Path Integration for Real Options

Path Integration for Real Options I Sebastian Grillo2 , Gerardo Blanco, Christian E. Schaerer1 Department of Computer Science, Polytechnic School, National University of Asuncion, P.O.Box: 2111 SL, San Lorenzo, Central, Paraguay.

Abstract Real Options were firstly formulated by using traditional financial option models; however, an investor can confront in practice with exotic dynamics. Nowadays, approaches based on simulations have been gaining relevance for solving complex options. This paper proposes the application of the Path Integral approach (PI) to multivariate Real Option problems. We discuss the viability of the proposal by a mathematical analysis of the problem and an application to a case study of control chart decision (CCD). The proposal is compared with the traditional approaches for solving Real Option problems. The results present the proposal as a competitive alternative for the simulation in low dimensional problems. Keywords: Real Option. Path Integration. Markov Process. Continuous state. Control Chart. European Option. American Option.

1. Introduction Real Options (RO) can be traced to [1], who first defined investments in real assets as mere options. Hence, the RO approach emerges from the idea of applying financial option appraisal theory to capital investment projects. However, financial options are mainly based on contracts, and conversely, RO are intrinsic features of strategic investments, which must be clearly identified and specified [2]. Several methods were developed to value financial options but their direct applications into the RO setting are conditioned to the particular characteristics of each particular problem. In this sense, the best-known approaches in option valuation, with several possible variants, are: partial differential equations [3], lattice methods [4] and Monte Carlo simulation [5]. The Partial Differential Equation (PDE) approach could be non trivial for RO problems due to its discontinuous formulation. In practice, most RO could have characteristics of American options coupled with several flexibilities and restrictions. The lattice approach is much more simpler than the PDE approach [4]. Its efficiency makes it a proficient appraisal for many RO problems. However, it still has some limitations, such as the difficulty of direct application to I This

document dated April 20, 2015 Email addresses: [email protected] ( Sebastian Grillo), [email protected] (Gerardo Blanco), [email protected] (Christian E. Schaerer) 1 CES acknowledges the support PRONII-CONACyT- Paraguay. 2 SG acknowledges the support given by CONACYT under Program 1698OC/PR, Paraguay. Preprint submitted to Applied Mathematics and Computation April 23, 2015

any stochastic dynamic. Finally, Monte Carlo-based methods, that could be considered the most versatile approach, always show a stochastic sampling error and a frequent lack of computational efficiency. Besides the above mentioned approaches, a less traditional one is the Path Integral (PI); which have been introduced in [6]. This approach was born in the context of the quantum mechanics and it has been applied recently to option pricing [7, 8, 9]. However, those applications of the PI are restricted to specific dynamics and usually pursue ad hoc formulas. Within this paper, a numerical procedure for RO valuation based on the PI approach is analyzed. A general formulation is applied with the aim to include a wide range of dynamics and possible RO. An additional goal in the application of this approach consists in proposing an approach in order to seize an intermediate trade-off between the computational efficiency of the lattice methods and the versatility of the Monte Carlo methods. In this context, this alternative is similar to lattice methods but with a significant difference: the discretization of the stochastic variables is avoided by using an interpolating or regressing procedure in order to ensure continuous stochastic variables. Thus, time is discretized by using numerical schemes applied to the corresponding equations, which defines the stochastic dynamics of the problem. Therefore, a typical dynamic programming is applied but considering infinite points within the stochastic variable interval rather than a finite number of states. The feasibility of this approach is not limited by the number of variables or a specific stochastic model, but by the requirement to ensure the existence of a system of stochastic differential equations with a unique solution that characterizes the RO problem and a numerical discretization which transforms the system in a stochastic recursive equation such that each path has a continuous density function almost everywhere. This paper formulates an application of the PI approach and its implementation in a problem which considers stochastic variables for both European and American RO. The results and efficiency of the approach are presented by a case study about a quality control chart problem [10]. We compared the numerical results of the proposal to the Black-Scholes formula, binomial method, and Monte Carlo simulation results for one variable of European options. In addition, the results of the binomial and Least Square Monte Carlo (LSM) [11] methods are applied for one variable of American options. Finally, the pentanomial method [12] and standard Monte Carlo simulation results are applied for two variables of European options. Similarly, pentanomial and Least Square Monte Carlo (LSM) methods for two variable American options are applied. Within the proposed approach, the Bellman equation formulation is applied under a riskneutral measure [13]. Thus, the PI can be employed for the numerical calculation of the conditional expectations that arise in the Bellman equations [14]. These calculations present the form of integrals involving density functions. In [8], the PI is used to the Geometric-Brownian dynamic [15] in the following way for the conditional expectations: h   i Z   k+1 k k E v s(t ), t /s(t ) = h s(tk ) v ezi , tk dzi (1)   where z(tk ) = ln s(tk ) and h s(tk ) denote the conditional density function of s(tk+1 ) given s(tk ). In this paper, we approximate the original stochastic process by a numerical scheme estimating the conditional expectations. This procedure is analyzed in detail in Section 3. This paper is organized as follows: in Section 2, we introduce the path integration approach algorithm for RO. In Section 3, theorems that generalize the estimation of the conditional expectation and analyze a partial error are introduced. Section 4 discusses the model chosen as a case study. Section 5 shows two implementations. Finally, Section 6 discusses and compares the performance of the proposal with traditional algorithms. The comparisons show that the PI 2

