z)} +\gamma(y,z)\tilde{\rho}_{12} +\xi(y,z)\rho_{2}, \end{align*} $$

where $\gamma (y,z)$ and $\xi (y,z)$ are bounded smooth functions of y and z. The volatility $f(y,z)$ is a positive function; $\epsilon $ and $\delta $ represent the time scales and allow us to study the problem in the fast and slow regimes at the same time. The fast factor Y is an ergodic process on J with a unique invariant distribution $\Pi $ . We take $J = \mathbb {R}$ , although in extensions to other cases, $\alpha (y)$ , $\beta (y)$ and $c(z)$ , $g(z)$ describe the dynamics of the process Y and Z, respectively, under the real-world measure $\mathbb {P}$ .

We employ this for the pricing of perpetual timer options. In the subsequent context, a timer option is understood as the perpetual timer option when there is no confusion. The timer call option has payoff $\max (S_{\tau }-K,0)$ at random maturity $\tau $ , where K is the strike price. Given the predetermined variance budget B, $\tau = \inf \{t>0,I_{t}=B\}$ . Define

$$ \begin{align*} I_{t}= \int_{0}^{t}f^2(Y_{s},Z_{s})\,ds \end{align*} $$

as the cumulative realized variance. Thus, the dynamics are now described by

$$ \begin{align*} \left\{ \begin{aligned} dS_{t} &=r S_{t}\,dt +f(Y_{t},Z_{t})S_{t}\,dW_{t}^{(0)\star }\\ dY_{t} &=\bigg(\frac{1}{\epsilon}\alpha(Y_t)- \frac{1}{\sqrt{\epsilon}}\Lambda(Y_{t},Z_{t})\beta(Y_t)\bigg)\,dt +\frac{1}{\sqrt{\epsilon}} \beta(Y_t)\,dW_{t}^{(1)\star}\\ dZ_{t} &=(\delta c(Z_{t})-\sqrt{\delta}\Gamma(Y_{t},Z_{t})g(Z_{t}))\,dt +\sqrt{\delta}g(Z_{t})\,dW_{t}^{(2)\star }\\ dI_{t} &= f^2(Y_{t},Z_{t})\,dt. \end{aligned} \right. \end{align*} $$

It is clear that $(S,Y,Z,I)$ is a four-dimensional Markov process. Denoting by $\mathbb {E}^{\star }$ the expectation with respect to $\mathbb {P}^{\star }$ , the price of a timer call option with payoff function $\max (S_{\tau }-K, 0)$ is

(2.2) $$ \begin{align} \begin{aligned} P^{\epsilon,\delta}(t,S_{t},Y_{t},Z_{t},I_{t})=\mathbb{E}^{\star}[e^{-r(\tau-t)}\max(S_{\tau}-K,0)\mid \mathcal{F}_{t}]. \end{aligned} \end{align} $$

Since the stochastic volatility setting under consideration is very general and complex, an analytic solution of (2.2) or (2.4) is difficult, if not impossible, to obtain. Thus, our goal is to derive an approximate price of the timer option. This approximation is of the form

(2.3) $$ \begin{align} P^{\epsilon,\delta}\approx \tilde{P}^{\epsilon,\delta}=P_{0,0}+\sqrt{\epsilon}P_{1,0}+\sqrt{\delta}P_{0,1}+\epsilon P_{2,0}+\delta P_{0,2}+\sqrt{\epsilon\delta}P_{1,1}, \end{align} $$

where $P_{0,0}$ is the leading term of the price; $P_{1,0}$ is the first-order fast scale correction; $P_{0,1}$ is the first-order slow scale correction; and $P_{2,0}, P_{0,2}, P_{1,1}$ are the second-order fast scale term, second-order slow scale term and first fast–slow term, respectively. In the notation $P_{i,j}$ , the subindex i corresponds to the power of $\sqrt {\epsilon }$ , and the subindex j corresponds to the power of $\sqrt {\delta }$ . The derivation of the expressions for these price approximation terms will be presented in the next section and summarized in Theorem 2.6.

2.2. Main result

In this section, we briefly introduce the asymptotic analysis of the timer option pricing and give the main result of this paper. First, we make the following assumptions, which are necessary throughout the paper.

The analysis is based on two small parameters $\epsilon $ and $\delta $ , which govern two time scales for the two volatility driving processes. The driver Y runs on a fast time scale and rapidly converges to stationarity, which leads to a singular perturbation analysis. On the contrary, the factor Z fluctuates on a slow time scale, and hence the corresponding expansion is a regular perturbation problem. Applying a combination of singular and regular perturbations, we derive a second-order approximate formula for the timer option.

By application of the Feynman–Kac formula [Reference Fouque, Papanicolaou, Sircar and Solna12, Ch. 1.9.3], the pricing function $P^{\epsilon ,\delta }$ in (2.2) is the solution of the following partial differential equation (PDE) with final condition

$$ \begin{align*} \left\{ \begin{aligned} &\mathcal{L}^{\epsilon,\delta}P^{\epsilon,\delta}=0\nonumber\\ &P^{\epsilon,\delta}(B,s,y,z)=\max(s-K, 0), \end{aligned} \right. \end{align*} $$

where the differential operator $\mathcal {L}^{\epsilon ,\delta }$ is the sum of components

$$ \begin{align*} \nonumber \mathcal{L}^{\epsilon,\delta}=\frac{1}{\epsilon}\mathcal{L}_{0}+\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}+\sqrt{\delta}\mathcal{M}_{1}+\delta\mathcal{M}_{2}+ \sqrt{\frac{\delta}{\epsilon}}\mathcal{M}_{3}, \end{align*} $$

and the operators $\mathcal {L}_{i} \,(i = 0, 1, 2)$ and $\mathcal {M}_{i}\,(i = 1, 2, 3)$ are defined by

(2.4) $$ \begin{align} \left\{ \begin{aligned} \mathcal{L}_{0}&=\alpha(y)\frac{\partial}{\partial y} +\frac{1}{2}\beta^2(y)\frac{\partial ^2}{\partial y^2}\\ \mathcal{L}_{1}&=\beta(y)\bigg(\rho _{1}f(y,z)s\frac{\partial ^2}{\partial s\partial y}-\Lambda (y,z)\frac{\partial}{\partial y}\bigg)\\ \mathcal{L}_{2}&=\frac{\partial}{\partial t} +rs\frac{\partial}{\partial s} +f^2(y,z)\frac{\partial}{\partial x}+ \frac{1}{2}f^2(y,z)s^2\frac{\partial ^2}{\partial s^2}-r\\ \mathcal{M}_{1}&=-g(z)\Gamma(y,z)\frac{\partial}{\partial z} +\rho_{2}f(y,z)g(z)s\frac{\partial^2}{\partial s \partial z}\\ \mathcal{M}_{2}&=c(z)\frac{\partial}{\partial z} +\frac{1}{2}g^2(z)\frac{\partial ^2}{\partial z^2}\\ \mathcal{M}_{3}&=\rho_{12}\beta(y)g(z)\frac{\partial ^2}{\partial y \partial z}.\\ \end{aligned} \right. \end{align} $$

