Explicit approximate analytic formulas for timer option pricing with stochastic interest rates

North American Journal of Economics and Finance 34 (2015) 1–21

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North American Journal of Economics and Finance

Explicit approximate analytic formulas for timer option pricing with stochastic interest rates Jingtang Ma a, Dongya Deng b, Yongzeng Lai c,∗ a School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, PR China b School of Finance, Southwestern University of Finance and Economics, Chengdu 611130, PR China c Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5

a r t i c l e

i n f o

Article history: Received 22 April 2015 Received in revised form 21 July 2015 Accepted 27 July 2015 Available online 5 August 2015 JEL classifications: G12 G13 C02 C63 Keywords: Timer options Stochastic interest rate models Stochastic volatility models Analytic methods

a b s t r a c t The interest rate risk is an important factor in the valuation of timer options. Since the valuation of timer options with interest rate risk is a four-dimensional problem, the dimensionality curse causes tremendous difficulty in finding analytic solutions to the pricing of timer options. In this paper, a fast approximate analytic method is developed to price power style timer options with Vasicek interest rate model. The valuation of timer options with interest rate risk is formulated as a four-dimensional partial differential equation (PDE) using -hedging approach. A dimension-reduction technique is then proposed to reduce the four-dimensional PDE into a two-dimensional nonlinear PDE. A perturbation approach is developed to solve the reduced two-dimensional nonlinear PDEs and then an explicit approximate analytic formula for the timer option is obtained. In particular, explicit approximate analytic formulas for timer options under both Heston and Hull–White models are further derived. Numerical examples of pricing timer options under the above two models are provided. Both the approximate analytic method and the crude Monte Carlo simulation method are used for the examples. The numerical results show that prices of timer options by both methods are close and the approximate analytic method is much faster than the crude Monte Carlo method. © 2015 Elsevier Inc. All rights reserved.

∗ Corresponding author. Tel.: +1 5198840710. E-mail addresses: [email protected] (J. Ma), [email protected] (D. Deng), [email protected] (Y. Lai). http://dx.doi.org/10.1016/j.najef.2015.07.002 1062-9408/© 2015 Elsevier Inc. All rights reserved.

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J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

1. Introduction Volatility is a very important factor in asset price modeling. Financial derivatives on asset volatilities are useful tools for trading, hedging, and risk management. Some information on asset volatility modeling and financial derivatives can be found in Swishchuk (2013), as well as references therein. Timer options are special types of financial derivatives on asset volatilities. Pricing timer options receives significant attention in recent mathematical finance literature. A timer option is similar to a vanilla European option with a random maturity date which is specified as the first time when the accumulated variance of the underlying asset price reaches a given budget level. As discussed in Sawyer (2007), this type of product is designed to give investors more flexibility and to ensure that they do not overpay for an option. In 2007 a timer option was first traded by Société Générale Corporate and Investment Banking (SG CIB). Due to the complexity, timer options were first sold to sophisticated investors such as hedge funds. These options are now becoming more and more popular. In fact, timer options first appeared in the academia, such as the “mileage option” studied by Neuberger (1990) and the continuous time models discussed by Bick (1995) many years ago, when such securities did not exist in financial market. Carr and Lee (2010) studied a robust replication for timer options where the risk-free rate is zero. For the Hull–White stochastic volatility model (see Hull & White, 1987), Geman and Yor (1993) established an explicit formula for the distribution related to random maturity date using some remarkable analytical properties of Bessel processes. In Saunders (2011) an asymptotic expansion was developed for fast mean reverting stochastic volatility models of timer options. Monte Carlo methods were used to simulate the prices of timer options for constant interest rates, e.g., Li (2010) and Bernard and Cui (2011). More attractively, Bernard and Cui (2011) proposed a technique of time change to reduce the computational cost of a single timer option from many hours to a few minutes. Liang, Lemmens, and Tempere (2011) used path integral technique developed in quantum field theory to evaluate timer options. Li and Mercurio (2013) developed closed-form approximation to timer option prices under general stochastic volatility models. Li (2013) obtained a Black–Scholes–Merton type formula for pricing timer options using joint distribution of the first-passage time of the realized variance and the corresponding variance characterized by Bessel processes with drift. To the best of our knowledge, it is assumed that interest rates are constants for all the studies on timer option pricing reported in the literature. However, the effect of the correlations between the variance process and the interest rate process is also an important risk factor for timer options as pointed out in Bernard and Cui (2011). The problem of pricing timer options becomes very difficult for the case with stochastic interest rates due to the complexity of the correlations among interest rate, volatility, underlying asset and random maturity. In this paper, we study timer option pricing with stochastic interest rates under Vasicek model and develop an efficient method, the approximate analytic method, to evaluate timer options with power-type payoffs and small volatility of volatility in the stochastic volatility models. We first derive a four-dimensional partial differential equation (PDE) for the timer option using -hedging approach. Then a dimension-reduction technique is proposed to reduce the four-dimensional PDE into a two-dimensional nonlinear PDE. A perturbation approach, which was used by Li and Mercurio (2013) for obtaining a solution of the two-dimensional PDEs for timer options with constant interest rates, is developed to solve the reduced two-dimensional nonlinear PDEs and obtain an approximate analytic formula for the timer options. The general form of the approximate analytic formula contains some definite integrals and a parameter which is determined by a simple nonlinear equation (see Theorem 3.3). For most popular volatility models like Heston models and Hull–White models, these integrals can be calculated explicitly. Alternatively these integrals can be calculated by numerical quadrature rules. The parameter involved in the formula can be fast calculated by solving the nonlinear equation using Newton’s iteration method. In particular, explicit approximate analytic formulas for timer options under both Heston and Hull–White models are further derived. Numerical results in comparing to the crude Monte Carlo simulation show that the approximate analytic formulas are accurate and much faster than the crude Monte Carlo method. The organization of the rest of this paper is as follows. First, a four-dimensional partial differential equation (PDE) satisfied by the timer option is derived using -hedging approach in Section 2. Next, the resulting four-dimensional PDE is reduced to a two-dimensional nonlinear PDE

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

3

using some technical variable transformations in Section 3. Then, the reduced two-dimensional nonlinear PDE is solved by a perturbation approach, finally an approximate analytic formula for timer option is derived in the same section. Two special cases of stochastic volatility models, Heston model and Hull–White model, for the underlying asset prices are discussed and explicit approximate analytic formulas for timer options under these two models are derived in Section 4. In Section 5, both the approximate analytic method and the crude Monte Carlo simulation method are applied to the timer options for asset prices under both Heston and Hull–White models. Numerical examples are provided to illustrate the efficiency of the approximate analytic method. 2. Timer options and PDE formulation Assume that the stock price St , volatility Vt and interest rate rt satisfy the following stochastic models under the risk neutral measure Q: dS t = rt St dt +



1

Vt St dW t ,

(1) 2

dV t = ˛(Vt )dt + ˇ(Vt )dW t , dr t = ( − rt )dt

(2)

3 + dW t ,

(3)

where , , ,  > 0 are constant model parameters and assume  is small, and Wt1 , Wt2 , Wt3 are Brownian motions under Q and satisfy 1

2

1

dW t dW t = 1 dt,

3

dW t dW t = 2 dt,

2

3

dW t dW t = 3 dt,

where 1 , 2 , 3 are correlation coefficients. In the Vasicek model (3), the dynamics for the zero-coupon bond with maturity T is given by (see Fabozzi, 2002) 3

dP t = rt Pt dt + P (t)Pt dW t ,

(4)

where P (t) = −

1 − e−(T −t) . 

(5)

Denote  as the random time remaining for a pre-specified variance budget B to be exceeded, i.e.,



 = inf





t

Vu du = B

t>0:

.

(6)

0

Consider a timer option with random maturity  and power payoff S , 0 < = / 1. The price at t = 0 of the timer option under Q is given by C = EQ [e−r  S ]. Denote the accumulated variance process by



t =

t

Vu du.