approach is a competitive alternative method and it shows good performance when compared to other methods. 2. The path integration approach Let us consider a system of stochastic differential equations with a unique strong solution as follows [22]: ds(t) = f (s (t) , t) dt + D (s (t) , t) dw (t) . (2) where s, ds, f ∈ Rn , dw ∈ Rm corresponds to the differential of a Wiener process w ∈ Rm and D ∈ Rn×m . At a specific time ti , the solution of (2) is denoted by s(ti ). For solving equation (2) numerically, we consider a uniform grid time discretization on [t0 , t f ] with kˆ time intervals such that τ = (t f − t0 )/kˆ and tk = kτ. Hence, employing the Euler-Maruyana discretization scheme, it is obtained [19]:     b sk , k 4wk , sk+1 = sk + b f sk , k τ + D (3) where k is the number of steps, sk ∈ Rn the discretizated vector of stochastic processes at step k which approximates s(tk ), ∆wk the differentiation of the Wiener process w at step k defined as ∆wk := wk+1 − wk with wk := w(kτ), b f ∈ Rn is a column vector of n functions fˆi : Rn × N → R. b D is a matrix n × m, where each entry is a function Dˆ i j : Rn × N → R. Considering an option value v depending on the underlying asset s(tk ) and the vector y containing particular constant parameters from the problem (the free interest rate or the strike), the corresponding Bellman equation takes the following form:     h   i  v s(tk ), tk = maxi∈I ( s(tk ),tk ) Hi s(tk ), y, E v s(tk+1 ), tk+1 /s(tk ) , (4)    where Hi : Rn+ˆy+1 → R, yˆ = dim(y), I s tk represents the set of all possible decisions on the   h   i state s tk and E v s(tk+1 ), tk+1 /s(tk ) is the conditional expectation of v(s(tk+1 )) under s(tk ). For a fixed k, the conditional expectation is approximated by: h      i h   i E v s tk+1 , tk+1 /s tk ≈ E v sk+1 , tk+1 /sk Z   = h sk (ζ) v ζ, tk+1 dζ, n ZR   ≈ h sk (ζ) v ζ, tk+1 dζ, (5) U sk

where ζ ∈ Rn , h sk : Rn → R is the joint density function of sk+1 given sk , and U sk ⊂ Rn . If it is the case the density function does not exist, we use a more general expression for the conditional expectation which is given by Theorem 1. Similarly to other algorithms for option pricing, this approach applies dynamic programming by backwardly taking  datafrom a specific  step to approximate an immediately previous step. In general terms, if e v ψ, tk+1 ≈ v ψ, tk+1 , then the expression (4) using (5) takes the following form:    Z           k  k k k+1 e v s , t = maxi∈I ( sk ,tk )  H  s , y, h sk (ψ)e v ψ, t dψ . (6)    i  U sk 3

By using expressions (5) and (6) v(sk−1 , tk−1 ). Finally, a n recursively,   it is possibleo to estimate e regression of the set of points ski ,e v ski , tk : 1 ≤ i ≤ m(k) (with m(k) ∈ N) is made based on e k ) instead of a predetermined family of functions basis. To simplify the notation, we use E(s j h   i E e v skj , tk /sk−1 . j ——————————————————– Algorithm 1: Path-Integration PI(T, τ) ——————————————————– t −t kˆ = f τ o number of iterations. m (k) is the number of points used for the regression for the calculation of e v(sk−1 , tk−1 ). 1. Discretize the SDE (2). 2. Determine the explicit expression for the approximated conditional expectation (5), obtained from expression (3). 3. For k = kˆ to 1, (a) For j = 1 to Rm(k − 1),   e k) = i. E(s h k−1 (ζ)e v ζ, tk dζ j U sk−1 s j n   j o k−1 e k ii. e v sk−1 = maxi∈I ( sk−1 ,tk−1 ) H j sk−1 j ,t j , y, E(s j ) . o n   v sik−1 , tk−1 : 1 ≤ i ≤ m(k − 1) . (b) Interpolate the points sik−1 ,e ———————————————————-

3. Discrete conditional expectations In this section, we discuss the conditional expectation for any given value of sk . To this end, k+1 we consider the system (3), where in order  to  simplify√the notation  we let s = s and fix values k k k k m b b for k and s . The constant µ = s + f s , k τ, B = τD s , k and g ∈ R is a column vector where each entry is a random variable following the standard normal distribution and the same correlation matrix ρ of w. Then the system (3) becomes: s = µ + Bg.

(7)

Let gI be a vector of m independent random variables (each of them with a standard normal distribution) and G a matrix m × m, where g = GgI . Then, the equation (7) can be written as: s = µ + FgI

(8)

where F is a matrix n × m such that F = BG. Next, we present results for obtaining an expression of the expectation of the value v(s). Lemma 1. Let C be the covariance matrix corresponding to the random vector s. If the rows of F are linearly independent then C −1 exists.

4

  D E Proof. Let Fi and Ci be the i-th row of F and C, respectively. Since Ci j = cov si , s j = Fi , F j ,  T then Ci = FFiT . Being the rows of F linearly independent, then n ≤ m and dim (ker (F)) =  T  m − n. Considering that FFiT generates the row space of C and using the rank-nullity theorem: dim (C) = n, finally, C is invertible because it is a n × n matrix. Theorem 1. Let v : Rn → R be a valuation function. A set {ei } ⊂ N with ncardinal number α o  such that Fei is a linearly independent basis for the row space of F, a set: di ⊂ Rα such that   α P Fi = dki Fek and Cˆ a α × α matrix such that Cˆ i j = cov sei , se j . In addition, vˆ is considered as k=1

the restriction of v with respect to the normal variables sei and µˆ i its mean value. Hence: ! Z 1 1 T ˆ −1 (x (x − µ) ˆ − µ) ˆ vˆ (x) dx exp − C E (v (s)) = 1/2 2 ˆ Rα (2π)α/2 C

(9)

  Proof. Since each member of sei follows a normal distribution and Fei is linearly independent, then Cˆ −1 exists (by the Lemma 1). Cˆ −1 is symmetric, semi-definite, and positive [17], hence by  using the Lemma 1, Cˆ −1 is symmetric, definite, and positive. The random variables sei follow a multivariate normal distribution [26] with the probability density function:

f (x) =

1 1/2 (2π)α/2 Cˆ

1 exp − (x − µ) ˆ T Cˆ −1 (x − µ) ˆ 2

! (10)

n o  Considering that Fei is a linearly independent basis for the row space of F, then di ⊂ α P  Rα exists such that Fi = dki Fek , and every s j , where j < {ei }, is in function of sei due to α P

k=1


 sj = µj + sek − µek . Finally, the valuation function v depends only on sei and the density k=1 R function f (x) can be applied into the formula: E (v (X)) = f (x) v (x) dx. dki

Remark 1. Notice that expression (9) becomes the conditional expectation (5), when we express equation (8) as (3). h      i An important error source is the difference between E v s tk+1 , tk+1 /s tk and its approxn o   imation using (5) with v˜ instead of v. Given ω/s tk = sk ⊂ Ω, this error could be estimated if the valuation function is a Lipschitz one and thus, we have the next considerations: a1 The approximation function v˜ is Lipschitz with constant Lv˜ , i.e. |˜v (x) − v˜ (y)| ≤ Lv˜ kx − yk.

