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Tempered stable process, first passage time, and path-dependent option pricing

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Abstract

In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Lévy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. For the numerical illustration, we calibrate risk neutral process parameters using S&P 500 index option prices and apply those parameters to find prices of perpetual American option and barrier option.

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Notes

  1. See Schoutens (2003) for additional details.

  2. See Hull (2015) for more details.

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Acknowledgements

I am grateful to Professor Kyuong Jin Choi, in Haskayne School of Business, University of Calgary, who gave the motivation to complete of this research. Also, all remaining errors are entirely my own.

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Correspondence to Young Shin Kim.

Appendix

Appendix

As “Appendix”, we discuss perpetual American option pricing and barrier option pricing under the Lévy market model.

1.1 Perpetual American option

The perpetual call and put option price on Lévy model can be obtained by the martingale method introduced in Gerber and Shiu (1994). In this section, we just follow the martingale method for the Lévy market price model. We consider a perpetual American call option with strike price K. If the option holder exercise the call at a time T, then the holder obtain \((S(T)-K)^+\) where \(x^+ = \max \{0,x\}\). Let L be a real number with \(L\ge K\). The holder will exercise the call when the asset price first become greater than or equal to the level L. We define the first passage time

$$\begin{aligned} \tau (l) = \inf \{t\ge 0|S(t)\ge L\} = \inf \{t\ge 0|X(t)\ge l\}, \end{aligned}$$

where \(l=\log (L/S(0))>0\). Then the current value of the perpetual American call is

$$\begin{aligned} C_{\mathrm{perpetual}}=\max _{L\ge K} E[e^{-r\tau (l)}(S(\tau (l))-K)^+]. \end{aligned}$$

Let

$$\begin{aligned} C(L) = E[e^{-r\tau (l)}(S(\tau (l))-K)^+]=(L-K)E[e^{-r\tau (l)}] \end{aligned}$$

which is the Laplace transform of \(\tau (l)\). Applying Lemma 1, we can obtain the Laplace transform as

$$\begin{aligned} E\left[ e^{-r\tau (l)}\right] = \phi _{\tau (l)}(ir) =e^{-l\eta ^+(ir)}, \end{aligned}$$

where \(\eta ^+(ir)\) is the value satisfying (2) and (3) for \(l>0\) and \(u=ir\). Hence we have

$$\begin{aligned} C(L) = (L-K)e^{-l\eta ^+(ir)}=(L-K)\left( \frac{S(0)}{L}\right) ^{\eta ^+(ir)}. \end{aligned}$$

By solving

$$\begin{aligned} \frac{\partial C}{\partial L}(L^+) = 0, \end{aligned}$$

we find the optimal value \(L^+\)

$$\begin{aligned} L^+ = \frac{\eta ^+(ir)K}{\eta ^+(ir)-1}. \end{aligned}$$

Hence, we obtain the maximum value

$$\begin{aligned} C(L^+)=\frac{K}{\eta ^+(ir)-1}\left( \frac{S(0)(\eta ^+(ir)-1)}{K\eta ^+(ir)}\right) ^{\eta ^+(ir)}. \end{aligned}$$

If \(L^+< S(0)\) then the call is immediately exercised so we have price \(S(0)-K\). Therefore the perpetual call price is equal to

$$\begin{aligned} C_{\mathrm{perpetual}}={\left\{ \begin{array}{ll} \displaystyle \frac{K}{\eta ^+(ir)-1}\left( \frac{S(0)(\eta ^+(ir)-1)}{K\eta ^+(ir)}\right) ^{\eta ^+(ir)} &{}\text { if } S(0)\le L^+ \\ S(0)-K &{}\text { if } S(0)>L^+ \end{array}\right. }. \end{aligned}$$