Observe that $\mathcal {L}_2 $ contains the derivative with t. In fact, given the value of the realized variance, the pricing problem for the perpetual timer option is independent of t [Reference Li15, Reference Saunders17]. Therefore, $ \mathcal {L}_2$ can be rewritten as

$$ \begin{align*}\mathcal{L}_2 = f^2 \frac{\partial}{\partial x}+rs\frac{\partial}{\partial s} +\frac {s^2 f^2}{2}\frac{\partial ^2}{\partial s^2}-r ,\end{align*} $$

which can be regarded as the Black–Scholes operator with x instead of the time variable t [Reference Saunders17]. Note that $\langle \mathcal {L}_2 \rangle $ plays an important part in the derivation of the approximate price, where the bracket notation means the integration with respect to the invariant distribution $\Pi $ of the process Y.

We expand $P^{\epsilon , \delta }$ as

$$ \begin{align*}P^{\epsilon, \delta}= \sum_{j\geq0}\sqrt{\delta}^jP_{j}^{\epsilon} \quad \text{where}\ P_{j}^{\epsilon} = \sum_{i\geq0}\sqrt{\epsilon}^{i}P_{i,j}.\end{align*} $$

Inserting this expression into (2.4), and equating both sides of the above equation with respect to the corresponding powers of $\sqrt {\delta }$ , we have a system of pricing equations. Then, carrying out a singular perturbation analysis with respect to $\sqrt {\epsilon }$ for these equations, we will obtain the terms $P_{0,0}$ , $P_{1,0}$ , $P_{0,1}$ , $P_{1,1}$ , $P_{0,2}$ and $P_{2,0}$ explicitly. Note that there are many Poisson equations of asymptotic expansion terms for the timer option price. By Assumption 2.5, the source term $\mathcal {X}(y,z)$ must satisfy the solvability condition $\langle \mathcal {X}(y,z)\rangle =\int \mathcal {X}(y,z)\pi (y)\, dy=0$ , where the bracket notation denotes integration with respect to the invariant probability density $\pi $ of the process Y. Thus, we derive equations about the operator $ \langle \mathcal {L}_2\rangle $ for the terms. Matching boundary conditions, we have the following PDEs for the terms $P_{0,0}$ , $P_{1,0}$ , $P_{0,1}$ , $P_{1,1}$ , $P_{2,0}$ and $P_{0,2}$ :

(2.5) $$ \begin{align} \mathcal{O}(1)&: \langle\mathcal{L}_2\rangle P_{0,0}=0, \quad P_{0,0}(B,s,z)= \max(s -K, 0),\nonumber\\[4pt] \mathcal{O}(\!\sqrt{\epsilon})&: \langle \mathcal{L}_2\rangle P_{1,0}= -\langle \mathcal{L}_1P_{2,0}\rangle, \quad P_{1,0}(B,s,z)=0,\nonumber\\[4pt] \mathcal{O}(\!\sqrt{\delta})&: \langle \mathcal{L}_2\rangle P_{0,1}= -\langle \mathcal{M}_1\rangle P_{0,0},\quad P_{0,1}(B,s,z)=0,\nonumber\\[4pt] \mathcal{O}(\!\sqrt{\epsilon \delta})&: \langle \mathcal{L}_2\rangle P_{1,1}= -\langle\mathcal{L}_1P_{2,1}\rangle-\langle\mathcal{M}_3P_{2,0}\rangle-\langle\mathcal{M}_1\rangle P_{1,0}, \quad P_{1,1}(B,s,z)=0,\nonumber\\[4pt] \mathcal{O}(\epsilon)&: P_{2,0}(x,s,y,z) = -\phi(y,z)(\partial_x +\tfrac{1}{2}\mathcal{D}_2)P_{0,0} +F_{2,0}(x,y,z),\nonumber\\[4pt] \langle\mathcal{L}_2\rangle F_{2,0}& =-\langle\mathcal{L}_1P_{3,0}\rangle + \langle\phi f^2\rangle( \partial_{xx} +\mathcal{D}_2\partial_x +\tfrac{1}{4}\mathcal{D}_2^2)P_{0,0}, \quad F_{2,0}(B,s,z)=0, \nonumber\\[4pt] \mathcal{O}(\delta)&:\langle\mathcal{L}_2\rangle P_{0,2}=-\langle\mathcal{M}_1\rangle P_{0,1}-\mathcal{M}_2P_{0,0}, \quad P_{0,2}(B,s, z)=0. \end{align} $$

Here, $\mathcal {D}_k = s^k\partial ^k/\partial s^k, k = 1,2,\ldots ,$ and $\phi (y,z)$ is the solution of $\mathcal {L}_{0}\phi = f^2-\langle f^2\rangle $ , with $\langle \phi \rangle = \int \phi (y,z)\pi (y)\, dy = 0$ . The equations are from (3.10), (3.11), (3.22), (3.23), (3.16) and (3.29), respectively. The formal derivation of the price approximation is described in Section 3 in detail.

As mentioned in Section 2.1, $\langle \mathcal {L}_2\rangle $ is regarded as the Black–Scholes operator. Obviously, for the leading term, $P_{0,0}(x,s,z) = P_{BS}(s,K,(B - x)/\bar {\sigma }^2(z), r, \bar {\sigma }(z))$ , that is, the Black–Scholes formula with volatility $\sigma $ set to $\bar {\sigma }(z)$ and the time for expiration set to $(B - x)/\bar {\sigma }^2(z)$ , is given by

(2.6) $$ \begin{align} P_{BS}\bigg(s,K,\frac{B-x}{\bar{\sigma}^2(z)}, r, \bar{\sigma}(z)\bigg) = sN(d_1) - Ke^{-r(B-x)/\bar{\sigma}^2(z)}N(d_2), \end{align} $$

where

$$ \begin{align*} d_1 = \frac{\log (s/k)+ \{r +\bar{\sigma}^2(z)/2\}(B-x)/\bar{\sigma}^2(z)}{\sqrt{B-x}},\quad d_2 = d_1 - \sqrt{B-x}, \end{align*} $$

$$ \begin{align*} N(z) = \int_{-\infty}^{z} e^{-y^2/2}\frac{1}{\sqrt{2\pi}}\,dy. \end{align*} $$

We denote $\bar {\sigma }^2(z)$ as $\bar {\sigma }^2$ , for short, in the following context. The other terms $P_{i,j} (0 < i+j \leq 2)$ turn out to be related to the leading term $P_{0,0}$ in the sense that they are combinations of various Greeks of $P_{BS}$ and some basis functions. The following theorem is the main result of this paper.