 

a2 E

s tk+1 − sk+1

is obtained from the analysis of the strong convergence in the numerical scheme [20]. 5

n o a3 A factor of error α which arises when considering U sk ⊂ Rn in (5). ϕ = ω : sk (ω) ∈ U sk and 1ϕ is its associated indicator function such that 1ϕ (x) = 1 for x ∈ ϕ and 1ϕ (x) = 0 for x < ϕ. a4 The error function is defined as e (x) := v (x) − v˜ (x). The last item is the most difficult to obtain. Nevertheless, it is important to consider this term within an analysis of the error propagation through iterations. The next results show how the integration error affects the propagation of the error e, that is, the result of the other errors accumulated on the previous steps. Theorem 2. Let be two n−dimensional random variables s and s˜ in (Ω, F ), a Lipschitz function v˜ : Rn → R with constant Lv˜ , other functions: v : Rn → R, and a region ϕ ⊂ Ω such that E 1ϕ v˜ ( s˜) := αE (˜v ( s˜)) for α ∈ R, then:   E 1ϕ v˜ ( s˜) − E (v (s)) ≤ |α| Lv˜ E (k s˜ − sk) + |α| E (|e (s)|) + |(α − 1) E (v (s))| (11)

Proof. Initially, we consider |E (˜v ( s˜)) − E (v (s))|

= |E (˜v ( s˜)) − E (˜v (s) + e (s))| ≤ E (|˜v ( s˜) − v˜ (s)|) + E (|e (s)|) ≤ Lv˜ E (k s˜ − sk) + E (|e (s)|)

(12)

In addition,   E 1ϕ v˜ ( s˜) − E (v (s))

=

|αE (˜v ( s˜)) − E (v (s))|

≤

|α| |E (˜v ( s˜)) − E (v (s))| + |(α − 1) E (v (s))|

(13)

Using (12) and (13), we obtain:   E 1ϕ v˜ ( s˜) − E (v (s)) ≤ |α| (Lv˜ E (k s˜ − sk) + E (|e (s)|)) + |(α − 1) E (v (s))|

(14)

    Corollary 1. Let be the random variables sa tk+1 = s tk+1 and sk+1 = sk+1 restricted to a n o   Ω0 = ω/s tk = sk = a ⊂ Ω. Then, for ω ∈ Ω0 :               E 1ϕ v˜ sk+1 /sk − E v s tk+1 /s tk ≤ |α| Lv˜ E

sk+1 − sa tk+1

a          + |α| E e sa tk+1 + (α − 1) E v sa tk+1

(15)

            Proof. By taking s = sa tk+1 , s˜ = sak+1 , and considering that E v sa tk+1 = E v s tk+1 /s tk (ω)          and E 1ϕ v˜ sk+1 = E 1ϕ v˜ sk+1 /sk (ω) for ω such that s tk , ω = sk (ω) = a, then the Corola lary follows using the Theorem 2. 6

     In practice, it is difficult to know E e sa tk+1 but it could be replaced by max(|v − v˜ |). The last corollary is just a partial estimation of the numerical error, because the interpolation error in each iteration is an independent source of error. 4. Real option formulation of Quality Control Charts Control chart is a statistical discipline that aims to improve the quality of a process [10], helping to find cyclical problems that arise under unpredictable circumstances. Given that an over cost of not using control charts exists due to either the scrapping by final inspection, returns of defective parts by the consumer, or the risk of a reputation of bad quality service or product, manufacturers use control charts to account for any low quality production in the system. Obviously, by using control charts has a cost as well that includes running equipments, software, and operators [16]. In the context of this paper, we consider that the RO problem is based on the following hypothesis: there is an option of applying the control chart approach. This profit depends on the market price and number of sales of the product, which are taken into account as uncertain variables. Thus, the proposed model considers that the number of sales and the price of the product are stochastic variables. To attain this goal, re(t) is defined as the total sales revenue of the product (the total amount of money received) per time interval beginning at time t, which depends on the stochastic variable number of product sales during the same time denoted by s1 (t) and the price of the product denoted by s2 (t). Therefore, the total sales revenue re(t) fulfills the following equation: re(t) = s1 (t)s2 (t). (16) Assuming that the number of sales and the number of units produced per time interval are equal, the total profit per time interval that begins at time t can be defined as a subtraction between revenues and expenses: p(t) := re(t) − pc − s1 (t)pu , (17) where pc is the fixed production cost per time interval and pu is the variable production cost per unit of product. The fraction of revenue which is lost when control charts are not applied is considered as a constant and denoted by θ. The cost of implementing control charts per time interval is denoted by K. Hence, the profit without control chart at a given time interval is formulated as: e p(t) = (1 − θ)s1 (t)s2 (t) − pc − s1 (t)pu ,

(18)

and the profit with chart at a given time interval is defined by b p(t) = s1 (t)s2 (t) − pc − s1 (t)pu − K.

(19)

The profit d (t) by the mere application of controls charts at a given time interval is the difference between equations (18) and (19) which reduces to: d(t) = b p(t) − e p(t) = θs1 (t)s2 (t) − K.