We consider a perpetual American put option with strike price K. If the option holder exercise the put at a time T, then the holder obtain \((K-S(T))^+\). Let L be a real number with \(0<L\le K\). The holder will exercise the put when the asset price first become less than or equal to the level L. We define the first passage time

$$\begin{aligned} \tau (l) = \inf \{t\ge 0|S(t)\le L\} = \inf \{t\ge 0|X(t)\le l\} \end{aligned}$$

where \(l=\log (L/S(0))<0\). Then the current value of the put is

$$\begin{aligned} P_{\mathrm{perpetual}} = \max _{0<L\le K} E[e^{-r\tau (l)}(K-S(\tau (l)))^+] \end{aligned}$$

which is the Laplace transform of \(\tau (l)\). For the same arguments as the call option case, we find the optimal value \(L^-\)

$$\begin{aligned} L^- = \frac{\eta ^-(ir)K}{\eta ^-(ir)-1}, \end{aligned}$$

where \(\eta ^-(ir)\) is the value satisfying (2) and (3) for \(l<0\) and \(u=ir\). Hence the perpetual put price is equal to

$$\begin{aligned} P_{\mathrm{perpetual}}={\left\{ \begin{array}{ll} \displaystyle \frac{K}{1-\eta ^-(ir)}\left( \frac{S(0)(\eta ^-(ir)-1)}{K\eta ^-(ir)}\right) ^{\eta ^-(ir)} &{}\text { if } S(0)\ge L^- \\ K-S(0) &{}\text { if } S(0)<L^- \end{array}\right. }. \end{aligned}$$

1.2 Barrier option

Let \(\varPi \) be the payoff function of European options. For example, the European call and put options with strike price K are given by \(\varPi (S(T))=(S(T)-K)^+\) and \(\varPi (S(T))=(K-S(T))^+\), respectively. The knock-in barrier option with the barrier level B, time to maturity T is priced by the following equation

$$\begin{aligned} V_{i}=e^{-rT}E\left[ \varPi (S(T))1_{\tau (l)<T}\right] \end{aligned}$$

where \(l = \log (B/S(0))\). Note that \(l<0\) for the down-and-in barrier option and \(l>0\) for the up-and-in barrier option. Since we have

$$\begin{aligned} \varPi (S(T)) = \varPi (S(T))1_{\tau (l)<T}+\varPi (S(T))1_{\tau (l)\ge T}, \end{aligned}$$

the knock-out barrier option price can be obtained by the following equation

$$\begin{aligned} V_{o}=e^{-rT}E\left[ \varPi (S(T))1_{\tau (l)\ge T}\right] = e^{-rT}\left( E\left[ \varPi (S(T))\right] - E\left[ \varPi (S(T))1_{\tau (l)< T}\right] \right) = V - V_{i}, \end{aligned}$$

where \(V = e^{-rT}E\left[ \varPi (S(T))\right] \). Note that \(l<0\) for the down-and-out barrier option and \(l>0\) for the up-and-out barrier option.

Case 1 \(\varPi (S(T)) = \varPi (S(T))1_{\tau (l)<T}\)

If \(\varPi (S(T)) = \varPi (S(T))1_{\tau (l)<T}\) then the barrier option price is the same as option prices without the barrier:

$$\begin{aligned} V_{i}=e^{-rT}E\left[ \varPi (S(T))1_{\tau (l)<T}\right] =e^{-rT}E\left[ \varPi (S(T))\right] . \end{aligned}$$

For example (1) up-and-in call option with \(K>B\), we have \((S(T)-K)^+=(S(T)-K)^+1_{\tau (l)<T}\), and (2) down-and-in put option with \(K<B\), we have \((K-S(T))^+=(K-S(T))^+1_{\tau (l)<T}\).