Theorem 2.6. Let $P_{i,j}, \,0 \leq i+j \leq 2$ , in (2.3) be the unique classical solutions of the linear ${PDEs}$ with terminal conditions given in (2.5). Then we have the following expressions for $P_{i,j}$ in terms of $P_{BS}(s,K,(B-x)/\bar {\sigma }^2,r,\bar {\sigma })$ in (2.6):

$$ \begin{align*} \left\{ \begin{aligned} P_{0,0}(x,s,z)&=P_{BS}(s,K,(B-x)/\bar{\sigma}^2,r,\bar{\sigma})\\ P_{1,0}(x,s,z)&= -\frac{B-x}{\bar{\sigma}^2}\bigg[C_1(z)\bigg(s\frac{\partial^2P_{0,0}}{\partial s\partial x} +s^2\frac{\partial^2 P_{0,0}}{\partial s^2}+\frac{s^3}{2}\frac{\partial ^3 P_{0,0}}{\partial s^3}\bigg)\\ &\quad- C_2(z) \bigg(\frac{\partial P_{0,0}}{\partial x}+\frac{s^2}{2}\frac{\partial ^2 P_{0,0}}{\partial s^2}\bigg)\bigg] \\ P_{0,1}(x,s,z)&=\frac{B-x}{\bar{\sigma}^2}(C_3(z)s\partial^2_{s\bar{\sigma}}+C_4(z)\partial_{\bar{\sigma}})P_{0,0}\\ P_{2,0}(x,s,y,z)&= -\phi(y,z)\bigg(\frac{\partial P_{0,0}}{\partial x}+\frac{s^2}{2} \frac{\partial ^2 P_{0,0}}{\partial s^2}\bigg) +F_{2,0}(x,s,z),\nonumber \end{aligned} \right. \end{align*} $$

where $F_{2,0}(x,\!s,\!z)$ is defined in (3.19), $P_{0,2}$ in (3.30), $P_{1,1}$ in (3.27), $C_1(z)$ and $C_2(z)$ in (3.15), and $C_3(z)$ and $C_4(z)$ in (3.24).

Buyers of vanilla option often overpay for their options, because the implied volatility in the market is usually higher than the realized volatility. This was shown by an empirical analysis by SG CIB, which found that 80% of 3-month calls that had matured in the money were overpriced [Reference Sawyer18]. The principal pricing term $P_{0,0}$ provides a theoretical justification of the argument that the investor of the timer option only pays the real cost and does not suffer from high implied volatility. As $\epsilon $ and $\delta $ are small enough, $\bar {\sigma }$ is a natural estimation of the realized volatility $I_t/t$ . In fact, we have

$$ \begin{align*} {\lim_{\epsilon\to 0\atop \delta \to 0}}\frac{1}{t}\int_0^t f^2(Y_s, Z_s)\,ds = \bar{\sigma}^2(Z_0), \end{align*} $$

where $\bar {\sigma }^2(z) = \int f^2(y,z)\pi (y)\,dy$ . It is not difficult to obtain this result by the time scales of fluctuation in the volatility process [Reference Fouque, Papanicolaou, Sircar and Solna12, Ch. 3]. The volatility function $f(Y_t, Z_t)$ in the model is driven by $Y_t$ and $Z_t$ , which are both ergodic processes, running on a fast and a slow time scale, respectively. Thus, the principal pricing term $P_{0,0}$ is dependent on the realized volatility $\bar {\sigma }$ . It shows that when the implied volatility in the market is higher than the realized volatility, the leading term $P_{0,0}$ in the timer option price is lower than the price of the corresponding vanilla option whose strike price is K and whose maturity is $(B-x)/\bar {\sigma }^2$ . This theoretical justification was also provided with Heston’s model (a review of Heston’s model can be found in [Reference Heston13]) under the assumption that the risk-free rate r was $0\%$ ; for $r> 0\%$ , a numerical example showed that a timer call option with variance budget $B = \sigma ^2_0T_0$ was less expensive than the European call option with maturity $T_0$ (using the notation of Li [Reference Li15]; see further details therein).

Furthermore, by inverting the timer option approximation formula, we arrive at a second-order effective implied volatility expansion.

Theorem 2.7. With the terms $P_{i,j}, 0 \leq i+j \leq 2$ , in Theorem 2.6 , the approximation effective volatility formula can be obtained as

$$ \begin{align*}\nonumber I^{\epsilon,\delta} \approx \tilde{I}^{\epsilon.\delta}=I_{0,0}+\sqrt{\epsilon}I_{1,0}+\sqrt{\delta}I_{0,1}+\epsilon I_{2,0}+\delta I_{0,2}+\sqrt{\epsilon\delta}I_{1,1}, \end{align*} $$

where $I_{0,0}$ is the leading term of the effective volatility; $I_{1,0}, I_{0,1}$ and $I_{1,1}$ are the first-order fast term, first-order slow term and first-order fast–slow term, respectively; $I_{2,0}$ and $I_{0,2}$ the are second-order fast term and second-order slow term, respectively; and $I_{i,j}, 0 \leq i+j \leq 2$ satisfy (3.31).

A proof of this theorem via a detailed asymptotic analysis will be given in Section 3.

3.1. Related lemmas

To obtain expressions for the higher-order terms $P_{i,j}, 1\leq i+j\leq 2$ , it is desirable to use the following two lemmas.

Lemma 3.1. The Black–Scholes pricing function $P_{BS}(s,K,(B-x)/\bar {\sigma }^2(z),r,\bar {\sigma }(z))$ of the timer call option, given by (2.6), satisfies the following relationship between its vega (that is, $\partial P_{BS}/\partial _{\bar {\sigma }}$ ) and $\mathcal {D}_1P_{BS}$ :

$$ \begin{align*}\nonumber \frac{\partial P_{BS}}{\partial_{\bar{\sigma}}}=\frac{2r}{\bar{\sigma}}\frac{B-x}{\bar{\sigma}^2}(P_{BS}-\mathcal{D}_1P_{BS}),Z, \end{align*} $$

where $P_{BS}=sN(d_1)-Ke^{-r(B-x)/\bar {\sigma }^2}N(d_2)$ .