(20)

To introduce the proposed RO appraisal, we consider initially a simplified model, where only the number of sales is stochastic. This is done by assuming that the price of the product s2 (t) is constant and denoted by s2 . 7

European Option model. This valuation is based on the underlying hypothesis that a decision made in an individual interval does not affect the subsequent intervals, i.e. all the possible decisions are reversible. Consequently, the manufacturer can make a modification of the decision applying control charts at any interval. For instance, if we assume that the manufacturer will decide to use or not the procedure every month during a year, then there are twelve decision points and twelve exercise costs for each one. Following this hypothesis, the RO problem can be modeled by a European option for each month because the execution of the option of using the control chart for a period only takes place at its corresponding decision point, where each option expires at a different time interval (at the decision points). The final RO value is the total sum of all European option values [10]. American Option model. In this case, we consider that the option will be exercised just once, and thereafter, the decision is going to remain until the investment horizon. In other words, if we choose to apply the control charts at a given time, then the control will be running until the maturity of the investment project. Therefore, along the investment maturity, the option can be executed just once. Thus, we have an American valuation model where a revenue re(t) is collected at each discrete period (months in this case) until the option expires. Under this assumption, it is necessary to define a total and unique exercise cost for the option. Then, we determine Ka as the total value cost K at the intermediate point between the beginning and the expiration of the option [10]: (erτ )n+1 − 1 , (21) Ka = K (erτ − 1) (erτ )n/2 where r is the risk-free interest rate and there are n intervals. 5. One variable case: implementation By using the study case discussed in [10] for one stochastic variable, we implement an American and European option pricing approach regarding the following parameters: the number of sales s1 (0) = $872, 640 per month; the price of the product is s2 = $5.678; the volatility of the number of sales is 0.930354. On the one hand, when control charts are not implemented for controlling the process, the loss revenue factor is equal to: θ = 0.018. On the other hand, the cost of implementing control charts is K = $11, 000 per month. Finally, the cost of using control charts for an American option is KA = $143, 044. The yearly risk-free interest rate is equal to r = 0.08 and the maturity is t f = 1. We define the stochastic revenue variable s := θs1 s2 which follows the same stochastic dynamic than s1 in a risk-neutral basis: ds = (µs) dt + (σs) dw

(22)

where w is a Wiener process, µ = r = 0.08 is the expected return in a risk-neutral basis, and σ = 0.930354 is the volatility. Using the Euler-Maruyama method with the time discretization τ = 1/24, then equation (8) takes the form: ! ! 0.930354 k I k+1 k k 0.08 s =s +s s g1 , (23) + √ 24 24 which is replaced by: ! ! 0.08 0.930354 k I sk+1 = sk + sk + s g √ 1 . 24 24 8

(24)

This consideration assures the desired property that sk never has negative values, and, as we will see, that it modifies the density function obtained with the original scheme. In fact, this affects the estimation of the conditional expectation only in the points ski near to zero. Let vE , vA , ve and vam be the European, American, and two auxiliary option values respectively, and (ve and vam are used to compute the European and American option values). Let be e vE , e vA , e ve , and e vam the estimations of the last option values. Hence, we define the following Bellman equations : n   h   io ve sk , tk = max 0, e−rτ E ve sk+1 , tk+1 /sk , (25) n   h   io vam sk , tk = max 0, e−rτ E vam sk+1 , tk+1 /sk , (26) (if k is an odd number) n   h   io vam sk , tk = max 0, sk + e−rτ E vam sk+1 , tk+1 /sk , (if k is an even number)

n o     vA sk , tk = max 0, vam sk , tk − KA ,

(27) (28)

k 2

   X  ve s2i , t2i , vE sk , tk =

(29)

i=0

(if k is an even number), where

and

n o   ve s24 , 1 = max 0, s24 − 11, 000 ,

(30)

n o   vam s24 , 1 = max 0, s24 = s24 .

(31)

Remark 2. Assuming that cash flow is monthly, Bellman equation (26) corresponds to an intermediate period within the month. Thus, equation (26) is used as an extra calculation point in order to improve the estimation of conditional expectations in (27) as well as for adapting the option pricing scheme to a discretization grid more dense than τ = 1/12. If τ = 1/12, then equation (27) can be directly applied. It can be seen in the Bellman equations that the American RO of this particular problem is not formulated as the financial American option. For further details see the original article that propose this model [10]. In this context, we must calculate the integral (9) in enough points to approximate the option values at each point ski . Thus, we must choose the points with a sufficient spacing among them with the objective of capturing a range which is wide enough for possible values so that it accurately approximates to the option function with an acceptable error in each iteration. Within this work, it has been selected 54 values for skj such that skj = 1.5 j and sk0 = 0.0001. By Theorem 1, the equation (23) implies that if sk = skj , then sk+1 has a normal distribution q   1 k (0.930354) 24 s j , so the modified discretized with average skj + skj 0.08 24 and standard deviation variable sk+1 , given by the expression (24), has the next density function: h ski (x) =

1(R+ ∩{0}) (x) (a1 (x) + a2 (x))  , pπ (0.930354) ski 12 9

(32)

where:

 √ 1 2 ! 2 k −( x−ski −ski ( 0.08 24 )) / 2 (0.930354)si 24

a1 (x) = e

,

 √ 1 2 ! 2 k −(−x−ski −ski ( 0.08 24 )) / 2 (0.930354)si 24

a2 (x) = e

and

,

where 1D (x) is the indicator function such that 1D (x) = 1 for x ∈ D and 1D (x) = 0 for x < D. Then, for each ski and the corresponding region of integration U ski which satisfies the expression below: U ski = [LI , LS ] with (33) q q         1 1 k k k 0.08 k LI = ski + ski 0.08 24 − 5 0.930354si 24 , LS = si + si 24 + 5 0.930354si 24 . n o Afterwards, we estimate the conditional expectation on the set ski using the following integral for the European and American options respectively: Z   ee (sk+1 ) = E h ski (x)e ve x, tk+1 dx, (34) i U sk i

eam (sk+1 ) = E i

Z U sk

  h ski (x)e vam x, tk+1 dx.

(35)

i

Finally, the integrals (34) and (35) on their respective Bellman equations are: n o   ee (sk+1 ) , e ve ski , tk = max 0, e−rτ E i

(36)

n o   eam (sk+1 ) , e vam ski , tk = max 0, ski + e−rτ E i

(37)

n o   eam (sk+1 , k) , e vam ski , tk = max 0, e−rτ E i

(38)

(if k is an even number),

(if k is an odd number), k



e vE ski , tk



=

2 X

  e ve ski , t2 j

and

(39)

j=0

n o     e vA ski , tk = max 0, vam ski , tk − KA .