(7)

0

Then the processes St , Vt , t , rt form a Markovian system which is sufficient to model the timer options. Therefore, the price at time t of the timer option can be written as a function f(S, V, , r), where S, V, , r are called dummy variables and the dependence on t is suppressed for convenience. Following (Gatheral, 2002 Chapter 1), we prove that the function f for valuing the timer option satisfies a four-dimensional PDE (see Theorem 2.1 below). Theorem 2.1. Under models (1)–(3), the price at t of the timer option, f(S, V, , r), satisfies the following four-dimensional PDE

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J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21 2

V

2

2

√ ∂ f √ ∂f ∂ f ∂ f ∂f ∂f ∂f + 1 Sˇ(V ) V + 2 S V P + 3 ˇ(V ) + ( − r) + rS + ˛(V ) ∂r ∂S ∂V ∂S ∂r ∂V ∂r ∂V ∂

∂S 2

2

+

2

1 ∂ f ∂ f 1 1 2∂ f + 2 − rf = 0, + 2 ˇ2 (V ) VS 2 ∂V 2 2 ∂r 2 ∂S 2 2

(8)

with terminal condition f (S, V, B, r) = S .

(9)

Proof See Appendix A. It is noted that compared to a standard European style option, variable plays the role of time. Therefore is often called “stochastic clock”. 3. Explicit approximate analytic formula for pricing timer options In this section, we perform a number of variable transforms to convert the four dimensional PDE (8) to a two dimensional one, and then use a perturbation approach to find an approximate analytic formula for the price of the timer option. Theorem 3.1. y=

S , P

Let f (S, V, , r) fˆ (y, V, ) = , P

(10)

where P = P(t, r). Then the four-dimensional PDE (8) with terminal condition (9) is equivalent to the following three-dimensional PDE

 ∂fˆ √ ∂fˆ  ∂ fˆ ∂ fˆ 1 + ˛(V ) + 3 ˇ(V )P + (1 V − 3 P )yˇ(V ) + 2 ˇ2 (V ) 2 ∂

∂V ∂ y∂ V ∂V 2 2

V

+

2

∂2 fˆ √ 1 2 P − 22 P P V + V y2 = 0, 2 ∂ y2

(11)

with terminal condition fˆ (y, V, B) = P −1 y ,

(12)

where  P is given by (5). Proof See Appendix B. Moreover, using a variable transform, the three-dimensional PDE (11) can be further reduced to a two-dimensional one. The result is summarized in the following theorem. Theorem 3.2.

Using the transform

fˆ (y, V, ) = P −1 y Veg(V, ) ,

(13)

the three dimensional PDE (11) with boundary condition (12) is transformed into



2

1 ∂g ∂g ∂ g ˛ ˛ + ˇ2 2 ) +V + (

+ + ˇ2 2 V V2 ∂

∂V V 2 ∂V 2 = 0,



∂g ∂V

2  +

√ ( − 1)  2 P − 22 P P V + V V 2 (14)

with boundary condition g(V, B) = − ln V, where

(15) √

˛(V ) = V (˛ + 1 ˇ V ) + (1 − )3 ˇ(V )P V.

(16)

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

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Proof See Appendix C. In the following we use a perturbation method to find an approximate analytic solution to the twodimensional nonlinear PDE (14). The procedure is as follows: Let

g (V, ) be the zeroth-order solution in  for (14). Then it satisfies V2

√

( − 1)  2 ∂

g ˛ ∂

g ˛ + ˇ2 2 ) P − 22 P P V + V V = 0, + + + (

2 ∂V V ∂

(17)

g (V, B) = − ln V . Write the solution of (14) with boundary condition (15) into with boundary condition

the form g (V, ) + 2 J(V, ). g(V, ) ≈

(18)

Then plugging (18) into (14) and using (17), we obtain the first-order PDE for J(V, ),



2

g ∂J ∂

1 ∂J +V + ˇ2 V ˛ + ˇ2 2 ) V + (

2 ∂V ∂V 2 ∂



2

g ∂

∂V

2 

= 0,

(19)

with boundary condition J(V, B) = 0 (notice that we have omitted the high-order term O(2 ).). The solutions of the first order linear equations (17) and (19) with corresponding boundary conditions can be found by the method of characteristics (see e.g., Polyanin, Zaitsev, & Moussiaux, 2002). The results are summarized in the following theorem. Theorem 3.3.

=

−1

Let ( (V0 ) + B),

(20)

where B is the pre-specified variance budget, V0 is the volatility at time zero, and function 

(x) =

x2

˛(x) + 2 ˇ2 (x)

satisfies that

.

(21)

Then the solution of (17) with boundary condition

g (V, B) = − ln V is given by



g (V, ) = ϕ(V, ) ≡ − ln − −

( − 1) 2



V



V

˛(u)

˛(u) + 2 ˇ2 (u)) u(

 2 √ P − 22 P P u + u u du,

˛(u) + 2 ˇ2 (u)

du

(22)

and the solution of (19) with boundary condition J(V, B) = 0 is given by 1 J(V, ) = (V, ) ≡ − 2

=−

1 2



V



V



uˇ2 (u)

˛(u) + 2 ˇ2 (u)

2

∂ ϕ (u, ) + ∂u2



2 

∂ϕ (u, ) ∂u



du



ˇ2 (u) 2

˛2 (u) + 2

˛(u)ˇ2 (u) + 22 u

˛(u)ˇ (u)ˇ(u) − 2 u

˛ (u)ˇ2 (u)





3

˛(u) + 2 ˇ2 (u) u

du.

(23)

Proof See Appendix D. Finally, we obtain the pricing formula for the timer option with stochastic interest rates as follows: f (S, V, , r) = S Veg(V, ) ,

0< = / 1, t ≤ T,

(24)

) + 2 (V,

where P is given by (B.10) in Appendix B, g(V, ) ≈ ϕ(V, ) with ϕ,  given by (22) and (23) respectively, and  being a small number. In formula (24), can be calculated by (20) directly if the inverse function −1 has explicit form. For most cases, we do not have the explicit expression for −1 , in which case, we shall use numerical methods such as Newton’s iteration method to solve ( ) =

(V0 ) + B

(25)

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J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

to obtain a numerical value of . For some specific volatility models, e.g., Heston model or Hull–White model, the expressions for in (21), for ϕ in (22), and for  in (23) can be calculated explicitly, as can be seen from the next section. 4. Two specific volatility models 4.1. Heston volatility model For Heston model, functions in (2) are given by √ ˛(V ) = (ω − V ), ˇ(V ) = V ,

(26)

where constants  and ω satisfy the usual Feller condition 2ω > 2 and it is also assumed that 1  −  < 0, where 1 is the correlation coefficient between the two Brownian motions Wt1 and Wt2 in the set-up models (1)–(3) (see Heston, 1993). ˛ defined by (16) becomes With (26),



˛(V ) = ωV + ( 1  − )V 2 + (1 − )3 P V 3/2 , and Eq. (21) for defining 

(x) =

(27)

becomes

x x √ , √ = (ω + 2 ) + ( 1  − )x + (1 − )3 P x D1 + D2 x + D3 x

(28)

where D1 ≡ ω + 2 ,

D2 ≡ 1  − ,

D3 ≡ (1 − )3 P .