Lemma 3.2. The Black–Scholes pricing function $P_{BS}(s,K,(B-x)/\bar {\sigma }^2,r,\bar {\sigma })$ of the timer call option, given by (2.6), satisfies, for nonnegative integers k and n,

$$ \begin{align*} &\langle\mathcal{L}_2\rangle\frac{((B-x)/\bar{\sigma}^2)^{n+1}}{n+1}P(\mathcal{D}_k)P_{BS} = -\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^{n}P(\mathcal{D}_k)P_{BS}, \\ &\langle\mathcal{L}_2\rangle\frac{((B-x)/\bar{\sigma}^2)^{n+1}}{n+2}P(\mathcal{D}_k)\partial_{\bar{\sigma}}P_{BS}= -\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^{n}P(\mathcal{D}_k)\partial_{\bar{\sigma}}P_{BS}, \\ &\langle\mathcal{L}_2\rangle\frac{((B-x)/\bar{\sigma}^2)^{n+1}}{n+3}P(\mathcal{D}_k)\bigg(\partial_{\bar{\sigma}\bar{\sigma}}^2\!-\! \frac{3}{\bar{\sigma}(n+2)}\partial_{\bar{\sigma}}\bigg)P_{BS}\!= -\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^{n} P(\mathcal{D}_k)\partial_{\bar{\sigma}\bar{\sigma}}^2P_{BS},\nonumber \end{align*} $$

where $P(\mathcal {D}_k)$ is some polynomial of $\mathcal {D}_1, \mathcal {D}_2, \mathcal {D}_3, \ldots , \mathcal {D}_k$ .

By virtue of the two lemmas above, we are able to derive explicit expressions for $P_{i,j}, 1\leq i+j\leq 2$ . We carry out singular and regular perturbation asymptotics and establish the second-order approximation formula for the timer option price in the subsequent derivation.

3.2. Second-order asymptotics for pricing

3.2.1. Zero-order term $P_{0,0}$ and first-order fast term $P_{1,0}$

In this section, we deal with $P_{0,0}$ and $P_{1,0}$ , which are defined in (2.3). We first expand $P^{\epsilon ,\delta }$ in the powers of $\sqrt {\delta }$ as

(3.1) $$ \begin{align} P^{\epsilon,\delta}=\sum_{j\geq0}\sqrt{\delta}^{j}P_{j}^{\epsilon}, \end{align} $$

where

(3.2) $$ \begin{align} P_{j}^{\epsilon}=\sum_{i\geq0}\sqrt{\epsilon}^{i}P_{i,j}. \end{align} $$

We substitute this expansion (3.1) into the PDE (2.4). With the decomposition (2.4) and collecting the terms based on increasing powers of $\sqrt {\delta }$ , we have

(3.3) $$ \begin{align} &\bigg(\displaystyle\frac{1}{\epsilon}\mathcal{L}_{0}+\displaystyle\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}\bigg)P_{0}^{\epsilon}+ \sqrt{\delta}\bigg\{\bigg(\displaystyle\frac{1}{\epsilon}\mathcal{L}_{0}+\displaystyle\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}\bigg)P_{1}^{\epsilon}+ \bigg(\displaystyle\frac{\mathcal{M}_{3}}{\sqrt{\epsilon}}+\mathcal{M}_{1}\bigg)P_{0}^{\epsilon}\bigg\}\nonumber\\ &\quad +\delta\bigg\{\bigg(\displaystyle\frac{1}{\epsilon}\mathcal{L}_{0}+\displaystyle\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}\bigg)P_{2}^{\epsilon}+ \bigg(\displaystyle\frac{\mathcal{M}_{3}}{\sqrt{\epsilon}}+\mathcal{M}_{1}\bigg)P_{1}^{\epsilon}+\mathcal{M}_{2}P_{0}^{\epsilon}\bigg\}+\cdots=0. \end{align} $$

Equating the corresponding powers of $\sqrt {\delta }$ on both sides of equation (3.3), we have the following expressions:

(3.4) $$ \begin{align} &\,\mathcal{O}(1):0=\bigg(\frac{1}{\epsilon}\mathcal{L}_{0}+\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}\bigg)P_{0}^{\epsilon}, \end{align} $$

(3.5) $$ \begin{align} &\,\mathcal{O}(\sqrt{\delta}):0=\bigg(\frac{1}{\epsilon}\mathcal{L}_{0}+\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}\bigg)P_{1}^{\epsilon}+ \bigg(\frac{\mathcal{M}_{3}}{\sqrt{\epsilon}}+\mathcal{M}_{1}\bigg)P_{0}^{\epsilon}, \end{align} $$

(3.6) $$ \begin{align} &\,\mathcal{O}(\delta):0=\bigg(\frac{1}{\epsilon}\mathcal{L}_{0}+\frac{1}{\sqrt{\epsilon}}\mathcal{L}_{1}+\mathcal{L}_{2}\bigg)P_{2}^{\epsilon}+ \bigg(\frac{\mathcal{M}_{3}}{\sqrt{\epsilon}}+\mathcal{M}_{1}\bigg)P_{1}^{\epsilon}+\mathcal{M}_{2}P_{0}^{\epsilon}. \end{align} $$

Substituting expansions (3.2) into (3.4), and matching terms in powers of $\sqrt {\epsilon }$ , we find that the terms $P_{0,0}$ and $P_{1,0}$ satisfy the equations

$$ \begin{align*}\nonumber \mathcal{O}(1/\epsilon)&:0=\mathcal{L}_0P_{0,0}, \nonumber\\ \mathcal{O}(1/\sqrt{\epsilon})&:0= \mathcal{L}_0P_{1,0}+\mathcal{L}_1P_{0,0}.\nonumber \end{align*} $$

Note that the operators $\mathcal {L}_0$ and $\mathcal {L}_1$ defined in (2.4) take derivatives with respect to y, and we see that they are ordinary equations in y. We choose $P_{0,0}$ and $P_{1,0}$ to be independent of y so that $P_{0,0}$ = $P_{0,0}(x,s,z)$ and $P_{1,0}$ = $P_{1,0}(x,s,z)$ satisfy the above equations; thus, we seek $P_{0,0}$ and $P_{1,0}$ which are independent of y. Continuing the asymptotic analysis, we have:

(3.7)

(3.8) $$ \begin{align} \mathcal{O}(\!\sqrt{\epsilon})&:0=\mathcal{L}_0P_{3,0}+\mathcal{L}_1P_{2,0}+\mathcal{L}_2P_{1,0}, \end{align} $$

(3.9) $$ \begin{align} \mathcal{O}(\epsilon)&:0=\mathcal{L}_0P_{4,0}+\mathcal{L}_1P_{3,0}+\mathcal{L}_2P_{2,0}. \end{align} $$