(40)

At the beginning, integrals (34) and (35) are computed using the conditions (30) and (31). All integrals are computed using Simpson’s   rule, and the proposed method runs backwards by using interpolated approximations of ve x, tk and vam x, tk for n < 24. In this sense, let be the family of sets: c j = [skj , skj+1 ) for 0 < j < 53 and c54 = [sk54 , ∞). We make a “local” approximation of the option value function in each c j using the resulting function by interpolating the next points:             { skj−1 ,e v skj−1 , tk , skj ,e v skj , tk , skj+1 ,e v skj+1 , tk , skj+2 ,e v skj+2 , tk }

10

n o , where b j = c j−1 ∪ c j ∪ c j+1 by the set of functions 1b j , x.1b j , x2 .1b j , x3 .1b j . Similarly, for c0 , an approximation of the option value function is obtained interpolating             v sk3 , tk } v sk2 , tk , sk3 ,e v sk1 , tk , sk2 ,e v sk0 , tk , sk1 ,e { sk0 ,e n o by the set of functions 1b1 , x.1b1 , x2 .1b1 , x3 .1b1 . Finally, for c53 and c54 an approximation of the option value function is estimated interpolating             { sk51 ,e v sk51 , tk , sk52 ,e v sk52 , tk , sk53 ,e v sk53 , tk , sk54 ,e v sk54 , tk } n o by the set of functions 1b52 , x.1b52 , x2 .1b52 , x3 .1b52 . 5.1. Two variables case: implementation Similarly to the one variable problem, the same case study reported in [10] is analyzed. For this problem, we impose a volatility of s2 (t), σ2 = 0.059634, and the correlation factor between s1 (t) and s2 (t) is ρ = 0.111344. Let w1 and w2 be independent Wiener processes. r = 0.08 is the expected return in a riskneutral basis. Then it follows that the dynamics for s1 (t) and s2 (t) are ds1

=

ds2

=

(0.08) s1 dt + (0.930354) s1 dw1

and ! q (0.08) s1 dt + (0.059634) s2 ρdw1 + 1 − ρ2 dw2 ,

respectively. Using the Euler-Maruyama method with τ = 1/24, we obtain ! ! 0.930354 k I 0.08 k k+1 k s + s1 g1 , s1 and = s1 + √ 24 1 24 ! ! ! q 0.059634 k 0.08 k I I k+1 k 2 s2 ρg1 + 1 − ρ g2 s + s2 = s2 + √ 24 2 24 Similarly to equation (24), we replace equations (43) and (44) by ! ! k 0.930354 k I 0.08 k k+1 s1 = s1 + s + s1 g1 , and √ 24 1 24 ! ! ! q k 0.08 k 0.059634 k k+1 I I 2 s2 = s2 + s + s2 ρg1 + 1 − ρ g2 . √ 24 2 24

(41) (42)

(43) (44)

(45) (46)

Since τ = 1/24, then the same Bellman equations and conditions at k = 24 are used in the one variable case and are applied for: sk = (0.018) sk1 sk2 . (47) i−1 Let the next sets {Mi } bensuch for 0 < i < 55, M0 = 0, and {Ni } such that Ni = i o that Mi =n (1.5) o k for i < 19. Considering si j = {Mi } × N j , then for each skij = Mi , N j we have:

h skij (x, y) = + + +

1(R+ ∩{0})×(R+ ∩{0}) (x, y) h (x, y) 1(R+ ∩{0})×(R+ ∩{0}) (x, y) h (−x, y) 1(R+ ∩{0})×(R+ ∩{0}) (x, y) h (x, −y) 1(R+ ∩{0})×(R+ ∩{0}) (x, y) h (−x, −y) , 11

(48)

  i √ Mi , σ0 := (0.930354)M , µ” := N j + 0.08 24 N j + Li j , with 24 √   (0.08) y−Mi − 24 Mi (0.059634) 1−(0.111344)2 N j √ Li j := (0.059634) (0.111344) N j (0.930354)M , σ” := and h (x, y) := i 24  (x−µ0 )2 2   − 0 2 − (y−Li j −µ”)  1 e 2(σ ) e 2(σ”)2 . 0

where µ0 := Mi +



0.08 24



2πσ σ”

Remark 3. Note that h (x, y) is the density function obtained directly from (43) and (44), while h skij is the density function obtained from (45) and (46). The conditional expectations to be used in the Bellman equations are: Z   k+1 e Ee (si j ) = h skij (x, y) ve (0.018)xy, tk+1 dx dy, U sk

ij

eam (sk+1 ) = E ij

Z U sk

  h skij (x, y) vam (0.018)xy, tk+1 dx dy,

(49)

ij

where: ! ! ! !# (5) (0.930354) (5) (0.930354) 0.08 0.08 − + , M 1 + √ √ i 24 24 24 24 ! ! ! !# " (5) (0.059634) (5) (0.059634) 0.08 0.08 − , Nj 1 + + . Nj 1 + √ √ 24 24 24 24

" U sk

ij

= ×

Mi 1 +

We have in each bi j = [Mi , Mi+1 ) × [N j , N j+1 ) using the set of n implemented an interpolation o functions 1bi j , x 1bi j , y 1bi j , xy 1bi j , but other interesting alternative to the interpolation consists n o in using a Delaunay triangulation with skij as vertices and applying a finite element function method (for details see [23]). 6. Numerical Results In this section, we discuss computational aspects of the proposal and its performance for the cases of one and two variables. Assessments are made through comparisons between numerical results obtained by the traditional algorithms as the Lattice Methods and Monte Carlo simulation. The European and American valuation problems are the same cases provided in [10]. Analyzing possible sources of error in Algorithm 1 (Section 2), it is possible to define three primary types of error: • Time discretization, which can be minimized by reducing the value of the time intervals τ. This type of error is in almost every (numerical) algorithm except those that perform analytical calculations like the Black-Scholes formula. • The approximation of the valuation function by doing an interpolation or regression on n o fixed points ski . This approximation error can be reduced by considering larger number of points as a better approximation of the desired curve. This type of error is inherent to the PI approach. 12

• Because the computation of the integral for estimating conditional expectations is numerical there is always an error attached to this approach. In our implementation, the discrete variable follows a normal distribution, then it is reasonable to integrate around the mean of the random variable. Therefore, the distance from the integration domain boundaries to the mean value is a variable which affects the accuracy (in our implementation we take a distance of five standard deviations).     R   Taking E 1ϕ v˜ sk+1 /sk = U k h sk (ζ)e v ζ, tk+1 dζ, the Corollary 1 analyzes the partial error: s

