A Finite Element Platform For Pricing Path-Dependent Exotic Options Zili Zhu1 Nick Stokes CSIRO Mathematical & Information Sciences Clayton, Victoria 3169 Australia Copyright CSIRO 1999
Introduction For many path-dependent options, such as Asian options, lookbacks, Parisian options, using partial differentiation equation (PDE) approach to price these exotic options can be straightforward and flexible. In addition, modified versions of the standard exotic path-dependent options can be easily accommodated in the same pricing algorithms. For example, American early exercise features, barriers (both constant or time dependent) on the underlying or the path-dependent parameters, and any pay-off functions can be readily implemented in an extended version of the same pricing scheme. Most importantly, the methods is simple and easy to understand, and it is fast to implement. Of course, volatility surfaces and interest-rate term structures are natural part of any PDE solutions. In a PDE approach, once the underlying is seen as the first dimension (first factor), and the path-dependent parameter is regarded as the second dimension (second factor), we can easily formulate the path-dependent option as a two-dimension problem (twofactor). The pay-off function is the initial condition, and we solve the Black-Scholes equation in time backwards toward present time, we can obtain prices for all different values of the underlying asset and the path-dependent parameter. The computation of delta and gamma values becomes a simple post-processing task. Such time-marching approach is flexible and robust, and discrete dividends can also be easily considered. In this paper, we will adopt a PDE approach to price path-dependent exotic options, and we will use finite-element formulation to numerically solve the partial-differential equations governing the exotic options prices. There are many advantages of using finite-element formulation, the most obvious ones are that 1). finite-element methods can easily handle irregular barriers such as time-dependent barriers, 2). The unstructured mesh system of finite-element formulations allows easy concentration of grid points where high accuracy of solution is desired, 3). We can develop more stable and robust solution algorithms by using finite-element approach. In this paper, we will use Galerkin finite-element method to price two types of pathdependent options: discrete/continuous Asian options, and continuous Parisian options. We will compare the numerical results from the current finite-element method with data 1
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obtained by other methods. From the examples shown in this paper, we will illustrate the advantages of using PDE approach for pricing path-dependent exotic options, as well as the robustness and flexibility of Finite-Element formulation for options pricing.
Asian Options Asian options are financial contracts giving the holder the right to buy a certain asset for a pay-off price related directly to its average price during a certain time period before expiry date. The averaging can be arithmetic or geometric, the average price can also be either discretely sampled, or continuously monitored.
Continuous Asian Options: For the continuously sampled Asian options, the arithmetic average value A(t) of asset t price S(t) is: 1
A=
t
S(τ )dτ
0
If the averaging value A(t) is computed only for a short time interval [ t1 , t2 ] within the contract, the arithmetic value is calculated as:
1 A = t − t1
t
S (τ ) d τ
t1
The partial-differential equation governing this continuously-monitored Asian option price V(S,A,t) is given in [5] as: ∂V 1 2 2 ∂ 2V ∂V S − A ∂V + σ S + rS + − rV = 0 (1) 2 ∂t 2 ∂S ∂S t ∂A Compared with standard Black-Scholes equation, this equation (1) has an extra term: S − A ∂V t ∂A This term provides the mechanism for satisfying the jump-condition continuously. When the average parameter A is defined as a new state variable (a dimension), we can solve the modified Black-Scholes equation (1) in a two-dimension space (S-A) in a two-factor model. We have developed two schemes: a fully-implicit scheme and a Crank-Nicholson scheme in the time direction to solve the equation, these two numerical schemes can be expressed as:
β 1 2 2 ∂ 2 ∆V k ∂∆V k S − A ∂∆V k k ∆V + σ S + rS + − r∆V k = 2 2 t ∆t ∂S ∂A ∂S 1 2 2 ∂ 2V k −1 ∂V k −1 S − A ∂V k −1 + rV k −1 − − σ S − rS 2 2 t ∂A ∂S ∂S
(2)
here when β = 1 , the fully-implicit scheme is used, and β = 2 corresponds to CrankNicholson scheme. ∆V k is the price increment from (k-1)th time step to the current kth time step, and the option price at kth time step is V k = V k −1 + β ⋅ ∆V k , t k = t k −1 − ∆t , and t 1 = T and t m = 0 , m is the total number of time steps.
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The differentiation with respect to S and A in equation (2) can be numerically implemented either in finite-difference or finite-element formulation. In the present paper, we use Galerkin finite-element approach, and implement the numerical algorithm through finite-element generic PDE package Fastflo. American early exercise feature can be easily implemented within the current timemarching scheme. At each time step k, we only need to compare V k = V k −1 + β ⋅ ∆V k with the intrinsic value of an American early exercise, and choose the larger value. Though this is an approximate American option value, more elaborate schemes can be developed ([7]).