Here, equations 3.7–3.9 are Poisson equations of the form $\mathcal {L}_0P + \chi = 0$ . In virtue of Assumption 2.5, the Poisson equation has a solution P if the following solvability condition holds [Reference Fouque, Papanicolaou, Sircar and Solna12, Ch. 3]: $\langle \chi \rangle =\int \chi (y)\pi (y) \,dy=0,$ where $\pi (y)$ is the invariant probability density of Y. Imposing the solvable condition on equations (3.7)–(3.9), we have:

(3.10) $$ \begin{align} \mathcal{O}(1)&: 0=\langle\mathcal{L}_2\rangle P_{0,0}, \end{align} $$

(3.11) $$ \begin{align} \mathcal{O}(\sqrt{\epsilon})&:0=\langle\mathcal{L}_1P_{2,0}\rangle+\langle\mathcal{L}_2\rangle P_{1,0}, \end{align} $$

(3.12) $$ \begin{align} \mathcal{O}(\epsilon)&:0=\langle\mathcal{L}_1P_{3,0}\rangle+\langle\mathcal{L}_2P_{2,0}\rangle. \end{align} $$

We expand the terminal condition in (2.4) and have $ P_{0,0}(B,s,z)= \max (s - K, 0)$ , and $P_{1,0}(B,s,z) = 0$ . We note that $P_{0,0}(x,s,z)=P_{BS}(s,K,(B-x)/\bar {\sigma }^2,r,\bar {\sigma })$ , given by (2.6).

Before getting $P_{1,0}$ from equation (3.11), we first compute $\langle \mathcal {L}_1P_{2,0}\rangle $ . Using equation (3.10), equation (3.7) can be rewritten as

$$ \begin{align*} \mathcal{L}_{0}P_{2,0} = -\mathcal{L}_2P_{0,0} = -(\mathcal{L}_2-\langle \mathcal{L}_2\rangle)P_{0,0} =(\bar{\sigma}^2(z)-f^2(y,z))(\partial_x + \mathcal{D}_2/2)P_{0,0}. \end{align*} $$

Let $\phi (y,z)$ be a solution to the Poisson equation

(3.13) $$ \begin{align} \mathcal{L}_{0}\phi = f^2-\langle f^2\rangle , \end{align} $$

and we derive the expression for $P_{2,0}$ as

(3.14) $$ \begin{align} P_{2,0}(x,s,y,z) = -\phi(y,z)(\partial_x +\mathcal{D}_2/2)P_{0,0} +F_{2,0}(x,s,z). \end{align} $$

Thus $P_{1,0}$ satisfies the PDE

$$ \begin{align*} \langle \mathcal{L}_2\rangle P_{1,0} =C_1(z)\bigg(s\frac{\partial^2\!P_{0,0}}{\partial s\partial x}\!+\!s^2\frac{\partial^2 \!P_{0,0}}{\partial s^2}\!+\!\frac{s^3}{2}\frac{\partial ^3 P_{0,0}}{\partial s^3}\bigg) -C_2(z)\bigg(\frac{\partial P_{0,0}}{\partial x}\!+\!\frac{s^2}{2}\frac{\partial ^2 P_{0,0}}{\partial s^2}\bigg) \end{align*} $$

with terminal condition $P_{1,0}(B,s,z)=0$ , where

(3.15) $$ \begin{align} C_1(z) =\langle \rho_1\beta(\cdot)f(\cdot,z)\partial_{y}\phi(\cdot,z)\rangle, \quad C_2(z) =\langle \beta(\cdot)\Lambda(\cdot,z)\partial_{y}\phi(\cdot,z)\rangle. \end{align} $$

Using Lemma 3.2, we obtain the following solution $P_{1,0}$ :

$$ \begin{align*} P_{1,0}\!=\!-\frac{B-x}{\bar{\sigma}^2}\bigg[C_1(z)\bigg(s\frac{\partial^2\!P_{0,0}}{\partial s\partial x}+s^2\frac{\partial^2 P_{0,0}}{\partial s^2}+\frac{s^3}{2}\!\frac{\partial ^3 P_{0,0}}{\partial s^3}\bigg) -\!C_2(z)\bigg(\frac{\partial P_{0,0}}{\partial x}\!+\!\frac{s^2}{2}\frac{\partial ^2\! P_{0,0}}{\partial s^2}\bigg)\bigg]. \end{align*} $$

3.2.2. Second-order fast term $P_{2,0}$

In this section, we deal with the term $P_{2,0}$ , which is defined in (2.3). From equation (3.14), it turns out that the natural terminal condition $P_{2,0}(B,s,y,z)=0$ is not possible. However, we can obtain the averaged terminal condition via the ergodicity, $\langle P_{2,0}(B,s,y,z)\rangle =0.$ In addition, the solution to the Poisson equation (3.13) is chosen here by imposing the condition $\langle \phi (\cdot ,z)\rangle =0$ . From (3.14), using the fact that $\mathcal {D}_2$ and $\mathcal {L}_2$ can commute and $\langle \phi (\cdot ,z) \rangle = 0$ , we have:

(3.16) $$ \begin{align} \!\langle\mathcal{L}_2P_{2,0}\rangle\! = -\langle\phi f^2\rangle(\partial_{xx} + \mathcal{D}_2\partial_x + \mathcal{D}_2^2/4)P_{0,0} +\langle\mathcal{L}_2\rangle F_{2,0}. \end{align} $$

Let $\phi _1(y,z)$ and $\phi _2(y,z)$ be the solutions of the Poisson equations

(3.17) $$ \begin{align} \mathcal{L}_0\phi_1\! &= \!\beta(y)f(y,\!z)\phi^{\prime}_y(y,\!z)-\langle \beta(\!\cdot\!)f(\cdot,\!z) \phi^{\prime}_y(\!\cdot,\!z)\rangle, \end{align} $$

(3.18) $$ \begin{align} \mathcal{L}_0\phi_2\! &=\!\beta(y)\Lambda(y,\!z)\phi^{\prime}_y(y,\!z)-\langle\beta(\!\cdot\!)\Lambda(\!\cdot,\!z)\phi^{\prime}_y(\!\cdot,\!z)\rangle, \end{align} $$

respectively. According to equations (3.8), (3.11), (3.13), and (3.14), we compute $\mathcal {L}_1P_{2,0}$ , $\mathcal {L}_2P_{1,0}$ , and have the solution

$$ \begin{align*} P_{3,0}=(\rho_1\phi_1\mathcal{D}_1 -\phi_2) (\partial_x + \mathcal{D}_2/2)P_{0,0} -\phi(\partial_x +\mathcal{D}_2/2)P_{1,0} +F_{3,0}, \end{align*} $$

for some $F_{3,0}$ independent of y. With equations (3.16) and (3.12), we have the following PDE for $F_{2,0}(x,s,z)$ and the terminal condition $F_{2,0}(B,s,z) = 0$ :

$$ \begin{align*} \langle\mathcal{L}_2\rangle F_{2,0}=-\langle\mathcal{L}_1P_{3,0}\rangle+\langle\phi f^2\rangle(\partial_{xx} + \mathcal{D}_2\partial_x +\mathcal{D}_2^2/4)P_{0,0}. \end{align*} $$