Z   h   k+1  k+1   k i k+1 ,t /s t − h sk (ζ)e v ζ, t dζ e p := E v s t Uk

(50)

s

which affects the total error e = (v − e v) directly with the interpolation error. The discretization and integrations errors affects the total error e only through (50), since e depends on (50). In addition, the interpolation error affects this partial error indirectly by the error e of the previous step. The analysis of the computational complexity of this implementation (Algorithm 1) considers the following input variables m = T/τ (where T is the expiration time of the option), n is the number of points Pi employed for the interpolation or regression, and p is the number of points evaluated for the numerical integration. If the nested loops of the pseudocode are considered (Algorithm 1), then the analysis yields the following order of complexity for computing an option O(m.n.p). 6.1. Comparative numerical results The following tables compare different approaches, for the study case of previous sections and according to ranges of time varying from six months to three years. For the single variable valuation case, it is performed a comparison of the percentage of absolute error for the calculations obtained with the binomial method, Monte Carlo simulation, and the proposal of this work: Path Integration (PI). This type of error is computed using ePr =

100 |A − E| E

(51)

where A is the approximate value given by numerical methods and E is the exact value. One variable European option. Tables 1, 2 and 3 analyze the European valuation problem of a single variable. Table 1 presents exact values obtained using the Black-Scholes formula. Table 2 compares the error given by expression (51) and Table 3 compares the elapsed time of each algorithm (Binomial, Path Integration and Monte Carlo method). The Monte Carlo valuation [24] simulates 60,000 realizations using a discretization τ = 1/1728. The PI and the Binomial method employ the same discretization τ = 1/12. It can be observed that the PI has worse performance than the Binomial method but better than the Monte Carlo Method in terms of the error (51) and elapsed time of the algorithm. One variable American option. Tables 4, 5 and 6 analyze the one variable American problem. Table 4 presents almost exact values obtained by overkilling of the binomial method using τ = 1/100. Table 5 compares the error 51 and Table 6 compares the elapsed time of each algorithm. For the Monte Carlo valuation (LSM) were employed simulations of 100 realizations, that were repeated until obtaining a final result of 1% relative error[21] with a 95% of interval of 13

confidence and a discretization τ = 1/1728. The proposal and the binomial method use the same discretization τ = 1/24. Here similar results than the European one variable case are obtained in terms of the relationship between the performances of elapsed time and the error (51). Two variable European option. In this case, due to the lack of obtaining an analytical solution to compare the error between the solution of the methods, we consider as the exact solution the one obtained by the Pentanomial method. Table 7 compares the error (51) of the PI and MC with respect to the Pentanomial method. On other hand, Table 8 shows the elapsed time of each algorithm. In the Monte Carlo valuation were employed simulations of 100 realizations, repeated until reaching 1% of relative error with a 95% of interval of confidence and a discretization τ = 1/1728. The proposal and the pentanomial method use the same discretization τ = 1/12. From Table 7, it can be seen that the error of PI is larger than MC. The same conclusion can observed at Table 8 in terms of elapsed times. This means, that for this case, the PI is inferior in performance than the MC. Two variable American option. Tables 9 and 10 show the error and elapsed time of the Pentanomial, PI and the Monte Carlo Least Squares (LSM). For the LSM implementation was employed simulations of 50 realizations, repeated until reaching 1% of relative error with a 95% of confidence and a discretization τ = 1/1728. The PI and the pentanomial method use the same discretization τ = 1/12. Observing Table 9 and 10, we notice that the PI has lesser error and elapsed time than the LSM, being the PI superior in terms of performance than the LSM. This result is important since the LSM is today widely used in real complex option problems due to its versality. In the tables above, we can find similarities between the results obtained by the implementation of PI and other implemented algorithms in literature [4, 12, 5, 11] (one and two variables). The effectiveness of the proposal is experimentally observed for RO problems involving one variable. In fact, there is an obvious better performance of the PI when compared with the MC in terms of error and computational effort (elapsed time). This conclusion can be extended to the two variables American valuation where the PI has better performance than the LSM in terms of error and elapsed time. It was observed that the PI was always inferior than lattice methods in terms of error, however, the range of application of lattice methods is more reduced. Therefore, PI serves as a complement because it should be used for problems where the lattice approach is not applicable. 7. Concluding Remarks This paper applied the path integration approach to assessing multivariate RO problems. The results for Geometric Brownian models of one and two variables suggest a good performance for low dimensional dynamics. It have been analytically proved, by mean of two theorems, the features of the proposal regarding issues such as error and convergence. This approach seems promising for application to problems where some traditional algorithms are no applicable or have not an adequate performance due to its low versatility of implementation. In this sense, PI approach can offer an interesting trade-off between the precision of the lattice methods and versatility of Monte Carlo methods. In this context, this approach may be an useful appraisal in some problems where the model versatility is required but the stochastic error is not desired. Future works could extend the proposed approach to more general dynamic involving Levy processes [25]. 14