The solution of (28) is calculated by

(x) =

√ √ 1 1 x + 3 [−2D2 D3 x + (D32 − D1 D2 ) ln |D1 + D2 x + D3 x|] D2 D2

   2D √x + D + D2 − 4D D  3 1 2  2 3 ln  −   . √ D23 D32 − 4D1 D2  2D2 x + D3 − D32 − 4D1 D2  (3D1 D2 − D32 )D3

(29)

Therefore, using Theorem 3.3 and after fundamental calculations (details are omitted here), we obtain the approximate formula for g g(V, ) ≈ ϕ(V, ) + 2 (V, ), where



ϕ(V, ) = − ln − ( − 1) − 2



 = − ln −

V

V

V

(30)

√ ω + ( 1  − )x + (1 − )3 P x √ dx x[ω + 2 + ( 1  − )x + (1 − )3 P x] √ P2 − 22 P P x + x

√ dx ω + 2 + ( 1  − )x + (1 − )3 P x √ ω + D2 x + D3 x ( − 1) √ dx − 2 x[D1 + D2 x + D3 x]





V

√ P2 − 22 P P x + x √ dx D1 + D2 x + D3 x

 

  √ √ 2ω  V  ( − 1) ( − 1) 2D3 42 P P ln  + ( V − ) = − ln −  − 2D (V − ) + 2 D1 2 D D2   2 2

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

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     2D √V + D + D2 − 4D D  D1 + D2 V + D3 √V  2 3 1 2  3 + 1 ln  √  + 2 ln   √ D1 + D2 + D3   2D V + D − D2 − 4D D 

with 1 ≡

( − 1) ω −1− 2 D1

 2 ≡ ×





P2 D2

( − 1) 3D3 − D3 2 2



3

2

  2D2 + D3 − D32 − 4D1 D2  ×  , √ 2D2 + D3 + D32 − 4D1 D2 

3



√

1 D32

− 4D1 D2

−



D1 D22

P2 D2

−

+

D23

D22

2

(31)

D32

D1

1

+

22 P PD3

−

D22 2D3 D22

 ,

42 P P + D2

D1 −

D32



2D2

,

and



√ 2[ω + ( 1  − )x + (1 − )3 P x] √ 2 dx x[ω + 2 + ( 1  − )x + (1 − )3 P x]  V  V √ √ ω + 2( 1  − )x + 32 (1 − )3 P x 2[ω + D2 x + D3 x] 1 2 dx = − dx + 2 x[ω + 2 + (   − )x + (1 − )  √x]3 2 x[D + D x + D √x]2 P 1 3 1 2 3  V √ ω + 2D2 x + 32 D3 x 2 + dx. (32) 2 x[D + D x + D √x]3 1 2 3

(V, ) = −

1 2

V

Explicit forms for the two integrals in (32) are too long to be given here. They are available upon request. Finally, using formula (24), we obtain the approximate analytic formula of the timer option for asset price under Heston’s model. 4.2. Hull–White volatility model For Hull–White model (see Hull & White, 1987), functions in (2) are given by ˛(V ) = V,

ˇ(V ) = V,

(33)

where  is a constant. With the setting in (33),

˛ defined by (16) becomes

˛(V ) = V 2 + 1 V 5/2 + (1 − )3 P V 2 , and equation (21) for defining 

(x) =

becomes

1 , √  + 2 + 1  x + (1 − )3 P

whose solution is (x) =

(34)





√ √ 2  + 2 + (1 − )3 P ln | + 2 + (1 − )3 P + 1  x| . x− 1  1 

(35)

Therefore, using Theorem 3.3 (formulas (22) and (23)) and after some calculations (details are omitted here again), we obtain the approximate formula for g g(V, ) ≈ ϕ(V, ) + 2 (V, ),

(36)

8

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

where



V

ϕ(V, ) = − ln −

−

( − 1) 2





√ P2 − 22 P P x + x

V

√ dx x[ + 2 + (1 − )3 P + 1  x]





√  + (1 − )3 P + 1  x √ dx x[ + 2 + (1 − )3 P + 1  x]

3 + 1 √x D

V

= − ln −

dx −

3 + 2 + 1 √x] x[D



( − 1) 2





V

√ P2 − 22 P P x + x dx

3 + 2 + 1 √x] x[D

    1/2 D

3 + 2 + 1 V 1/2  22   ln = − ln V −

3 + 2  V

3 + 2 + 1  1/2  D D    √  D

3 + 2 + 1  V 

3 + 2 P2 D 22 P P   + + ( − 1) + ln  2 1 

3 + 2

3 + 2 + 1 √  ( 1 ) D D    − ( − 1)

3 D

P2

+ 2

ln

V



+

√ 1 √ ( V − ) , 1 

(37)

and

(V, ) = −

1 2



V

3 + 1 √x)2 + 32 (D

3 + 1 √x) − 2 (2D

3 + 52 1 √x) 2(D dx

3 + 2 + 1 √x]3 x[D    1/2

D3 + 2 + 1 V 1/2 

3 + 2 + 1  1/2  D

3 3 + 1 5 + 4D

32 + 2D

3 2  1 1 D = ln  3 V 2

3 + 2 ) (D −

3 2 − 24 1 1 3 − 5D 2

3 + 2 )2 (D

3 + 22 ) 1 2 (D − 4

3 + 2 D





1

3 + 2 + 1 V 1/2 D 1

3 + 2 + 1 V 1/2 ) (D

2

−

−

1

3 + 2 + 1  1/2 D 

1

3 + 2 + 1  1/2 ) (D

2

,

(38)

where

3 ≡  + (1 − )3 P . D Finally, using formula (24), we obtain the approximate analytic formula of the timer option for asset price under the Hull–White model. 5. Numerical examples In this section, we implement the formulas derived by this paper for Heston model and Hull–White model and compare the results by both the approximate analytic method and the Monte-Carlo method. To simulate the price of a timer option by the plain Monte-Carlo (MC) method, the life-time interval [0, T] is evenly divided into m subintervals [ti , ti+1 ], ti = ih, h = T/m, i = 0, 1, . . ., m. The integral in (7) is approximated by the composite trapezoidal rule as follows

⎡

⎤

 1

ti ≈ h ⎣ (V0 + Vti ) + Vtj ⎦ . 2 i−1

j=1

Table 1 Prices of timer option under Heston model. = 1.01

= 1.06

1 , 2 , 3

Formula

MC (std)

0.5, −0.3, −0.8 Time (s) −0.5, −0.3, 0.8 Time (s) −0.5, −0.3, 0 Time (s) 0.5, −0.3, 0 Time (s) 0, −0.3, 0.8 Time (s) 0, −0.3, −0.8 Time (s) 0.5, 0.3, 0.8 Time (s) −0.5, 0.3, −0.8 Time (s) −0.5, 0.3, 0 Time (s) 0.5, 0.3, 0 Time (s) 0, 0.3, 0.8 Time (s) 0, 0.3, −0.8 Time (s)

104.72 0.27 (s) 104.72 0.25 (s) 104.72 0.11 (s) 104.72 0.11 (s) 104.72 0.24 (s) 104.72 0.24 (s) 104.73 0.23 (s) 104.73 0.25 (s) 104.73 0.14 (s) 104.73 0.13 (s) 104.72 0.24 (s) 104.72 0.25 (s)

104.79 (0.04) 56.94 (s) 104.73 (0.04) 56.80 (s) 104.75 (0.04) 59.85 (s) 104.77 (0.04) 59.80 (s) 104.78 (0.04) 59.97 (s) 104.86 (0.04) 59.96 (s) 104.64 (0.04) 59.51 (s) 104.77 (0.04) 61.69 (s) 104.83 (0.04) 75.87 (s) 104.78 (0.04) 60.28 (s) 104.81 (0.04) 59.42 (s) 104.80 (0.04) 62.50 (s)

Rel-Err 6.46e−4 1.09e−4 2.91e−4 4.65e−4 5.78e−4 1.35e−3 6.84e−4 3.60e−4 9.70e−4 4.46e−4 8.28e−4 7.75e−4

= 1.10

Formula

MC (std)

131.91 0.24 (s) 131.91 0.24 (s) 131.91 0.11 (s) 131.91 0.10 (s) 131.91 0.24 (s) 131.91 0.24 (s) 131.98 0.30 (s) 131.98 0.33 (s) 131.98 0.11 (s) 131.98 0.11 (s) 131.91 0.23 (s) 131.91 0.30 (s)

132.25 (0.05) 56.96 (s) 132.17 (0.05) 56.90 (s) 132.07 (0.05) 59.94 (s) 132.18 (0.05) 60.18 (s) 132.18 (0.05) 59.94 (s) 132.24 (0.05) 59.91 (s) 131.98 (0.05) 91.03 (s) 132.94 (0.05) 62.79 (s) 132.17 (0.05) 59.52 (s) 132.19 (0.05) 59.78 (s) 132.23 (0.05) 75.09 (s) 132.21 (0.05) 68.82 (s)

