An analytical approximation for single barrier options under stochastic volatility models

Abstract

The aim of this paper is to derive an approximation formula for a single barrier option under local volatility models, stochastic volatility models, and their hybrids, which are widely used in practice. The basic idea of our approximation is to mimic a target underlying asset process by a polynomial of the Wiener process. We then translate the problem of solving first hit probability of the asset process into that of a Wiener process whose distribution of passage time is known. Finally, utilizing the Girsanov’s theorem and the reflection principle, we show that single barrier option prices can be approximated in a closed-form. Furthermore, ample numerical examples will show the accuracy of our approximation is high enough for practical applications.

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Fig. 1

Notes

  1. 1.

    The up-and-out, up-and-in, down-and-in, and down-and-out barrier cases is considered in Sect. 5.5.

  2. 2.

    The existence of \(\omega _{i}(t)\) is discussed in Sect. 4.

  3. 3.

    For example, we have \(h_{1}(x)=x\), \(h_{2}(x)=x^{2} - 1\), \(h_{3}(x)=x^{3} - 3x\), etc.

  4. 4.

    See, e.g., Emanuel and MacBeth (1982) for more details.

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Acknowledgements

The authors is grateful to the anonymous referees for invaluable comments that improved the original manuscript considerably. Funahashi also thanks Tetsuhiro Takeshita, QDS Consulting, for his careful reading of this manuscript. Needless to say, all errors and confusions are ours.

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Correspondence to Hideharu Funahashi.

Appendices

Appendix A: Proof of Proposition 3.1

In order to calculate the probability distribution of \(X_t\), we derive an approximated characteristic function of \(X_t\) and invert it back to derive an approximation of the probability distribution of \(X_t\).

Let the characteristic function of \(X_t\) be \(\Psi (\xi ) = \mathbb {E}[ \mathrm{e}^{i \xi X_t} ] \). We approximate it as

$$\begin{aligned} \Psi (\xi ) \approx \left\{ \begin{array}{lll} \displaystyle 1 + i \xi \mathbb {E}[ a_{2}(t) ],&{} \quad \ if \ a_1(t) = 0, \ a.s., \\ \mathbb {E}[ \mathrm{e}^{ i \xi a_{1}(t)} ] + i \xi \mathbb {E}\left[ \mathrm{e}^{ i \xi a_{1}(t)} \mathbb {E}[ a_{2}(t)\, |\, a_{1}(t) ] \right] ,&{} \quad otherwise. \end{array} \right. \end{aligned}$$

where \(R_3\) consists of the forth or higher-order multiple stochastic integrals. Note that, since

$$\begin{aligned} \left| \mathbb {E}\left[ \mathrm{e}^{ i \xi a_{1}(t) } R_3 \right] \right|\le & {} \mathbb {E}\left[ |\mathrm{e}^{ i \xi a_{1}(t) } R_3 |\right] \\\le & {} \left( \mathbb {E}\left[ |\mathrm{e}^{ i \xi a_{1}(t) }|^2\right] \right) ^{\frac{1}{2}} \left( \mathbb {E}\left[ |R_3 |^2\right] \right) ^{\frac{1}{2}} \\= & {} \left( \mathbb {E}\left[ |R_3 |^2\right] \right) ^{\frac{1}{2}} \ \approx \ 0 , \end{aligned}$$

we regard \(\mathbb {E}\left[ \mathrm{e}^{ i \xi a_{1}(t) } R_3 \right] \approx 0\) as for the previous case.

Taking the conditional expectation on \(a_1(t)\), which follows normal distribution with 0 mean and variance \(\Sigma _t\), we then have

$$\begin{aligned} \Psi (\xi ) \approx \left\{ \begin{array}{ll} \displaystyle 1, &{}\quad \Sigma _t = 0, \\ \mathbb {E}[ \mathrm{e}^{ i \xi a_{1}(t)} ] + i \xi \mathbb {E}\left[ \mathrm{e}^{ i \xi a_{1}(t)} \mathbb {E}[ a_{2}(t)\, |\, a_{1}(t) ] \right] , &{}\quad otherwise . \end{array} \right. \end{aligned}$$

Further, if X follows a normal distribution with zero mean and variance \(\Sigma \), then by differentiating both sides of

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

w.r.t. y, we have for any polynomial functions f(x) and g(x)

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

where n(xab) denotes the normal density function with mean a and variance b.