Acknowledgement The authors acknowledge fruitful discussions with Benjam´ın Bar´an Cegla and Ernesto Mordecki on topics of this work. References [1] Myers, S. C. Determinants of Corporate Borrowing. Journal of Financial Economics. Vol. 5, No. 2. 1977. [2] Dixit, A. K. Pindyck R. S. Investment under uncertainty. Princeton University Press. 1994. [3] Black F. and Scholes M. “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, Vol. 81, No. 3. 1977. [4] Cox J.C. and Ross S.A., and Rubinstein M. ”Option Pricing: A Simplified Approach.” Journal of Financial Economics 7. 1979. [5] Boyle, P.P. Options: A Monte Carlo approach. Journal of Financial Economics 4, no. 3. Journal of Financial Economics: 323-338. 1977. [6] Feynman R.P. and Hibbs A.R. Quantum Mechanics and Path Integrals. New York: McGraw-Hill, 1965. [7] Baaquie B.E. A path integral approach to option pricing with stochastic volatility: some exact results. Arxiv preprint cond-mat/9708178 (1997). [8] Montagna G., Nicrosini O. and Moreni N. A path integral way to option pricing. Physica A: Statistical Mechanics and its Applications 310, n. 3-4 (2002): 450–466. [9] Lemmens D., Wouters M., Tempere J. and Foulon S. Path integral approach to closed-form option pricing formulas with applications to stochastic volatility and interest rate models. Physical Review E 78, n. 1 (2008): 016101. [10] Nembhard H.B., Shi L. and Aktan M. A real options design for quality control charts. Proceedings of the 32nd conference on Winter simulation. 2002. [11] Schwartz E.S. and Longstaff F.A. Valuing american options by simulation: a simple least-squares approach. Review of Financial Studies, 2001. [12] Boyle P.P. A Lattice Framework for Option Pricing with Two State Variables. Journal of Financial and Quantitative Analysis, Vol. 23, No. 1, 1988. [13] Bingham N.H. and Kiesel R. Risk-neutral valuation: pricing and hedging of financial derivatives. Springer, 2004. [14] Bellman R. E. Dynamic Programming. Courier Dover Publications. 2003. [15] Dave H. Numerical Methods for Pricing Exotic Options. Accessed 22/12/12 www3.imperial.ac.uk/pls/portallive/docs/1/55071696.PDF [16] Montgomery D.C. Introduction to Statistical Quality Control, Fourth Edition, John Wiley & Sons, New York, NY, 2000. [17] Horn R.A. and Johnson C.R. Topics in Matrix Analysis. Cambridge University Press, 1994. [18] Kollo T. and Von Rosen D. Advanced Multivariate Statistics With Matrices. Springer, 2005. [19] Higham D. J. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. SIAM Review 43. 2001. 17. [20] Higham D.J., Mao X. and Stuart. A.M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM Journal on Numerical Analysis (2003): 10411063. [21] Fisherman G. Monte Carlo: Concepts, Algorithms, and Applications (Stochastic Modeling). Kluwer Academic Publishers, Springer, 2003. [22] Oksendal B.K. Stochastic differential equations: an introduction with applications. Springer.2003. [23] Pepper D.W. and Heinrich J.C. The finite element method: basic concepts and applications. Taylor & Francis. 1992. [24] Bampou D. Numerical Methods for Pricing Exotic Options. Accessed 22/12/12 www3.imperial.ac.uk/pls/portallive/docs/1/45415707.PDF [25] Huang X. Numerical Valuation of American Options under Exponential Levy Processes. Accessed 22/12/12 ta.twi.tudelft.nl/mf/users/oosterle/oosterlee/huang.pdf [26] Grillo S., Blanco G. and Schaerer C.E. Real options using a continuous-state markov process approximation. In 15th Real Option Conference, 2011.

Appendix: A simple numerical example Here we present a simple numerical example for a financial American call option. The aim of the example is to just illustrate the procedure of the proposed approach. We choose the following 15

parameters: riskless rate r = 0.05, strike price K = 5, the asset initial value s0 = 10, the volatility σ = 0.3, the time t f = 1 and the time discretization τ = 13 , with tk = kτ, kˆ = 3 and k = 0, 1, 2, 3. Assuming that the process followed by the underlying variable S in a risk-neutral basis is: ds = rsdt + σsdw = (0.05) sdt + (0.3) sdw, where w is a Wiener process, if we apply the Euler-Maruyama method yields the following discretization: ! 1 k s + (0.3) sk ∆wk . sk+1 = sk + rsk τ + σsk ∆wk = sk + 60   1 Observe that for a given sk = x, sk+1 has a normal distribution with mean m x = x + 60 x and standard deviation d x =

(0.3)x √ , 3

which implies the following density function: h x (y) =

1 √

!

  2 − (m x −y) 2

2d x . d x 2π   Here we use the condition in time t f : v s3 , 1 = max(s3 − 5, 0) to approximate the option value   function in the previous instant t = 23 and using the Bellman equation v sk , tk τ = max(sk −     k K, e−rτ E v sk+1 , k+1 3 /s ) backwards. Following the idea put forward earlier an estimation of the function using the next integral is performed in a finite number of values of sk given by {0.1, 12, 24, 36, 48}:

e

∞

Z     E v sk+1 , (k + 1) τ /sk (ω) ≈ v (y, (n + 1) τ) h x (y) dy. 0

where ω belongs to a sample space Ω and sk (ω) = x. For ω such that s2 (ω) = 0.1, then m x = 0.1017 and d x = 0.0173: 76

Z     3 2 E v s , 1 /s (ω) ≈ max(y − 5, 0) 0

⇒ v 0.1,

!

1 √

 −

e

(m x −y)2 2d2x

d x 2π



dy ≈ 0

! 2 ≈ max (0.1 − 5, 0) = 0. 3

As the integrals are calculated using some numerical method, it is considered a finite interval of integration [0, 76] accurate enough for our requirements. For ω such that s2 (ω) = 12, then m x = 12.2 and d x = 2.0785: 76

Z     E v s3 , 1 /s2 (ω) ≈ max(y − 5, 0) 0

!

⇒ v 12,

1 √ d x 2π

! e

  2 − (m x −y) 2 2d x

dy ≈ 12.1988

2 ≈ max (12 − 5, 11.9972) = 11.9972. 3 16

For ω such that s2 (ω) = 24, then m x = 24.4 and d x = 4.1569: 76

Z     3 2 E v s , 1 /s (ω) ≈ max(y − 5, 0) 0

!

1 √

e

  2 − (m x −y) 2 2d x

d x 2π

dy ≈ 24.4

! 2 ⇒ v 24, ≈ max (24 − 5, 23.9967) = 23.9967 3 For ω such that s2 (ω) = 36, then m x = 36.6 and d x = 6.2354: 76

Z     3 2 E v s , 1 /s (ω) ≈ max(y − 5, 0) 0

⇒ v 36,

!