Rel-Err 2.63e−3 1.97e−3 1.20e−3 2.09e−3 2.09e−3 2.56e−3 1.39e−4 3.54e−4 1.43e−3 1.57e−3 2.43e−3 2.32e−3

Formula

MC (std)

158.66 0.25 (s) 158.66 0.27 (s) 158.66 0.12 (s) 158.66 0.10 (s) 158.66 0.26 (s) 158.66 0.23 (s) 158.82 0.2631 (s) 158.82 0.27 (s) 158.82 0.12 (s) 158.82 0.11 (s) 158.66 0.30 (s) 158.66 0.24 (s)

159.11 (0.07) 60.10 (s) 159.11 (0.07) 56.35 (s) 159.13 (0.07) 60.09 (s) 159.05 (0.07) 60.83 (s) 159.20 (0.07) 59.36 (s) 159.17 (0.07) 59.23 (s) 159.95 (0.06) 60.19 (s) 158.85 (0.06) 59.92 (s) 159.18 (0.07) 59.89 (s) 159.28 (0.07) 63.07 (s) 159.32 (0.07) 75.81(s) 159.03 (0.07) 60.32 (s)

Rel-Err 2.85e−3 2.85e−3 2.94e−3 2.44e−3 3.38e−3 3.25e−3 8.16e−4 2.40e−4 2.25e−3 2.89e−3 4.12e−3 2.37e−3

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Correl. coeff.

Model parameters are set as follows:  = 1,  = 0.04,  = 0.1,  = 0.000625,  = 2, ω = 0.0324, S0 = 100, V0 = 0.0625, B = 0.0265, r0 = 0.04, T = 1.

9

10

Table 2 Prices of timer option under Heston model. = 1.01

1 , 2 , 3

Formula

MC (std)

= 1.06

0.5, −0.3, −0.8 Time (s) −0.5, −0.3, 0.8 Time (s) −0.5, −0.3, 0 Time (s) 0.5, −0.3, 0 Time (s) 0, −0.3, 0.8 Time (s) 0, −0.3, −0.8 Time (s) 0.5, 0.3, 0.8 Time (s) −0.5, 0.3, −0.8 Time (s) −0.5, 0.3, 0 Time (s) 0.5, 0.3, 0 Time (s) 0, 0.3, 0.8 Time (s) 0, 0.3, −0.8 Time (s)

20.61 0.32 (s) 20.61 0.24 (s) 20.61 0.12 (s) 20.61 0.12 (s) 20.61 0.27 (s) 20.61 0.23 (s) 20.61 0.23 (s) 20.61 0.26 (s) 20.61 0.11 (s) 20.61 0.10 (s) 20.61 0.22 (s) 20.61 0.23 (s)

20.63 (0.01) 56.09 (s) 20.62 (0.01) 55.86 (s) 20.63 (0.01) 56.06 (s) 20.63 (0.01) 55.85 (s) 20.63 (0.01) 56.20 (s) 20.63 (0.01) 55.90 (s) 20.61 (0.01) 70.58 (s) 20.59 (0.01) 58.80 (s) 20.63 (0.01) 58.95 (s) 20.64 (0.01) 58.61 (s) 20.64 (0.01) 58.61 (s) 20.61 (0.01) 58.63 (s)

= 1.10

Rel-Err

Formula

MC (std)

1.11e−3

23.95 0.23 (s) 23.95 0.23 (s) 23.95 0.11 (s) 23.95 0.11 (s) 23.95 0.22 (s) 23.95 0.23 (s) 23.97 0.23 (s) 23.97 0.25 (s) 23.97 0.11 (s) 23.97 0.10 (s) 23.95 0.26 (s) 23.95 0.29 (s)

24.00 (0.01) 55.97 (s) 24.03 (0.01) 55.96 (s) 24.01 (0.01) 55.90 (s) 24.00 (0.01) 55.99 (s) 24.01 (0.01) 56.13 (s) 23.99 (0.01) 55.93 (s) 23.99 (0.01) 58.65 (s) 23.98 (0.01) 58.68 (s) 24.01 (0.01) 58.68 (s) 24.02 (0.01) 61.01 (s) 24.00 (0.01) 67.37 (s) 24.00 (0.01) 62.11 (s)

3.29e−3 9.37e−4 7.48e−3 9.65e−3 8.76e−4 3.26e−4 1.18e−3 1.05e−3 1.58e−3 1.30e−3 1.64e−3

Rel-Err

Formula

MC (std)

2.04e−3

27.01 0.26 (s) 27.02 0.25 (s) 27.01 0.11 (s) 27.01 0.10 (s) 27.02 0.22 (s) 27.01 0.23 (s) 27.04 0.23 (s) 27.04 0.29 (s) 27.04 0.10 (s) 27.04 0.12 (s) 27.02 0.22 (s) 27.01 0.22 (s)

27.09 (0.01) 55.81 (s) 27.10 (0.01) 55.99 (s) 27.08 (0.01) 55.99 (s) 27.10 (0.01) 55.93 (s) 27.12 (0.01) 56.53 (s) 27.09 (0.01) 57.43 (s) 27.05 (0.01) 56.18 (s) 27.07 (0.01) 56.16 (s) 27.11 (0.01) 61.69 (s) 27.10 (0.01) 72.62 (s) 27.09 (0.01) 57.46 (s) 27.09 (0.01) 56.32 (s)

3.12e−3 2.17e−3 1.87e−3 2.22e−3 1.50e−3 8.83e−4 7.04e−4 1.70e−3 2.06e−3 2.09e−3 1.90e−3

Model parameters are set as follows:  = 1,  = 0.04,  = 0.1,  = 0.000625,  = 2, ω = 0.0324, S0 = 20, V0 = 0.0625, B = 0.0265, r0 = 0.04, T = 1.

Rel-Err 2.62e−3 3.08e−3 2.55e−3 3.09e−3 4.04e−3 2.96e−3 3.38e−4 9.35e−4 2.57e−3 2.15e−3 2.77e−3 2.82e−3

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Correl. coeff.

Table 3 Prices of timer option under Heston model. = 1.01

= 1.06

1 , 2 , 3

Formula

MC (std)

0.5, −0.3, −0.8 Time (s) −0.5, −0.3, 0.8 Time (s) −0.5, −0.3, 0 Time (s) 0.5, −0.3, 0 Time (s) 0, −0.3, 0.8 Time (s) 0, −0.3, −0.8 Time (s) 0.5, 0.3, 0.8 Time (s) −0.5, 0.3, −0.8 Time (s) −0.5, 0.3, 0 Time (s) 0.5, 0.3, 0 Time (s) 0, 0.3, 0.8 Time (s) 0, 0.3, −0.8 Time (s)

115.31 0.34 (s) 115.31 0.27 (s) 115.31 0.13 (s) 115.31 0.13 (s) 115.31 0.25 (s) 115.31 0.24 (s) 115.32 0.23 (s) 115.32 0.26 (s) 115.32 0.12 (s) 115.32 0.18 (s) 115.31 0.23 (s) 115.31 0.26 (s)

115.46 (0.04) 59.95 (s) 115.44 (0.04) 59.05 (s) 115.44 (0.04) 60.32 (s) 115.44 (0.04) 60.73 (s) 115.41 (0.04) 60.47 (s) 115.38 (0.04) 59.39 (s) 115.24 (0.04) 59.75 (s) 115.29 (0.04) 60.13 (s) 115.42 (0.04) 85.62 (s) 115.45 (0.04) 72.72 (s) 115.43 (0.04) 60.27 (s) 115.48 (0.04) 60.27 (s)

Rel-Err 1.36e−3 1.18e−3 1.13e−3 1.21e−3 9.02e−4 6.64e−4 6.22e−4 1.87e−4 9.44e−4 1.13e−3 1.10e−3 1.49e−3

= 1.10

Formula

MC (std)