Initial and boundary conditions: The initial condition for the solution is the pay-off function at maturity time T, equation (2) is solved in time by marching backwards from expiry date T to the starting date of the contract at t=0. In the current set-up of PDE Finite-Element approach, not only fixed/floating strike options, actually any pay-off function of S and A can be specified. In addition, there is no limit on the types of barriers we can impose on both the underlying S and its average value A. We can specify irregular knock-out or knock-in barriers on S or A. For fixed/floating strike Asian options, the initial conditions at t=T are: 1. floating strike call: V ( S , A(T ), T ) = max( S (T ) − A(T ),0) 2. floating strike put: V ( S , A(T ), T ) = max( A(T ) − S (T ),0) 3. fixed strike call:
V ( S , A(T ), T ) = max( A(T ) − X ,0 )
4. fixed strike put:
V ( S , A(T ), T ) = max( X − A(T ),0 )
Here, X is the strike price. For the boundary conditions at S=0 and as S→ ∞ and A=0 and as A → ∞ , there are no obvious rules to follow, certain conditions have been proposed such as in [2][4][5]. However, all the derivations themselves involve certain assumptions on the option price, particularly for S→ ∞ and A → ∞ . These assumptions can often introduce unwanted errors. From a numerical point of view, if the boundary conditions do not come from a clear financial argument, it is better to impose no boundary conditions at all. In the case of Asian options, in the present algorithm, we do not specify any boundary conditions for S→ ∞ , A → ∞ and S = 0, A = 0. The cut-off point for S→ ∞ is generally 2-3 times the spot price. Figure 1 illustrates the finite-element representation (discretization) of the two-factor state-variable space (S-A) for an Asian option with a spot price S=100. Mesh points are concentrated around S=100, A=100 where we need the option price. The flexibility of imposing concentration points where high accuracy is needed is a distinctive advantage of finite-element methods over finite-difference methods. An example of fixed-strike call-options price is plotted in Figure 2 as the third axis.
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Figure 1. Two-factor S-A domain
Figure 2. Fixed-strike call-option price
The numerical results from the present finite-element method are compared with other methods in the following table: σ 0.10 0.10
T-t 0.25 1.0
European Fixed Strike Call Option ( r=0.1, X=100 ) Implicit C-N Lower 1-D Z&F&V 1.876 1.880 1.851 1.841 1.793 5.267 5.298 5.255 5.254 5.261
B&P 1.869 5.279
σ 0.10 0.10
European Floating Strike Put Option ( r=0.1, S=100 ) T-t Implicit C-N Lower 1-D Z&F&V 0.25 0.6219 0.6252 0.628 0.636 0.582 1.0 0.5992 0.5979 0.598 0.598 0.589
B&P 0.632 0.614
σ 0.10 0.10
American Floating Strike Put Option ( r=0.1, S=100 ) T-t Implicit C-N Z&F&V 0.25 1.186 1.192 1.359 1.0 1.832 1.822 1.952
B&P 1.194 1.799
Table 1 “implicit” refers to the implicit scheme implemented in the present paper, C-N refers to the Crank-Nicholson scheme of this paper. Lower and 1-D refer to Rogers and Shi[1], Z&F&V refers to Zvan, Forsyth and Vetzal [2]. B&P refers to Barraquand and Pudet [3]. X is stike price for fixed strike opions, S is the current asset spot price.
Discrete Asian Options: When the underlying S is discretely sampled to calculate its average A, arithmetic average value A at the N-th sampling time is: AN =
1 N
i= N
S ( ti )
i =1
For the discretely sampled Asian option, the familiar Black-Scholes equation stands:
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∂V σ 2 S 2 ∂ 2V ∂V + + rS − rV = 0 (3) 2 2 ∂S ∂t ∂S Now, for the Black-Scholes equation (3), the average value A is simply a parameter, and extra jump conditions across the sampling dates will have to be satisfied. The jump conditions are to update the arithmetic average A at the sampling date
Across the N-th sampling date, the average value A satisfies the jump condition: 1 AN −1 = AN + ( AN −S ) ( 4) N −1 Here AN is the average value of underlying after the N-th sampling time. AN −1 is the average underlying value immediate before the N-th sampling time t N , (we time-step backwards from the N-th step to (N-1)-th step). The option value V(S,A,t) immediately before and after the N-th sampling time t N should satisfy the continuous condition: V ( S , AN −1 , t N− ) = V ( S , AN , t N+ ) (5) The jump conditions (4) and (5) produce the option price V ( S , AN −1 , t N− ) at t N− by way of interpolation and extrapolation of options price V ( S , AN , t N+ ) at t N+ (such as in [4]). However, the interpolation can be a source of numerical inaccuracy, it can also cause stability problems. In this paper, to avoid problems associated with interpolation across the N-th sampling time t N , we move the coordinate value AN to AN −1 for each nodal point in the (S,A) domain. This is effectively to satisfy jump conditions (4) and (5) by changing the coordinates of all the mesh points in the state variable A direction through formulae (4). As the option price V ( S , AN −1 , t N− ) is fixed with the mesh, it moves with the mesh points, jump condition (5) is thus automatically satisfied. Figure 3 shows a jumped mesh after a few sampling dates from the maturity date T when the mesh looks as shown in Figure 1.