1 √

e

  2 − (m x −y) 2 2d x

d x 2π

dy ≈ 36.6

! 2 ≈ max (36 − 5, 35.9951) = 35.9951. 3

For ω such that s2 (ω) = 48, then m x = 48.8 and d x = 8.3138: 76

Z     3 2 E v s , 1 /s (ω) ≈ max(y − 5, 0) 0

1 √

! e

  2 − (m x −y) 2 2d x

d x 2π

dy ≈ 48.7582

! 2 ⇒ v 48, ≈ max (48 − 5, 47.9523) = 47.9523. 3 n o   Then we approximate the whole function v s2 , 23 in R by a regression in 1, x, x2 , obtaining v(x, 32 ) ≈ −0.093 + 1.0076x − 0.0001x2 and repeat the above procedure with the instant t = 31 : For ω such that s1 (ω) = 0.1, then m x = 0.1017 and d x = 0.0173: ! ! !  (mx −y)2  Z76 2 1 − 1 2 E v s , /s (ω) ≈ (−0.093 + 1.0076y − 0.0001y ) e 2d2x dy ≈ 0.0094 √ 3 d x 2π 2

0

⇒ v 0.1,

! 1 ≈ max (−4.9, 0.0093) = 0.0093. 3

For ω such that s1 (ω) = 12, then m x = 12.2 and d x = 2.0785: ! ! !  (mx −y)2  Z76 2 1 − 1 2 E v s , /s (ω) ≈ (−0.093 + 1.0076y − 0.0001y ) e 2d2x dy ≈ 12.1844 √ 3 d x 2π 2

0

! 1 ⇒ v 12, ≈ max (7, 11.9830) = 11.9830. 3 For ω such that s1 (ω) = 24, then m x = 24.4 and d x = 4.1569: ! ! 2 2 E v s , /s1 (ω) 3 17

Z76 ≈ (−0.093 + 1.0076y − 0.0001y2 ) 0

⇒ v 24,

1 √

! e

  2 − (m x −y) 2 2d x

d x 2π

dy ≈ 24.4312

! 1 ≈ max (19, 24.0274) = 24.0274. 3

For ω such that s1 (ω) = 36, then m x = 36.6 and d x = 6.2354: ! ! 2 E v s2 , /s1 (ω) 3 Z76 ≈ (−0.093 + 1.0076y − 0.0001y2 ) 0

⇒ v 36,

1 √

! e

  2 − (m x −y) 2 2d x

d x 2π

dy ≈ 36.6473

! 1 ≈ max (31, 36.0416) = 36.0416 3

For ω such that s1 (ω) = 48, then m x = 48.8 and d x = 8.3138: ! ! 2 E v s2 , /s1 (ω) 3 Z76 ≈ (−0.093 + 1.0076y − 0.0001y2 ) 0

1 √

!

d x 2π

e

  2 − (m x −y) 2 2d x

dy ≈ 48.7911

! 1 ≈ max (43, 47.9523) = 47.9846. 3 n o   Again we approximate the whole function v s1 , 13 in R by a regression in 1, x, x2 , obtaining v(x, 13 ) ≈ −0.0951 + 1.0087x − 0.0001x2 and repeat exactly the same procedure with the instant t = 0. ⇒ v 48,

Table 1: The exact values for the one variable European case, using the Black-Scholes formula 0.5 years 1 years 1.5 years 2 years 2.5 years 3 years Black-Scholes 548833 1022240 1499027 1979683 2464361 2952997

Table 2: Comparison of absolute errors for the one variable European case. 0.5 years 1 years 1.5 years 2 years 2.5 years 3 years PI Binomial MC

0.0098 0.0007 0.1970

0.0311 0.0085 0.1804

0.0560 0.0136 0.4604

0.0777 0.0155 0.7582

18

0.0947 0.0159 0.0057

0.1064 0.0155 0.9024

PI Binomial MC

0.5 years

Table 3: Comparison of times for the one variable European case 1 years 1.5 years 2 years 2.5 years 3 years

0.7600 0.0005 2.2435

1.7980 0.0010 4.2014

3.3462 0.0017 6.2458

5.3742 0.0032 8.3590

7.7691 0.0043 10.6537

10.8831 0.0063 12.5599

Table 4: Accurate exact values for the one variable American case by the overkilling of the binomial method 0.5 years 1 years 1.5 years 2 years 2.5 years 3 years Over-killing 481267 1016391 1551515 2086638 2621762 3156886

Table 5: Comparison of absolute errors for the one variable American case 0.5 years 1 years 1.5 years 2 years 2.5 years 3 years PI

0.0048

0.0084

0.0121

0.0159

0.0194

0.0236

Binomial

0.0012 1010

0.0057 1010

0.1405 1010

0.1892 1010

0.1032 1010

0.2705 1010

LSM

0.1814

0.3660

0.2151

1.1600

0.2825

0.1114

0.5 years

Table 6: Comparison of times for the one variable American case 1 years 1.5 years 2 years 2.5 years 3 years

1.8721 0.0002 2.4579

5.0148 0.0003 6.8740

PI Binomial LSM

PI MC

9.7565 0.0006 16.2938

15.8717 0.0010 39.4460

23.6049 0.0015 74.2814

32.7805 0.0021 123.7513

0.5 years

Table 7: Comparison of absolute errors for the two variable European case 1 years 1.5 years 2 years 2.5 years 3 years

0.0219 0.2707

0.1833 0.1809

0.5 years PI Pentanomial MC

5.4387 0.0014 0.9383

0.5 years PI LSM

0.0372 0.0089

0.3894 0.6335

0.5804 0.0109

0.7373 0.0017

0.8574 0.1417

Table 8: Comparison of times for the two variable European case 1 years 1.5 years 2 years 2.5 years 3 years

14.2018 0.0095 4.3679

31.6447 0.0377 12.6151

54.3607 0.1072 26.4696

83.9196 0.1726 49.4256

140.3083 0.2354 75.0726

Table 9: Comparison of absolute errors for the two variable American case 1 years 1.5 years 2 years 2.5 years 3 years

0.0660 0.2003

0.0954 0.3358

0.1253 0.9538

0.1557 0.7398

0.1864 1.3027

Table 10: Comparison of times for the two variable American case 0.5 years 1 years 1.5 years 2 years 2.5 years 3 years PI Pentanomial LSM

4.7036 0.0009 7.6787

11.8743 0.0035 28.8681

26.5217 0.0111 65.3780

46.0178 0.0263 135.9327 19

71.3046 0.0505 325.8369

114.7276 0.1184 555.2581