Therefore, we can apply (A.1) to obtain the approximation of the density function as

$$\begin{aligned} f_{X_t}(x) \approx \left\{ \begin{array}{lll} \displaystyle \delta (x),&{} \quad \Sigma _t = 0, \\ n\left( x; 0, \Sigma _{t} \right) - \frac{\partial }{\partial {x}} \left\{ \mathbb {E}[ a_{2}(t) | a_{1}(t) = x ] n\left( x; 0, \Sigma _{t} \right) \right\} + \cdots ,&{} \quad otherwise. \end{array} \right. \end{aligned}$$

Appendix B: Formulas for conditional expectation

Let \(W^p_t\), \(W^q_t\), and \(W^r_t\) be standard Brownian motions with correlation \(\rho _{i,j} \mathrm{d}t = \mathrm{d}W^i_t \mathrm{d}W^j_t\), and let \(y_{i}(x)\) for \(i=1,2,3\) be some deterministic functions. Moreover, let \(\Sigma := \int _{0}^{T} y^2_{1}(t) \mathrm{d}t\), and denote \(J_T(y_1)= \int _{0}^{T} y_{1}(t) \mathrm{d}W^p_t\).

Then, the following formulas are derived:

$$\begin{aligned} E\left[ \int _{0}^{T} y_{3}(t) \left( \int _{0}^{t} y_{2}(s) \mathrm{d}W^q_s \right) \mathrm{d}W^r_t \bigg | J_T(y_1) = x \right] = v_{1} \left( \frac{x^{2}}{\Sigma ^{2}}- \frac{1}{\Sigma } \right) , \end{aligned}$$
(B.1)

where

$$\begin{aligned} v_{1} = \int _{0}^{T} \rho _{p,r} y_{3}(t) y_{1}(t) \left( \int _{0}^{t} \rho _{p,q} y_{2}(s) y_{1}(s) \mathrm{d}s \right) \mathrm{d}t . \end{aligned}$$

The interested reader can find more details in Funahashi and Kijima (2015).

Appendix C: Proof of Theorem 5.1

If r, \(\kappa \), and \(\theta \) are zero, we have \(F(0,t) = S_0\) and \(V(0,t) = v_0\). Therefore, \({\bar{\sigma }} := \sigma (S_0,v_0)\), \({\bar{\sigma }}_S := \sigma _S(S_0,v_0)\), \({\bar{\sigma }}_v := \sigma _v(S_0,v_0)\), and \({\bar{\gamma }} := \gamma _0\) become constants.

The variance of the 1st-order Wiener–Ito chaos expansion \(V_1(t) := \mathbb {E}[\left( a_1(t) - \widetilde{a}_1(t) \right) ^2]\) is

$$\begin{aligned} V_1(t) = \int _0^t \left( \sigma ^{(0)}(s) - \sqrt{\Sigma _t / t} \right) ^2 \mathrm{d}s. \end{aligned}$$

But, since \(\sigma ^{(0)}(s) - \sqrt{\Sigma _t / t} = 0\), we get \(V_1(t) = 0\).

Similarly, let \({\bar{p}}_1 := {\bar{\sigma }} + S_0 {\bar{\sigma }}_S\), \({\bar{p}}_2 := {\bar{\sigma }} _v\), \({\bar{p}}_3 := {\bar{\gamma }}\), \(\Sigma _t = {\bar{p}}_1^2 t\), and \(q(t) = \left( {\bar{\sigma }}^3 {\bar{p}}_1 + \rho {\bar{p}}_2 {\bar{p}}_3 {\bar{\sigma }}^2 \right) t^2 / 2\) the variance of the 2st-order Wiener–Ito chaos expansion \(V_2(t):= \mathbb {E}[\left( a_1(t) - \widetilde{a}_1(t) \right) ^2]\) can be computed as

$$\begin{aligned}&V_2(t) = \left( {\bar{p}}^2_1 {\bar{\sigma }}^2 + {\bar{p}}^2_2 {\bar{p}}^2_3 + 4 \left( \frac{q(t)}{t \Sigma _t} \right) ^2 + 2 \rho \bar{p}_1 \bar{p}_2 \bar{p}_3 \bar{\sigma } - 4 \rho \frac{q(t)}{t \Sigma _t} {\bar{p}}_2 {\bar{p}}_3 - 4 \frac{q(t)}{t \Sigma _t} {\bar{p}}_1 {\bar{\sigma }} \right) \int _0^t \nonumber \\&\quad \left( \int _0^s \mathrm{d}u \right) \mathrm{d}s . \end{aligned}$$

Inserting \(q(t)/(t \Sigma _t) = \left[ {\bar{p}}_1 {\bar{\sigma }} + \rho {\bar{p}}_2 {\bar{p}}_3\right] /2\) and \(\rho ^2 = 1\) into the left-hand side of the last equation, we obtain \(V_2(t) = 0\).

Hence, from the definition of \(V_Y(t)\), we obtain the desired result.

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Funahashi, H., Higuchi, T. An analytical approximation for single barrier options under stochastic volatility models. Ann Oper Res 266, 129–157 (2018). https://doi.org/10.1007/s10479-017-2559-3

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Keywords

  • Single barrier option
  • Analytical approximation
  • Local and stochastic volatility models
  • Wiener–Ito chaos expansion