Case 2 \(\varPi (S(T)) \ne \varPi (S(T))1_{\tau (l)<T}\)

If \(\varPi (S(T))\ne \varPi (S(T))1_{\tau (l)<T}\), we have

$$\begin{aligned} V_{i}&=e^{-rT}E\left[ \varPi (S(T))1_{\tau (l)<T}\right] \\&=e^{-rT}E\left[ E\left[ \varPi (S(0)e^{X(T)-X(\tau (l))+X(\tau (l))})1_{\tau (l)<T}|\tau (l)\right] \right] \\&=e^{-rT}E\left[ E\left[ \varPi (S(0)e^le^{X(T-\tau (l))})1_{\tau (l)<T}|\tau (l)\right] \right] \\&=e^{-rT}\int _0^T E\left[ \varPi (S(0)e^le^{X(T-t)})1_{t<T}\right] f_{\tau (l)}(t)dt \end{aligned}$$

where \(f_{\tau (l)}\) is the pdf of \(\tau (l)\). Since we have \(S(0)e^l = B\) and

$$\begin{aligned} f_{\tau (l)}(t)=\frac{1}{2\pi }\int _{-\infty }^\infty e^{-ivt}\phi _{\tau (l)}(v)dv, \end{aligned}$$

the \(c_{i}\) becomes

$$\begin{aligned} V_{i}&=\frac{e^{-rT}}{2\pi }\int _0^T E\left[ \varPi (Be^{X(T-t)})\right] \int _{-\infty }^\infty e^{-ivt}\phi _{\tau (l)}(v)dv\, dt \nonumber \\&=\frac{e^{-rT}}{2\pi }\int _{-\infty }^\infty \int _0^T E\left[ \varPi (Be^{X(T-t)})\right] e^{-ivt}dt\phi _{\tau (l)}(v)dv. \end{aligned}$$
(10)

By European option pricing formula using Fourier transform [see Carr and Madan (1999), Lewis (2001) and Rachev et al. (2011)], we have

$$\begin{aligned} E\left[ \varPi (Be^{X(T-t)})\right] =\frac{1}{2\pi }\int _{-\infty }^\infty B^{i(u+i\rho )}e^{(T-t)\psi _X(u+i\rho )}\hat{\varPi }(u+i\rho )du, \end{aligned}$$

where \(\hat{\varPi }(z)=\int _{-\infty }^\infty e^{-izx}\varPi (e^x)dx\) for complex number z and \(\rho \) is a real constant such that \(\psi _X(u+i\rho )\) and \(\hat{\varPi }(u+i\rho )\) are well defined for all \(u\in \mathbb {R}\). Hence we have

$$\begin{aligned} V_{i}&=\frac{e^{-rT}}{2\pi }\int _{-\infty }^\infty \int _0^T \frac{1}{2\pi }\int _{-\infty }^\infty B^{i(u+i\rho )}e^{(T-t)\psi _X(u+i\rho )}\hat{\varPi }(u+i\rho )du\, e^{-ivt}dt\, \phi _{\tau (l)}(v)dv\\&=\frac{e^{-rT}}{(2\pi )^2}\int _{-\infty }^\infty \int _{-\infty }^\infty B^{i(u+i\rho )}\hat{\varPi }(u+i\rho )\int _0^T e^{(T-t)\psi _X(u+i\rho )} e^{-ivt}dt\, du\, \phi _{\tau (l)}(v)dv\\&=\frac{e^{-rT}}{(2\pi )^2}\int _{-\infty }^\infty B^{i(u+i\rho )}\hat{\varPi }(u+i\rho )\int _{-\infty }^\infty \frac{e^{T\psi _X(u+i\rho )}-e^{-ivT}}{\psi _X(u+i\rho )+iv}\phi _{\tau (l)}(v)dv\, du\\ \end{aligned}$$

Let

$$\begin{aligned} H(u) = \int _{-\infty }^\infty \frac{e^{T\psi _X(u+i\rho )}-e^{-ivT}}{\psi _X(u+i\rho )+iv}\phi _{\tau (l)}(v)dv \end{aligned}$$

then

$$\begin{aligned} V_i=\frac{e^{-rT}}{(2\pi )^2}\int _{-\infty }^\infty B^{i(u+i\rho )}\hat{\varPi }(u+i\rho )H(u) du. \end{aligned}$$