145.93 0.24 (s) 145.93 0.26 (s) 145.93 0.12 (s) 145.93 0.10 (s) 145.93 0.23 (s) 145.93 0.24 (s) 146.02 0.24 (s) 146.01 0.26 (s) 146.01 0.19 (s) 146.01 0.12 (s) 145.93 0.26 (s) 145.93 0.26 (s)

146.27 (0.06) 59.57 (s) 146.11 (0.06) 59.69 (s) 146.32 (0.06) 59.62 (s) 146.17 (0.06) 59.82 (s) 146.28 (0.06) 59.09 (s) 146.21 (0.06) 59.46 (s) 145.97 (0.06) 60.54 (s) 146.03 (0.06) 82.12 (s) 146.25 (0.06) 76.16 (s) 146.20 (0.06) 59.58 (s) 146.30 (0.06) 69.22 (s) 146.27 (0.06) 69.22 (s)

Rel-Err 2.33e−3 1.20e−3 2.64e−3 1.63e−3 2.38e−3 1.95e−3 3.33e−4 1.14e−4 1.64e−3 1.24e−3 2.50e−3 2.33e−3

Formula

MC (std)

176.19 0.26 (s) 176.20 0.27 (s) 176.20 0.11 (s) 176.20 0.13 (s) 176.20 0.25 (s) 176.19 0.24 (s) 176.38 0.31 (s) 176.37 0.27 (s) 176.37 0.12 (s) 176.37 0.11 (s) 176.20 0.26 (s) 176.19 0.26 (s)

176.76 (0.07) 59.56 (s) 176.73 (0.07) 59.23 (s) 176.78 (0.07) 63.40 (s) 176.74 (0.07) 75.54 (s) 176.20 (0.07) 60.55 (s) 176.67 (0.07) 59.59 (s) 176.48 (0.07) 69.41 (s) 176.51 (0.07) 58.97 (s) 176.63 (0.07) 60.65 (s) 176.85 (0.07) 70.37 (s) 176.71 (0.07) 59.54 (s) 176.84 (0.07) 59.54 (s)

Rel-Err 3.23e−3 3.04e−3 3.33e−3 3.08e−3 2.83e−3 2.71e−3 5.82e−4 8.15e−4 1.45e−3 2.72e−3 2.89e−3 3.69e−3

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Correl. coeff.

Model parameters are set as follows:  = 1,  = 0.04,  = 0.1,  = 0.000625,  = 2, ω = 0.0324, S0 = 110, V0 = 0.0625, B = 0.0265, r0 = 0.04, T = 1.

11

12

Correl. coeff.

= 1.01

= 1.06

1 , 2 , 3

Formula

MC (std)

0.5, −0.3, −0.8 Time (s) −0.5, −0.3, 0.8 Time (s) −0.5, −0.3, 0 Time (s) 0.5, −0.3, 0 Time (s) 0.5, 0.3, 0.8 Time (s) −0.5, 0.3, −0.8 Time (s) −0.5, 0.3, 0 Time (s) 0.5, 0.3, 0 Time (s)

104.22 0.07 (s) 104.13 0.03 (s) 104.13 0.03 (s) 104.22 0.03 (s) 104.25 0.07 (s) 104.16 0.03 (s) 104.16 0.03 (s) 104.25 0.03 (s)

104.79 (0.04) 29.59 (s) 104.72 (0.04) 29.62 (s) 104.76 (0.04) 29.67 (s) 104.75 (0.04) 29.70 (s) 104.65 (0.04) 29.57 (s) 104.73 (0.04) 29.70 (s) 104.83 (0.04) 29.67 (s) 104.78 (0.04) 29.58 (s)

Rel-Err 5.38e−3 5.62e−3 6.03e−3 5.08e−3 3.84e−3 5.54e−3 6.53e−3 5.13e−3

= 1.10

Formula

MC (std)

132.16 0.03 (s) 132.04 0.03 (s) 132.04 0.03 (s) 132.16 0.03 (s) 132.32 0.03 (s) 132.21 0.03 (s) 132.21 0.03 (s) 132.32 0.03 (s)

132.11 (0.05) 29.53 (s) 132.11 (0.05) 29.62 (s) 132.05 (0.05) 29.62 (s) 132.11 (0.05) 29.65 (s) 131.94 (0.05) 29.70 (s) 132.08 (0.05) 29.54 (s) 132.06 (0.05) 29.64 (s) 132.09 (0.05) 29.42 (s)

Model parameters are set as follows:  = 1,  = 0.04,  = 0.1,  = 0.3,  = 2, S0 = 100, V0 = 0.0625, B = 0.0265, r0 = 0.04, T = 1.

Rel-Err 4.08e−4 5.31e−4 5.07e−5 4.21e−4 2.88e−3 9.50e−4 1.11e−3 1.79e−3

Formula

MC (std)

159.81 0.03 (s) 159.67 0.03 (s) 159.67 0.03 (s) 159.81 0.03 (s) 160.14 0.03 (s) 160.00 0.03 (s) 160.00 0.03 (s) 160.14 0.03 (s)

158.90 (0.07) 29.66 (s) 158.97 (0.07) 29.66 (s) 158.86 (0.07) 29.78 (s) 159.01 (0.07) 29.65 (s) 158.82 (0.06) 29.62 (s) 158.95 (0.06) 29.58 (s) 158.89 (0.07) 29.61 (s) 158.93 (0.06) 29.43 (s)

Rel-Err 5.68e−3 4.37e−3 5.05e−3 5.02e−3 8.24e−3 6.58e−3 6.91e−3 7.56e−3

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Table 4 Prices of timer option under Hull–White model.

Correl. coeff.

= 1.01

1 , 2 , 3

Formula

MC (std)

0.5, −0.3, −0.8 Time (s) −0.5, −0.3, 0.8 Time (s) −0.5, −0.3, 0 Time (s) 0.5, −0.3, 0 Time (s) 0.5, 0.3, 0.8 Time (s) −0.5, 0.3, −0.8 Time (s) −0.5, 0.3, 0 Time (s) 0.5, 0.3, 0 Time (s)

20.51 0.04 (s) 20.49 0.06 (s) 20.49 0.03 (s) 20.51 0.06 (s) 20.52 0.03 (s) 20.50 0.04 (s) 20.50 0.03 (s) 20.52 0.03 (s)

20.63 (0.01) 28.53 (s) 20.62 (0.01) 28.27 (s) 20.61 (0.01) 28.70 (s) 20.63 (0.01) 30.91 (s) 20.59 (0.01) 35.18 (s) 20.60 (0.01) 37.03 (s) 20.62 (0.01) 34.60 (s) 20.62 (0.01) 29.36 (s)

= 1.06

= 1.10

Rel-Err

Formula

MC (std)

5.69e−3

24.00 0.04 (s) 23.98 0.04 (s) 23.98 0.03 (s) 24.00 0.03 (s) 24.03 0.07 (s) 24.01 0.03 (s) 24.01 0.03 (s) 24.03 0.03 (s)

24.00 (0.01) 37.27 (s) 23.97 (0.01) 32.59 (s) 23.99 (0.01) 28.97 (s) 23.97 (0.01) 28.39 (s) 23.97 (0.01) 28.41 (s) 23.97 (0.01) 28.50 (s) 24.00 (0.01) 28.38 (s) 23.97 (0.01) 28.54 (s)

5.98e−3 5.70e−3 5.57e−3 3.81e−3 5.00e−3 5.72e−3 5.17e−3

Model parameters are set as follows:  = 1,  = 0.04,  = 0.1,  = 0.3,  = 2, S0 = 20, V0 = 0.0625, B = 0.0265, r0 = 0.04, T = 1.

Rel-Err

Formula

MC (std)

1.28e−4

27.21 0.03 (s) 27.19 0.06 (s) 27.19 0.03 (s) 27.21 0.03 (s) 27.27 0.03 (s) 27.24 0.03 (s) 27.24 0.03 (s) 27.27 0.03 (s)

27.06 (0.01) 28.44 (s) 27.05 (0.01) 28.57 (s) 27.06 (0.01) 28.43 (s) 27.07 (0.01) 29.40 (s) 27.02 (0.01) 29.08 (s) 27.05 (0.01) 33.10 (s) 27.07 (0.01) 37.31 (s) 27.08 (0.01) 36.15 (s)

1.59e−4 3.39e−4 1.04e−3 2.43e−4 1.46e−3 4.35e−4 2.25e−3

Rel-Err 5.52e−3 4.90e−3 4.59e−3 5.12e−3 8.97e−3 6.95e−3 6.31e−3 6.83e−3

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Table 5 Prices of timer option under Hull–White model.