Calculating the $\langle \mathcal {L}_1P_{3,0}\rangle $ and using Lemma 3.2, we obtain:

(3.19) $$ \begin{align} F_{2,0} &=\frac{B\!-\!x}{\bar{\sigma}^2}\![(C_5(z)\mathcal{D}_1^2-C_6(z)\mathcal{D}_1 -C_7(z)\mathcal{D}_1 +C_8(z))(\partial_x+\mathcal{D}_2/2)P_{0,0}] \nonumber\\ &\quad-\{\rho_1C_1(z)\mathcal{D}_1 - C_2(z)\}\bigg[\frac{1}{\bar{\sigma}^2}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg) \mathcal{A}_{1,0}^{\epsilon}P_{0,0}-\frac{1}{2}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^2\mathcal{A}_{1,0}^{\epsilon}\frac{\partial P_{0,0}}{\partial x}\bigg] \nonumber \\ &\quad-\frac{1}{4}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^2\{\rho_1C_1(z)\mathcal{D}_1\mathcal{D}_2\mathcal{A}_{1,0}^{\epsilon} - C_2(z)\mathcal{D}_2\mathcal{A}_{1,0}^{\epsilon}\}P_{0,0} \nonumber \\ &\quad-\frac{B-x}{\bar{\sigma}^2}\langle \phi f^2\rangle\{\partial_{xx} + \mathcal{D}_2\partial_x + \mathcal{D}_2^2/4\}P_{0,0}, \end{align} $$

where $C_1(z),C_2(z)$ are defined in (3.15), and

(3.20) $$ \begin{align} \mathcal{A}_{1,0}^{\epsilon}&=C_1(z)\bigg(s\frac{\partial^2}{\partial s\partial x}+s^2\frac{\partial^2}{\partial s^2}+\frac{s^3}{2}\frac{\partial ^3}{\partial s^3}\bigg)-C_2(z)\bigg(\frac{\partial}{\partial x}+\frac{s^2}{2}\frac{\partial ^2}{\partial s^2}\bigg), \nonumber\\ C_5(z)& = \rho_1^2\langle\beta(\cdot)f(\cdot,z)\phi_1^{\prime}\rangle, C_6(z) = \rho_1 \langle\beta(\cdot)\Lambda(\cdot,z)\phi_1^{\prime}\rangle, \nonumber\\ C_7(z)& = \rho_1\langle\beta(\cdot)f(\cdot,z)\phi_2^{\prime}\rangle, C_8(z) = \langle\beta(\cdot)\Lambda(\cdot,z)\phi_2^{\prime}\rangle. \end{align} $$

Thus, from (3.14), the second-order term $P_{2, 0}$ has been obtained.

3.2.3. First-order slow term $P_{0,1}$ and fast–slow term $P_{1,1}$

The first-order slow term ${P_{0,1}}$ and fast–slow term $P_{1,1}$ are defined in (2.3). Substituting the expansions (3.2) into (3.5) and collecting terms in powers of $\sqrt {\epsilon }$ , we have

We see that $\mathcal {L}_0P_{0,1} = 0$ is an ordinary differential equation in y, which has constants in solutions (there are also exponentially growing solutions). Just as in the previous sections, we still look for solutions $P_{0,1}$ and $P_{1,1}$ , which are independent of y. We analyse and obtain Poisson equations of the form in Assumption 2.5:

(3.21)

Using the solvable condition, we have equations about $\langle \mathcal {L}_2\rangle $ for $P_{0,1}$ and $P_{1,1}$ :

(3.22) $$ \begin{align} \mathcal{O}(\sqrt{\delta})&: 0=\langle\mathcal{L}_2\rangle P_{0,1}+\langle\mathcal{M}_1\rangle P_{0,0}, \end{align} $$

(3.23) $$ \begin{align} \mathcal{O}(\sqrt{\delta}\sqrt{\epsilon}) &:0=\langle\mathcal{L}_1P_{2,1}\rangle+\langle \mathcal{L}_2\rangle P_{1,1}+\langle \mathcal{M}_{3}P_{2,0}\rangle+\langle \mathcal{M}_1\rangle P_{1,0}, \end{align} $$

with terminal conditions $P_{0,1}(B,s,z)=0$ and $P_{1,1}(B,s,z)=0$ . By Lemma 3.2, we can obtain an expression for $P_{0,1}$ :

$$ \begin{align*} P_{0,1}&=\frac{B-x}{\bar{\sigma}^2}(C_3(z)\mathcal{D}_1\partial_{\bar{\sigma}} +C_4(z)\partial_{\bar{\sigma}})P_{0,0}, \end{align*} $$

where

(3.24) $$ \begin{align} C_3(z)=\tfrac{1}{2}\rho_2g(z)\langle f(\cdot,z)\rangle\bar{\sigma}'(z),\quad C_4(z)=-\tfrac{1}{2}g(z)\langle\Gamma(\cdot,z)\rangle\bar{\sigma}'(z). \end{align} $$

Note that $P_{1,1}$ can be obtained using equation (3.23), if we compute $\langle \mathcal {L}_1P_{2,1}\rangle $ and $\langle \mathcal {M}_3P_{2,0}\rangle $ . Using (3.21) and (3.22), $P_{2,1}$ is given by

$$ \begin{align*} P_{2,1}=-\phi(y,z)(\partial_x+\tfrac{1}{2}\mathcal{D}_2)P_{0,1}-\{\rho_2g(z)\phi_3(y,z)\mathcal{D}_1\partial_z - g(z)\phi_4(y,z)\partial_z\}P_{0,0}+F_{2,1}(x,s,z), \end{align*} $$

where $F_{2,1}(x,s,z)$ does not depend on y, and $\phi _3(y,z)$ and $\phi _4(y,z)$ satisfy Poisson equations

(3.25) $$ \begin{align} \mathcal{L}_0\phi_3&=f-\langle f\rangle, \end{align} $$

(3.26) $$ \begin{align} \mathcal{L}_0\phi_4&=\Gamma-\langle \Gamma \rangle. \end{align} $$