European call and put options

For the call option payoff \(\varPi (S(T))=(S(T)-K)^+\), we have

$$\begin{aligned} \hat{\varPi }(u+i\rho ) = \int _{\log K}^\infty e^{-i(u+i\rho )x}(e^x-K)dx = \frac{K^{\rho +1-iu}}{(\rho -iu)(\rho +1-iu)}, ~~~\rho <-1 \end{aligned}$$

and for the put option payoff \(\varPi (S(T))=(K-S(T))^+\), we have

$$\begin{aligned} \hat{\varPi }(u+i\rho ) = \int ^{\log K}_{-\infty } e^{-i(u+i\rho )x}(K-e^x)dx = \frac{K^{\rho +1-iu}}{(\rho -iu)(\rho +1-iu)}, ~~~\rho >0. \end{aligned}$$

The down-and-in call option price (\(c_{di}\)) and up-and-in put option price (\(p_{ui}\)) are always in Case 2. Therefore, we have their prices as

$$\begin{aligned} c_{di} = \frac{e^{-rT}K^{1+\rho }}{(2\pi )^2 B^\rho }\int _{-\infty }^\infty \left( \frac{B}{K}\right) ^{iu} \left( \frac{H(u)}{(\rho -iu)(1+\rho -iu)}\right) du , ~~~\rho <-1 \end{aligned}$$

and

$$\begin{aligned} p_{ui} = \frac{e^{-rT}K^{1+\rho }}{(2\pi )^2 B^\rho }\int _{-\infty }^\infty \left( \frac{B}{K}\right) ^{iu} \left( \frac{H(u)}{(\rho -iu)(1+\rho -iu)}\right) du , ~~~\rho >0 \end{aligned}$$

For the up-and-in call option price (\(c_{ui}\)) and the down-and-in put option price (\(p_{di}\)), we consider Case 1, and finally obtain

$$\begin{aligned} c_{ui} = {\left\{ \begin{array}{ll} \frac{e^{-rT}K^{1+\rho }}{(2\pi )^2 B^\rho }\int _{-\infty }^\infty \left( \frac{B}{K}\right) ^{iu} \left( \frac{H(u)}{(\rho -iu)(1+\rho -iu)}\right) du &{} \text { if } K\le B \\ \frac{e^{-rT}K^{1+\rho }}{2\pi S(0)^\rho }\int _{-\infty }^\infty \left( \frac{S(0)}{K}\right) ^{iu} \left( \frac{\phi _{X(T-t)}(u+i\rho )}{(\rho -iu)(1+\rho -iu)}\right) du &{} \text { if } K> B \end{array}\right. } , ~~~\rho <-1. \end{aligned}$$

and

$$\begin{aligned} p_{di} = {\left\{ \begin{array}{ll} \frac{e^{-rT}K^{1+\rho }}{(2\pi )^2 B^\rho }\int _{-\infty }^\infty \left( \frac{B}{K}\right) ^{iu} \left( \frac{H(u)}{(\rho -iu)(1+\rho -iu)}\right) du &{} \text { if } K\ge B \\ \frac{e^{-rT}K^{1+\rho }}{2\pi S(0)^\rho }\int _{-\infty }^\infty \left( \frac{S(0)}{K}\right) ^{iu} \left( \frac{\phi _{X(T-t)}(u+i\rho )}{(\rho -iu)(1+\rho -iu)}\right) du &{} \text { if } K< B \end{array}\right. } , ~~~\rho >0. \end{aligned}$$

The up-and-out and down-and-out calls corresponding to the up-and-in and down-and-in calls above are priced by \(c_{uo} = c - c_{ui}\), and \(c_{do} = c - c_{di}\), respectively, where c is the vanilla call option price with strike price K, and time to maturity T. The up-and-out and down-and-out puts corresponding to the up-and-in and down-and-in puts above are priced by \(p_{uo} = p - p_{ui}\), and \(p_{do} = p - p_{di}\), respectively, where p is the vanilla put option price with strike price K, and time to maturity T.

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Kim, Y.S. Tempered stable process, first passage time, and path-dependent option pricing. Comput Manag Sci 16, 187–215 (2019). https://doi.org/10.1007/s10287-018-0326-9

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