13

14

Correl. coeff.

= 1.01

= 1.06

1 , 2 , 3

Formula

MC (std)

0.5, −0.3, −0.8 Time (s) −0.5, −0.3, 0.8 Time (s) −0.5, −0.3, 0 Time (s) 0.5, −0.3, 0 Time (s) 0.5, 0.3, 0.8 Time (s) −0.5, 0.3, −0.8 Time (s) −0.5, 0.3, 0 Time (s) 0.5, 0.3, 0 Time (s)

114.76 0.10 (s) 114.66 0.03 (s) 114.66 0.03 (s) 114.76 0.03 (s) 114.78 0.03 (s) 114.68 0.03 (s) 114.68 0.03 (s) 114.78 0.03 (s)

115.41 (0.04) 29.79 (s) 115.28 (0.04) 29.67 (s) 115.30 (0.04) 29.69 (s) 115.34 (0.04) 29.54 (s) 115.19 (0.04) 29.72 (s) 115.31 (0.04) 29.67 (s) 115.39 (0.04) 29.50 (s) 115.33 (0.04) 29.74(s)

Rel-Err 5.66e−3 5.45e−3 5.65e−3 5.10e−3 3.60e−3 5.45e−3 6.19e−3 4.79e−3

= 1.10

Formula

MC (std)

146.21 0.03 (s) 146.08 0.03 (s) 146.08 0.03 (s) 146.21 0.03 (s) 146.39 0.03 (s) 146.26 0.03 (s) 146.26 0.03 (s) 146.39 0.03 (s)

146.16 (0.06) 29.54 (s) 146.07 (0.06) 29.66 (s) 146.08 (0.06) 29.62 (s) 146.21 (0.06) 29.70 (s) 145.85 (0.06) 29.67 (s) 146.03 (0.06) 29.94 (s) 146.14 (0.06) 29.58 (s) 146.11 (0.06) 29.54 (s)

Model parameters are set as follows:  = 1,  = 0.04,  = 0.1,  = 0.3,  = 2, S0 = 110, V0 = 0.0625, B = 0.0265, r0 = 0.04, T = 1.

Rel-Err 3.81e−4 6.11e−5 2.24e−5 1.22e−5 2.99e−4 1.59e−4 8.54e−4 1.94e−3

Formula

MC (std)

177.48 0.03 (s) 177.32 0.03 (s) 177.31 0.03 (s) 177.48 0.03 (s) 177.84 0.03 (s) 177.68 0.03 (s) 177.69 0.03 (s) 177.84 0.03 (s)

176.72 (0.07) 29.60 (s) 176.46 (0.07) 29.76 (s) 176.57 (0.07) 29.70 (s) 176.55 (0.07) 29.67 (s) 176.25 (0.07) 29.76 (s) 176.43 (0.07) 29.72 (s) 176.46 (0.07) 29.83 (s) 176.56 (0.07) 30.86 (s)

Rel-Err 4.28e−3 4.85e−3 4.19e−3 5.21e−3 8.95e−3 7.07e−3 6.91e−3 7.23e−3

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Table 6 Prices of timer option under Hull–White model.

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

15

Then ti is compared with the threshold B. The first time moment tk such that tk ≥ B is taken as  defined by (6). For convenience, the SDEs (1)–(3) are written as the following equations dS t = rt St dt +



1

2

3

Vt St (a11 dBt + a12 dBt + a13 dBt ), 2

(39)

3

dV t = ˛(Vt )dt + ˇ(Vt )(a22 dBt + a23 dBt ),

(40)

3

dr t = ( − rt )dt + dBt , where a11 =



1 − 12 − 22 − 32 + 21 2 3 1 − 32

(41)

,

a12 =

1 − 2 3



1 − 32

 ,

a13 = 2 ,

a22 =

1 − 32 ,

a23 = 3 , and Bt1 , Bt2 , Bt3 are independent 1-dim Brownian motions. In this paper, the use of MC method is mainly for the purpose of comparison of the timer option prices obtained by the approximate analytical method. Therefore, we use the simple Euler method for SDEs (39)–(41). In the approximate analytic formula of the timer options, the value of is computed by solving nonlinear equation (25), that of  P by (5), and that of P by (B.10). Numerical results are presented in Tables 1–6, where numbers under the column of ‘Formula’ are option values computed by the approximate analytic formula (30), those under the column of ‘MC’ are option values and standard errors (in parentheses) by Monte-Carlo simulation using 200,000 paths and 300 time steps, those under the column of ‘Rel-Err’ is the relative error between option values by the approximate analytic formula and by MC, i.e.,

   Formula − MC  . Formula

Rel-Err ≡ 

Our numerical results show that the standard errors of the crude MC simulations are at the magnitude of 0.0x, where x is a number in {1, 2, . . ., 9}. All numerical results are computed by Matlab on a PC with Intel(R) CoreTM 2 Duo CPU. From Tables 1–6, we observe that option values by the approximate analytic formulas and by MC simulation are close. So, we believe that our approximate analytic formulas are correct. Moreover, we observe that it takes about less than 1 second to implement the approximate analytic formulas and almost 60 s to implement MC simulation with 200,000 paths and 300 time steps for Heston models. For Hull–White models, it takes about less than 0.1 seconds to compute the approximate analytic formulas and more than 20 s to implement MC simulation with 200,000 paths and 300 time steps. Therefore, the approximate analytic method proposed by this paper is very efficient. 6. Conclusions Pricing formulas or methods for timer options with constant interest rates have been well developed. However it is still very challenging for pricing timer options with stochastic interest rates by approximate analytic methods since the PDE governing the price of a timer option is a four-dimensional PDE. In this paper, we develop an approximate analytic method for pricing power style timer options with Vasicek interest rate model using a PDE-dimension-reduction approach. Explicit approximate analytic formulas for timer options are derived under both the Heston model and the Hull–White model. Numerical examples under these two models are presented. Numerical results show that option values by the approximate analytic formula are close to those by MC simulation while the approximate analytic method is much faster than the MC simulation method. The method proposed in this paper could be extended to other style timer options. This will be one of our future research topics.

16

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

Acknowledgements The work of Jingtang Ma was supported by Program for New Century Excellent Talents in University (No. NCET-12-0922). The work of Yongzeng Lai was supported by an NSERC Discovery Grant (No. RGPIN-2014-03574). The authors are grateful to the anonymous referees for their valuable comments that have led to a greatly improved paper. Appendix A. Proof of Theorem 2.1 Consider a dynamic portfolio on the time interval [0, T] which contains a timer option with value Ct = f(St , Vt , t , rt ), a quantity −1t of the stock St , −2t of another risky asset whose value depends only on the volatility process with maturity T and is denoted by Gt = g(t, Vt , t ), and −3t of zero-coupon bond Pt . Denote the value of the portfolio as t . Then t = Ct − 1t St − 2t Gt − 3t Pt , and dt = dC t − 1t dS t − 2t dGt − 3t dP t = df − 1t dS t − 2t dg − 3t dP t .

(A.1)

For brevity, the subscript t is suppressed in the notations and (A.1) is written as d = df − 1 dS − 2 dg − 3 dP.