Using (3.14), the above term $P_{2,1}$ , and operators $\mathcal {L}_1$ and $\mathcal {M}_3$ in (2.4), we obtain an equation about $\mathcal {L}_2$ for $P_{1,1}$ . By Lemma 3.2,

(3.27) $$ \begin{align} P_{1,1} &= -\frac{1}{2}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^2\frac{4r}{\bar{\sigma}^3} \{C_1(z)\mathcal{D}_1+C_2(z)\}\{C_3(z)\mathcal{D}_1+C_4(z)\}(P_{0,0}-\mathcal{D}_1P_{0,0})\nonumber \\ &\quad+\frac{1}{3}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^3\frac{2r}{\bar{\sigma}} \{C_1(z)\mathcal{D}_1+C_2(z)\}\{C_3(z)\mathcal{D}_1+C_4(z)\}(\partial_x-\mathcal{D}_1 \partial_x)P_{0,0} \nonumber \\ &\quad-\frac{1}{6}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^3\frac{2r}{\bar{\sigma}}\{C_1(z)\mathcal{D}_1\mathcal{D}_2 -C_2\mathcal{D}_2\}\{C_3(z)\mathcal{D}_1+C_4(z)\}(P_{0,0}-\mathcal{D}_1P_{0,0}) \nonumber \\ &\quad+\frac{1}{2}\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^2\frac{2r}{\bar{\sigma}} \{\tilde{C}_2(z)\mathcal{D}_1^2+\tilde{C}_1(z)\mathcal{D}_1+\tilde{C}_0(z)\}(P_{0,0}-\mathcal{D}_1P_{0,0})\nonumber \\ &\quad+2\bigg(\frac{B-x}{\bar{\sigma}^2}\bigg)^2\{C_3(z)\mathcal{D}_1\!+\!C_4(z)\}\bigg[\frac{1}{\bar{\sigma}}\mathcal{A}_{1,0}^\epsilon \!-\!\frac{1}{3}\mathcal{A}_{1,0}^{\epsilon}\partial_{\bar{\sigma}} \!-\! \frac{1}{3}\frac{1}{\bar{\sigma}'(z)}\partial_z \mathcal{A}_{1,0}^{\epsilon}\bigg]P_{0,0}\nonumber \\ &\quad-\frac{B\!-\!x}{\bar{\sigma}^2}\{\tilde{C_3}(z)+\tilde{C_4}(z)\partial_{\bar{\sigma}}\}(\partial_x +\mathcal{D}_2/2)P_{0,0}, \end{align} $$

where $C_1(z)$ and $C_2(z)$ are defined in (3.15), $C_3(z)$ and $C_4(z)$ are given in (3.24), $\mathcal {A}_{1,0}$ is given in (3.20), and

$$ \begin{align*} \tilde{C_0}(z)&=-\bar{\sigma}'(z)g(z)\langle\beta(\cdot)\Lambda(\cdot,z)\partial_y\phi_4(\cdot,z)\rangle, \\ \tilde{C_1}(z)&=\rho_2\bar{\sigma}'(z)g(z)\langle\beta(\cdot)\Lambda(\cdot,\!z)\partial_y\phi_3(\cdot,z)\rangle\!+\!\rho_1g(z)\langle\beta(\cdot)f(\cdot,z)\partial_y\phi_4(\cdot,z)\rangle, \\ \tilde{C_2}(z)&=-\rho_1\rho_2\bar{\sigma}'(z)g(z)\langle\beta(\cdot)f(\cdot,z)\partial_y\phi_3(\cdot,z)\rangle, \\ \tilde{C_3}(z) &= \rho_{12}g(z)\langle\beta(\cdot,\!z)\partial_{yz}\phi(\cdot,z)\rangle, \\ \tilde{C_4}(z) & = \frac{1}{2}\rho_{12}g(z)\langle\beta(\cdot,z)\partial_y \phi(\cdot,\!z)\rangle\bar{\sigma}'\!(z),\\ \frac{\partial \mathcal{A}_{1,0}^{\epsilon}}{\partial z} &= \frac{\partial C_1(z)}{\partial z}\bigg(s\frac{\partial^2}{\partial s\partial x}+s^2\frac{\partial^2}{\partial s^2}+\frac{s^3}{2}\frac{\partial ^3}{\partial s^3}\bigg)-\frac{\partial C_2(z)}{\partial z}\bigg(\frac{\partial}{\partial x}+\frac{s^2}{2}\frac{\partial ^2}{\partial s^2}\bigg). \end{align*} $$

3.2.4. Second-order slow term $P_{0,2}$

We insert the expansion (3.2) into (3.6) and obtain equations in powers of $\sqrt {\epsilon }$ . Similarly, based on the equations, we seek the solution of $P_{0,2}(x,y,z)$ . The $\mathcal {O}(\delta )$ equation is

(3.28)

Equation (3.28) is a Poisson equation with solvable condition

(3.29) $$ \begin{align} \mathcal{O}(\delta):0=\langle\mathcal{L}_2\rangle P_{0,2}+\langle\mathcal{M}_1\rangle P_{0,1}+\mathcal{M}_2P_{0,0}. \end{align} $$

The associated terminal condition is $P_{0,2}(B,s,z)=0.$ By computing $\langle \mathcal {M}_1\rangle P_{0,1}$ and $\mathcal {M}_2P_{0,0}$ , and by Lemma 3.2, the solution $P_{0,2}$ is given as

(3.30) $$ \begin{align} P_{0,2}&=-\bigg[\frac{4}{3\bar{\sigma}}\bigg(\frac{B\!-\!x}{\bar{\sigma}^2}\bigg)^2N_1^2\!\!-\!\! \frac{2}{3\bar{\sigma}'}\bigg(\frac{B\!-\!x}{\bar{\sigma}^2}\bigg)^2\!N_1N_1^{\prime}\bigg]\partial_{\bar{\sigma}}P_{00}\!+\!\frac{1}{2}\bigg(\frac{B\!-\!x}{\bar{\sigma}^2}\bigg)^2 N_1^2\bigg(\partial^2_{\bar{\sigma}\bar{\sigma}}\!-\!\frac{3}{\bar{\sigma}}\partial_{\bar{\sigma}}\bigg)P_{0,0} \nonumber\\ &\quad+\frac{1}{6}\frac{B\!-\!x}{\bar{\sigma}^2}g^2\bar{{\sigma}^{\prime}}^2\bigg(\partial ^2_{\bar{\sigma}\bar{\sigma}}\!-\!\frac{3}{2\bar{\sigma}}\partial_{\bar{\sigma}}\bigg)P_{0,0}-\bigg( \frac{1}{4}\frac{B\!-\!x}{\bar{\sigma}^2}g^2\!\bar{\sigma}''\!-\!\frac{1}{2}\frac{B\!-\!x}{\bar{\sigma}^2}c(z)\bar{\sigma}'\bigg)\partial _{\bar{\sigma}}P_{0,0}, \end{align} $$

where

$$ \begin{align*} &N_1=C_3(z)\mathcal{D}_1+C_4(z), \quad N_1^{\prime}=C_3^{\prime}(z)\mathcal{D}_1+C_4^{\prime}(z),\\ &C_3^{\prime}(z)=\partial_z C_3(z), \quad C_4^{\prime}(z)=\partial_z C_4(z),\\ & \bar{\sigma}^{\prime}(z)=\partial_z\bar{\sigma}(z), \quad \bar{\sigma}^{\prime\prime}(z)=\partial^2_{zz}\bar{\sigma}(z). \end{align*} $$