Figure3. Modified mesh by jump conditions:
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The following Table 2 compares prices for discretely monitored floating/fix strike options for both calls and puts between the finite-element formulation of this paper and those of Zvan, Forsth and Vetzal [4]. European Fixed Strike Call Option ( r=0.1, σ=0.2, T-t=0.25, X=100 ) Timestep Continuous 0.001 0.002 0.004 Present method 2.9409 2.9305 2.9284 2.9243 Z&F&V 2.928 2.9287 2.9151 2.8961 European Floating Strike Put Option ( r=0.1, σ=0.2, T-t=0.25, S=100) Timestep Continuous 0.001 0.002 0.004 Present method 1.7148 1.7151 1.7129 1.7085 Z&F&V 1.679 1.7121 1.6853 1.6942 Table 2. Pricing from the present numerical method are compared with data from [4]
Parisian Options Parisian options are knock-in/out barrier options for which the knock-in/out only occurs if the underlying asset price remains in breach of the barriers for a pre-specified time period Tp continuously. The requirement of being in breach of the barriers for a certain period of time continuously can reduce the effect of possible manipulation by traders to trigger a knock-in/out through moving the market briefly. If we define a new state variable J as the total time the underlying asset price stays beyond the barriers continuously, in the case of down-and-out barrier, we have: J = 0, dJ = 0, if S ≥ S down dJ = dt , if S < Sdown V ( S , J , t ) = 0, if J ≥ Tp and S < S down here, Sdown is the down-and-out barrier price level. These expressions state that when the asset price is below Sdown , the state variable J starts to accumulate value at the same rate as the passing time, and when the asset price is equal to or above Sdown , J is reset to zero, and remains zero. These features of the down-and-out barrier of the Parisian option can be reflected through governing equations and their boundary conditions. The state variables are now: S, J and t. The options price V is a function of S, J and t, i.e. V = V(S,J,t). We will solve the governing equation in the two-dimension space of S and J, and march the solution from maturity date t=T to present time t=0. Figure 4 shows a two-dimensional (S,J) domain representing a down-and-out Parisian options, the barrier level is Sdown = 8.0 , and we only allow the maximum asset price to be S=15.0. The maximum value for J is Tp , in this case, Tp = 0.1 years. However, we are free to scale J according to resolution requirement in the J direction. In Figure 4, we have scaled J by a factor of 40.
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When S ≥ Sdown , and 0 ≤ J ≤ Tp (shown as Region 1 of Figure 4), the governing equation is the standard Black-Scholes equation (3). Actually, since J is irrelevant to the options price, we can reduce this Region 1 from a two dimensional (S-J) domain into a one-dimensional (S) domain as is the case in [6]. However, for modification of the Parisian options into “Parasian” options ([6]), and to maintain a uniform scheme for pricing both Parisian and “Parasian” options, we will still use the two-dimensional (S-J) Region 1 even though J does not influence the options price in this domain. When S < Sdown and 0 ≤ J ≤ Tp (shown as the Region 2 in Figure 4), the governing equation is: ∂V 1 2 2 ∂ 2V ∂V ∂V + σ S + rS + − rV = 0 (6) 2 ∂t 2 ∂S ∂S ∂J The boundary conditions for V(S,J,t) are: V (S , J , t ) = 0 At S < Sdown , J = Tp , At S ≥ Sdown , J = Tp , At S ≥ Sdown , J = 0 , At S = Smax , At S = Smax (for S → ∞ ), At S = 0 ,
∂V ( S , J , t ) =0 ∂J ∂V ( S , J , t ) =0 ∂J ∂V ( S , J , t ) =0 ∂J V (S , J , t) = 0 no boundary condition is specified.
Figure 4. mesh representation of a down-and-out Parisian option when Sdown = 8 The initial conditions for the down-and-out Parisian options are: V ( S , J , T ) = max( E − S (T )) For J < Tp or ( J = Tp , S ≥ S doan ) For ( J = Tp , S < S down ),
V (S , J ,T ) = 0
In this paper, we solve equation (3) in Region 1 and equation (6) in Region 2 concurrently. Both the fully-implicit scheme and the Crank-Nicholson scheme described by equation (2) are similarly implemented for the coupled equations (3) and (6). We consider a simple Parisian down-and-out European put option, volatility σ=0.2, and
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interest rate r=0.08, a strike price E=10.0, and the down-barrier level is at Sdown = 8.0 , the barrier time Tp = 0.1 years. We consider this Parisian option with two expiry dates: T = 0.25 years and T = 1.0 years.
The Parisian put option price with T = 0.25 is shown here in Figure 5 in the vertical direction as a function of S and J. As can be expected, when J