(A.2)

Applying Ito’s lemma and considering d = Vdt, we obtain that



df =

2

2

2

√ ∂ f √ ∂ f ∂f ∂f ∂f ∂f ∂ f dS + dV + dr + V + 1 Sˇ(V ) V + 2 S V + 3 ˇ(V ) ∂S ∂V ∂r ∂

∂S ∂V ∂S ∂r ∂V ∂r

 dt +

2

2

2

1 1 1 2∂ f ∂ f ∂ f + 2 ˇ2 (V ) + 2 VS 2 ∂S 2 2 ∂V 2 2 ∂r 2



2

 dt,

∂g ∂g ∂g 1 2 2 ∂ g dt, dV + +V +  ˇ (V ) ∂V ∂t ∂ 2 ∂V 2

dP =

∂P ∂P 1 2 ∂ P dt. dr + +  ∂r ∂t 2 ∂r 2

2

(A.3)



dg =





(A.4)



(A.5)

Then using (A.3), (A.3) and (A.5), we derive that



2

d = V

 +

2

 

2

2

1 1 1 2∂ f ∂ f ∂ f + 2 ˇ2 (V ) + 2 VS 2 ∂S 2 2 ∂V 2 2 ∂r 2

− 2

+

2

2



√ ∂ f √ ∂ f ∂f ∂ f dt + 1 Sˇ(V ) V + 2 S V + 3 ˇ(V ) ∂

∂S ∂V ∂S ∂r ∂V ∂r



 dt 2

 

∂g ∂g ∂g 1 2 2 ∂ g dt + dV + +V +  ˇ (V ) ∂V ∂t ∂ 2 ∂V 2





2

dS +

∂f ∂g − 2 ∂V ∂V

dV



∂f ∂P ∂P 1 2 ∂ P dr − 3 dt. − 3 +  ∂r ∂r ∂t 2 ∂r 2

Then using (A.3)–(A.5), we derive that

∂f − 1 ∂S

(A.6)

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21



2

2

17



2

√ ∂ f √ ∂ f ∂ f ∂f + 3 ˇ(V ) + 2 S V d = V dt + 1 Sˇ(V ) V ∂S ∂r ∂V ∂r ∂S ∂V ∂

 +

 2

−

+

2

2

2

∂ f ∂ f 1 1 1 2∂ f + 2 + 2 ˇ2 (V ) VS 2 ∂V 2 2 ∂r 2 ∂S 2 2



 dt 2

 

∂g ∂g 1 2 2 ∂ g ∂g dt + +  ˇ (V ) dV + +V ∂ 2 ∂V 2 ∂V ∂t

∂g ∂f − 2 ∂V ∂V

 dV +



∂f − 1 ∂S



2

dS



∂f ∂P ∂P 1 2 ∂ P dr − 3 dt. +  − 3 ∂r ∂t 2 ∂r 2 ∂r

(A.7)

Since the bond price P = P(r, t) satisfies the following PDE (see e.g., Kwok, 1998) 2

∂P  2 ∂ P ∂P + ( − r) − rP = 0, + 2 ∂r 2 ∂t ∂r

0 ≤ t < T,

r > 0,

(A.8)

with P(t, T) = 1, we have 2

∂P 1 2 ∂ P ∂P = rP − ( − r) +  . ∂t 2 ∂r 2 ∂r

(A.9)

To make the portfolio instantaneously risk-free, i.e.,





d = rdt = r f − 1 S − 2 g − 3 P dt,

(A.10)

we must choose 1 =

∂f , ∂S

2 =

∂f ∂g / , ∂V ∂V

3 =

∂f . ∂P

(A.11)

Combing (A.6) and (A.9)–(A.11) together, we obtain



1 ∂f ∂V

V

2

2

2

√ ∂ f √ ∂ f ∂f ∂f ∂f ∂ f + rS + ( − r) + 1 Sˇ(V ) V + 2 S V + 3 ˇ(V ) ∂

∂S ∂r ∂S ∂V ∂S ∂r ∂V ∂r 2

2

2

1 1 ∂ f ∂ f ∂ f 1 + 2 ˇ2 (V ) + 2 − rf + VS 2 2 ∂S 2 2 ∂V 2 2 ∂r 2

 =

1 ∂g ∂V



2



∂g ∂g 1 2 2 ∂ g (1 − r) . +V +  ˇ ∂t ∂ 2 ∂V 2 (A.12)

According to the discussion in Gatheral (2002) and Bernard and Cui (2011), both sides of (A.12) are equal to −˛(V). Therefore, we obtain PDE (8) for the price of the timer option and thus complete the proof of Theorem 2.1. Appendix B. Proof of Theorem 3.1 Transform (10) gives that f (S, V, , r) = P fˆ (y, V, ), and fˆ (y, V, B) = P −1 y . Using transform (10) and the identity 1 dt = , V d

(B.1)

18

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

we calculate the following derivatives

∂f ∂fˆ , = ∂S ∂y

∂f ∂fˆ =P , ∂V ∂V ∂f =P ∂



(B.2)

∂fˆ ∂fˆ ∂y ∂P dt + ∂ ∂y ∂P ∂t d



1 ∂P ∂fˆ ∂fˆ 1 ∂P ∂P dt ˆ + fˆ , f =P −y V ∂t ∂y V ∂t ∂

∂t d

+

(B.3)

∂f ∂P ∂fˆ ∂y ∂P ˆ ∂P ∂fˆ ∂P = fˆ +P =f −y , ∂r ∂r ∂y ∂P ∂r ∂r ∂y ∂r 2

2

2

∂ f ∂ fˆ = , ∂S ∂V ∂ y∂ V 2

∂ f ∂ = ∂V ∂r ∂r 2



P

2

2

∂fˆ ∂V

2

2

2

=

=

∂fˆ ∂y ∂y ∂P

2

=

∂P ∂fˆ ∂P fˆ −y ∂r ∂y ∂r



∂P ∂r

∂fˆ fˆ − y ∂y



2

2

2

∂ f ∂ = ∂V 2 ∂V

2

P

∂fˆ ∂V

2



∂ fˆ , ∂V 2

2

∂P ∂r

2



(B.6)

2

=P

∂fˆ ∂P ˆ ∂ P ∂fˆ ∂ P ∂P ∂ −y − +f 2 ∂r ∂r ∂r ∂y ∂r 2 ∂r ∂r

∂ P ∂fˆ ∂ P ∂P −y − 2 ∂r ∂y ∂r 2 ∂r

∂ P 1 2 + y ∂r 2 P



2

=

2

+ fˆ

(B.5)

∂P ∂fˆ ∂ fˆ ∂y ∂P ∂P ∂fˆ ∂ fˆ ∂P +P = −y , ∂r ∂V ∂V ∂y ∂P ∂r ∂r ∂V ∂V ∂y ∂r

2



2

∂ f ∂ fˆ ∂ fˆ ∂y ∂P 1 ∂P ∂ fˆ , = = =− y 2 P ∂ r ∂ y2 ∂S ∂r ∂ y∂ r ∂y ∂P ∂r

∂ f ∂ fˆ ∂ fˆ ∂y 1 ∂ fˆ = , = = 2 ∂S ∂ y∂ S ∂ y2 ∂ S P ∂ y2 ∂ f ∂ = ∂r 2 ∂r

(B.4)

(B.7)

y

∂fˆ ∂y

2 ∂y ∂P ∂fˆ ∂ fˆ ∂y ∂P +y ∂P ∂r ∂y ∂ y2 ∂ P ∂ r



2

∂ fˆ . ∂ y2

(B.8)

Putting (B.1) and (B.2)–(B.8) into the original four dimensional PDE (8) and dividing by P give that 2

V

2

√ ∂ fˆ 1 ∂P ∂fˆ 1 ∂P ∂ fˆ ∂fˆ ∂fˆ + 1 yˇ(V ) V + 3 ˇ(V ) − 3 ˇ(V )y + ˛(V ) P ∂r ∂V P ∂r ∂V ∂y ∂V ∂ y∂ V ∂

2



√ ∂P 1 1 ∂ fˆ 1 + 2 ˇ2 (V ) + −2  V + V + 2 2 2 2 ∂V ∂r 2



2

1 ∂P ∂P 1 2 ∂ P + y rP − − ( − r) −  P ∂t ∂r 2 ∂r 2





1 ∂P P ∂r

∂fˆ 1 + ∂y P



2 

2

y2

∂ fˆ ∂ y2 2



∂P ∂P 1 2 ∂ P − rP fˆ = 0. + ( − r) +  ∂t ∂r 2 ∂r 2 (B.9)

It can be verified that PDE (A.8) with P(t, T) = 1 has the following solution P(r, t) = A(t)e−rB(t) , where



A(t) = exp [B(t) − (T − t)]

B(t) =

1 − e−(T −t) . 