3.3. Second-order asymptotics for effective implied volatility

The investigation of implied volatility is essential, as option prices rarely conform to the idealized assumption of constant volatility in the Black–Scholes pricing framework, and option prices are quoted in terms of it in practice. For timer options, there is no fixed maturity and hence the “implied volatility” does not exist in the usual sense. Recall that in our pricing formula $(B-x)/\bar {\sigma }^2$ plays the part of the corresponding fixed time to maturity. Therefore, this effective maturity is a reasonable candidate for exploring a similar concept of “implied volatility” for timer options, which we mentioned as “effective implied volatility” in the Introduction.

In this section, we convert the expansion of the time option price obtained in the previous section into an expansion of effective implied volatility of the form $I^{\epsilon ,\delta }=\sum _{i\geq 0}\sum _{j\geq 0}\sqrt {\epsilon }^i\sqrt {\delta }^jI_{i,j}$ such that $P^{\epsilon ,\delta }=P_{BS}(I^{\epsilon ,\delta })$ . Taking the Taylor expansion of $P_{BS}(I^{\epsilon ,\delta })$ about $I_{0,0}$ and rearranging terms yields

$$ \begin{align*} \begin{aligned} P_{0,0}+&\sqrt{\epsilon}P_{1,0}+\sqrt{\delta}P_{0,1}+\sqrt{\epsilon\delta}P_{1,1}+\epsilon P_{2,0}+\delta P_{0,2} +\cdots \\ &=P_{BS}(I_{0,0}+\sqrt{\epsilon}I_{1,0}+\sqrt{\delta} I_{0,1}+\sqrt{\epsilon\delta}I_{1,1}+\epsilon I_{2,0}+\delta I_{0,2}+\cdots) \\ &=P_{BS}(I_{0,0})+\sqrt{\epsilon}I_{1,0}\partial_{\bar{\sigma}}P_{BS}(I_{0,0})+\sqrt{\delta}I_{0,1}\partial_{\bar{\sigma}}P_{BS}(I_{0,0})\\ &\quad+\sqrt{\epsilon\delta}\{I_{1,0}I_{0,1}\partial_{\bar{\sigma}\bar{\sigma}}^2P_{BS}(I_{0,0})+ I_{1,1}\partial_{\bar{\sigma}}P_{BS}(I_{0,0})\}\\ &\quad +\epsilon\big\{\tfrac{1}{2}I_{1,0}^2\partial_{\bar{\sigma}\bar{\sigma}}^2P_{BS}(I_{0,0})+I_{2,0}\partial_{\bar{\sigma}}P_{BS}(I_{0,0})\big\}\\ &\quad+\delta\big\{\tfrac{1}{2}I_{0,1}^2\partial_{\bar{\sigma}\bar{\sigma}}^2P_{BS}(I_{0,0})+I_{0,2}\partial_{\bar{\sigma}}P_{BS}(I_{0,0})\big\}+\cdots .\nonumber \end{aligned} \end{align*} $$

Equating the same powers of $\sqrt {\epsilon }$ and $\sqrt {\delta }$ and applying $P_{0,0}=P_{BS}(\bar {\sigma })$ , we obtain

(3.31) $$ \begin{align} \mathcal{O}(1)&:I_{0,0}=\bar{\sigma},\nonumber\\ \mathcal{O}(\sqrt{\epsilon})&:I_{1,0}=\frac{P_{1,0}}{\partial_{\bar{\sigma}}P_{0,0}},\nonumber\\ \mathcal{O}(\epsilon)&:I_{2,0}=\frac{P_{2,0}}{\partial_{\bar{\sigma}}P_{0,0}}- \frac{1}{2}I_{1,0}^2\frac{\partial_{\bar{\sigma}\bar{\sigma}}^2P_{0,0}}{\partial_{\bar{\sigma}}P_{0,0}},\nonumber\\ \mathcal{O}(\sqrt{\delta})&:I_{0,1}=\frac{P_{0,1}}{\partial_{\bar{\sigma}}P_{0,0}},\nonumber\\ \mathcal{O}(\delta)&:I_{0,2}=\frac{P_{0,2}}{\partial_{\bar{\sigma}}P_{0,0}}-\frac{1}{2}I_{0,1}^2\frac{\partial_{\bar{\sigma}\bar{\sigma}}^2 P_{0,0}}{\partial_{\bar{\sigma}}P_{0,0}},\nonumber\\ \mathcal{O}(\sqrt{\epsilon\delta})&:I_{1,1}=\frac{P_{1,1}}{\partial_{\bar{\sigma}}P_{0,0}}-I_{1,0}I_{0,1}\frac{\partial_{\bar{\sigma}\bar{\sigma}}^2P_{0,0}} {\partial_{\bar{\sigma}}P_{0,0}}. \end{align} $$

Therefore, the desired second-order effective implied volatility approximation $I_{0,0}+\sqrt {\epsilon }I_{1,0}+\sqrt {\delta }I_{0,1}+\sqrt {\epsilon \delta }I_{1,1} +\epsilon I_{2,0}+\delta I_{0,2}$ of the effective implied volatility $I^{\epsilon ,\delta }$ is established. It is clear that the zero-order term of the approximation is the effective volatility $\bar {\sigma }$ ; $P_{0,0}$ and its derivatives with respect to $\bar {\sigma }$ appear in the corrected terms. We consider the “forward log-moneyness” with effective maturity for timer options, $d = \log (K/Se^{r(B-x)/\bar {\sigma }^2})$ , which is the analogue of the “forward log-moneyness” for vanilla options in [Reference Fouque, Lorig and Sircar8]. We show an effective implied volatility surface with d and effective maturity in Figure 1 in the next section.

Figure 1 Effective implied volatility surface.