(B.10)



2 − 22



2 2 B (t) , − 4

(B.11)

(B.12)

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

19

Therefore, by (5), we have 1 ∂P 1 = −B(t) = P . P ∂r 

(B.13)

It follows from (A.8) that the last two terms in (B.9) are zeros. Furthermore, using (B.13), we obtain the three-dimensional PDE (11). Thus the proof of Theorem 3.1 is complete. Appendix C. Proof of Theorem 3.2 Let fˆ (y, V, ) = P −1 y Veg(V, ) .

(C.1)

Then g(V, B) = − ln V and we calculate

∂g ∂fˆ , = P −1 y Veg(V, ) ∂

∂

2

∂ fˆ = P −1 y −1 ∂ y∂ V

∂fˆ = P −1 y ∂V

1+V



2

∂g ∂V 2

1+V

∂g ∂V

eg(V, ) ,

(C.2)

2

∂ fˆ = ( − 1)P −1 y −2 Veg(V, ) , ∂ y2

eg(V, ) ,

∂ fˆ ∂g ∂ g = P −1 y 2 +V +V ∂V 2 ∂V ∂V 2



∂g ∂V

(C.3)

2  eg(V, ) .

(C.4)

Inserting (C.2)–(C.4) into (11) gives (14). Hence Theorem 3.2 is proved. Appendix D. Proof of Theorem 3.3 Rewrite Eq. (17) into the form √



∂

g ˛ + 2 ˇ2 ∂

g ˛ ( − 1) P2 − 22 P P V + V + = 0. + + 2 V V2 ∂V V 3 ∂

(D.1)

Define the characteristics ( , V( )) of PDE (D.1) by

dV ˛ + 2 ˇ2 , = d

V2

(D.2)

with V(0) = V0 . Let G( ) =

g (V ( ), ). Then dG ∂

g ∂

g dV ∂

g ∂

g

˛ + 2 ˇ2 = . = + + d

V2 ∂ ∂V d

∂ ∂V

(D.3)

Using PDE (17), we have √

˛ ( − 1) P2 − 22 P P V + V dG =− 3 − , 2 V d

V

(D.4)

with condition G(B) =

g (V (B), B) = − ln V (B) = − ln , where denotes V(B). Integrating both sides of (D.4) leads to





G( )

dG = − G(B)

B



˛(V ())

( − 1) d − 2 V 3 ()

 B



P2 − 22 P P



V () + V ()

V ()

d,

20

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

which gives that





G( ) = − ln −

˛(V ())

( − 1) d − 2 V 3 ()

B





P2 − 22 P P



V () + V ()

V ()

B

d.

(D.5)

Now using (D.2) we calculate that



V ( )

G( ) = − ln −

˛(V ()) 1



( − 1) dV () − 2 V 3 () V  ()



1 dV () = − ln − V  () ( − 1) − 2





V ( )

V ( )

˛(V ()) V 3 ()

P2 − 22 P P



V ( )

= − ln −

V ( )

P2 − 22 P P



˛(V ()) + 2 ˇ2 (V ())

V () + V ()

V 2 ()

( − 1)  du −  2 2 u

˛(u) + 2 (ˇ(u))

V () + V ()

dV ()

˛(V ()) + 2 ˇ2 (V ())  V ( )  2

˛(u)



V ()

V 2 ()

V ()





dV ()

√ P − 22 P P u + u u

˛(u) + 2 ˇ2 (u)



du

≡ ϕ(V, ).

(D.6)

Consequently we obtain that

g (V, ) = G( ) = ϕ(V, ). which is formula (22). To complete the proof of (22), we discuss how to calculate now. Let 

(u) =

u2 2

˛(u) + 2 (ˇ(u))

.

Then integrating both sides of (D.2) gives that



V ( )

V0



u2

˛(u) + 2 (ˇ(u))

du = 2



d = , 0

which is (V ( )) −

(V0 ) = .

Letting = B in the above equation and noticing that ≡ V(B), we obtain that = Hence formula (22) is fully proved. Now we prove formula (23) in the theorem. Rewrite (19) into the form ˛ + 2 ˇ2 ∂J ∂J

1 ˇ2 + + 2 V ∂V 2 V ∂



2

∂

g + ∂V 2



∂

g ∂V

−1 (

(V0 ) + B).

2 

= 0.

(D.7)

Write J in the characteristics form as

J( ) = J(V ( ), ), with the characteristics defined by (D.2). Then equation (D.7) is written into d

1 ˇ2 J( ) =− 2 V d



2

g ∂

+ ∂V 2



g ∂

∂V

2 

1 ˇ2 =− 2 V



2

ϕ ∂

(V, ) + ∂V 2



2 

ϕ ∂

(V, ) ∂V

.

(D.8)

J. Ma et al. / North American Journal of Economics and Finance 34 (2015) 1–21

21

Then integrating both sides of (D.8) with

J(0) = 0 gives that



J( ) = − 1 2 1 =− 2 1 =− 2



B





B



(ˇ(V ())) V ()

(ˇ(V ())) V ()

V ( )

2

(ˇ(u)) u

2

2







2

ϕ ∂

(V (), ) + ∂V 2

2

∂

ϕ (V (), ) + ∂V 2 2

∂

ϕ (u, ) + ∂u2







2 

∂

ϕ (V (), ) ∂V

d

2 

∂

ϕ (V (), ) ∂V

2 

∂

ϕ (u, ) ∂u

1 dV () V  ()

u2 2

˛(u) + 2 (ˇ(u))

du ≡ (V, ).

(D.9)

Therefore, formula (23) in the theorem is obtained. Finally, using (18) and the transforms (10) and (13), we obtain the approximate analytic formula (24) for the value of the timer option. Thus the proof of Theorem 3.3 is complete. References Bernard, C., & Cui, Z. (2011). Pricing timer options. J. Comput. Finance, 15, 69–104. Bick, A. (1995). Quadratic-variation-based dynamic strategies. Manag. Sci., 41, 722–732. Carr, P., & Lee, R. (2010). Hedging variance option on continuous semimartingales. Finance Stoch., 14, 179–207. Fabozzi, F. I. (2002). Interest Rate, Term Structure, and Valuation. Hoboken, NJ: John Wiley and Sons. Gatheral, J. (2002). The Volatility Surface: a Practitioner’s Guide. Hoboken, NJ: John Wiley and Sons. Geman, H., & Yor, M. (1993). Bessel processes, Asian options and perpetuities. Math. Finance, 3, 349–375. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 6, 327–343. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatility. J. Finance, 42, 281–300. Kwok, Y. K. (1998). Mathematical Models of Financial Derivatives. Singapore: Springer-Verlag. Liang, L. Z. J., Lemmens, D., & Tempere, J. (2011). Path integral approach to the pricing of timer options with the Duru–Kleinert time transformation. Phys. Rev. E, 83, 056112. Li, C. X. (2010). Doctoral dissertation. In Managing Volatility Risk. Columbia University. Li, C. X. (2013). Bessel processes, stochastic volatility and timer options. Math. Finance (online). Li, M. Q., & Mercurio, F. (2013). Closed-Form Approximation of Timer Option Prices under General Stochastic Volatility Models. Available from: http://ssrn.com/abstract=2243629 Neuberger, A. (1990). Volatility Trading, working paper. London Business School. Polyanin, A. D., Zaitsev, V. F., & Moussiaux, A. (2002). Handbook of First Order Partial Differential Equations. London: Taylor & Francis. Saunders, D. (2011). Pricing Timer Options under Fast Mean-Reverting Stochastic Volatility, working paper. Sawyer, N. (2007). SG CIB launches timer options. Risk, 20. Swishchuk, A. (2013). Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities. World Scientific.