Pricing Parisian Options

Transcription

1 Pricing Parisian Options Richard J. Haber Mathematical Institute, St Giles, Oxford OX1 3LB Philipp J. Schönbucher University of Bonn, Statistical Department, Adenaueralle 24-42, Bonn, Germany Paul Wilmott Mathematical Institute, St Giles, Oxford OX1 3LB July Introduction Parisian options are barrier options for which the knock-in/knock-out feature is only activated after the price process has spent a certain prescribed, consecutive time beyond the barrier. This specification has two motivations: First, there is the need to make the option more robust against short-term movements of the share price. This is achieved in Parisian options where it is ensured that a single outlier cannot trigger the barrier. In particular, it is far harder to affect the triggering of the barrier by manipulation of the underlying (see Taleb [4]). Second, classical barrier options present hedging problems close to the barrier because their Gamma becomes very large. To some extent, these problems are reduced, or at least smoothed, in the Parisian contract. We present a flexible approach to valuing such options using the numerical solution of a partial differential equation. This approach can price a variety of modifications of the basic Parisian contract including Parasian options (activation of the barrier conditional on the total time spent above the barrier), American early exercise rights, discrete sampling of the barrier and 1

2 more general payoffs. The approach readily accommodates features, such as early exercise, that render the traditional Monte Carlo approach impractical. To demonstrate the flexibility of this method we have written a program for valuing many types of the Parisian option contract. This program may be downloaded free of charge from In January 1997 Risk Magazine published an article by Chesney, et al.[1] on the pricing of Parisian contracts by Laplace transform methods. That article provided the motivation for the present work since we believe our method to be superior for at least four reasons: It is relatively simple to understand, requiring no more than a knowledge of elementary partial differential equations. It is flexible, and can be extended to price many more general contracts. It is easy to program. Great care is needed with the numerical inversion of the Laplace transform but the finite difference method is robust. It is fast. 2 The State Space The crucial point to note about Parisian options is that they are strongly path dependent; not only does the payoff depend on the value of the underlying at expiry but also on the path taken to get there. Yet this path dependence is perfectly manageable within the partial differential equation framework: We do not need to know all the details of the path taken. The only further information we require is the value of the new state variable τ, the barrier time. The barrier time is defined as the length of time the asset price has been beyond the barrier in the current excursion τ := t sup { t t S(t ) S }. (1) This expression gives the difference between the current time t and the last time the share price S was below the barrier S. This is the correct definition of the barrier time for an up option. If the share price is currently below the barrier, τ is zero. The analogous expression for a down barrier is obtained by reversing the second inequality in relation (1) τ := t sup { t t S(t ) S }. (2) 2

3 The dynamics of τ, for the up barrier, are given by the simple expression dt if S(t) > S, dτ(t) = τ(t ) if S(t) = S, (3) 0 if S(t) < S, where τ(t ) is the left limit of τ. The barrier time increases at the same rate as t if the share price S(t) is beyond the up barrier S(t) > S, it is reset to zero if the share price hits the barrier S(t) = S, and it does not change if the share price is below the barrier S(t) < S. The new state variable τ can be viewed as a clock that starts ticking as soon as the share price crosses the barrier level, S, and is immediately reset when the share price returns below S. The knock-in/knock-out feature is only activated if the length of the current excursion exceeds a given level τ T, where T is the barrier-time triggering parameter. This specification exactly covers the covenants in the Parisian option s payoff. The other two state variables needed are the share price S, and time t. We assume that the share pays dividends at rate D, and its price follows a lognormal Brownian motion given by ds = µs dt + σs dw (4) where dw denotes the increment of a standard Brownian motion. A typical sample path of the state vector in the state space for an up option is shown in exhibit 1. As long as the barrier is not exceeded (S < S) the sample path moves along the τ = const-plane, after it crosses the barrier at the transition point τ and t increase at the same rate. This is shown as a movement along the diagonal grey plane in the state space. The path will be continued on the τ = 0 plane should the share price move back down below S. In terms of the state variables, the specification of a Parisian option with a knockout is as follows. The option has a payoff F (S) at expiry T, unless at some time before T the state variable τ reaches an upper barrier T, in which case the option expires worthless. Given this setup we can write the value of a Parisian option V as a function of the three state variables S, t and τ as V (S, t, τ). The governing equation can be derived formally but here we will just state it and give an intuitive justification. There are two distinct situations to consider. The first is when the asset is below the barrier, S < S. In this case the variable τ remains 3

4 S - S t 0 τ Exhibit 1: Characteristics for an up Parisian option. The τ-axis describes the time spent beyond the barrier so far (in this excursion), time runs along the t-axis and the current share price level is denoted on the S-axis. As long as the barrier is not exceeded (S < S) the sample path moves along the τ = const-plane, after the share price crosses the barrier S the barrier time τ and time t increase at the same rate. 4

5 unchanged and we must solve the basic Black Scholes equation with a continuous dividend rate D V t + σ2 S 2 2 V V + (r D)S 2 S2 S rv = 0. (5) The second case occurs when the asset rises above the barrier, S > S, and so the clock, τ, is ticking. Here we must solve V t + σ2 S 2 2 V V + (r D)S 2 S2 S rv + V τ = 0, (6) where the new state variable τ gives rise to a modified form of the Black Scholes equation. We will show below how the solutions are linked in these two regions. Both regions are shown in exhibit 2. The region where equation (5) governs the price is the plane denoted by C in exhibit 2. This is the area where the barrier has not been passed yet (S S), hence τ = 0 here. The region where the modified Black-Scholes equation (6) has to be satisfied is the area where S has exceeded the barrier (S S). In the exhibit this is the box bounded by the rectangles A and B. Here the barrier time τ can take values between 0 and T. For Parisian down options the positions of the regions are reversed. 3 Parisians and Parasians In a standard Parisian option the clock variable τ is reset to zero once the share price moves below S. This will be the case even if the excursion lasts for a very short time. For example, in the case of an up barrier the share price can reside above the barrier for almost the entire time without triggering the option, provided that it returns below S often enough. However, the constraint on the time τ being consecutive may defeat one of the original intentions of the Parisian option which was to make the triggering of the knock-in/knock-out less susceptible to one-off outliers and very short-term movements of the share price (or even manipulation). With the given specification the triggering of the barrier is fairly robust, but the resetting of τ is still very much subject to short-term price movements. In the light of this it seems natural to introduce a variation of the Parisian contract, an option where τ is not reset and where the knock-in/knock-out feature is only activated if the cumulative time spent beyond S exceeds some 5

6 D S A B - S C T t t 0 τ Exhibit 2: The domain for a Parisian up option. The τ-axis describes the time spent beyond the barrier so far (in this excursion), time runs along the t-axis and the current share price level is denoted on the S-axis. In the lower box (with the rectangle denoted by C as left side) equation (5) must be satisfied. In the upper box (bounded by the rectangles A and B) the modified Black-Scholes equation (6) has to be satisfied. (This box is unbounded in the S -direction.) 6

7 prescribed value. This aggregation feature resembles closely the averaging feature of an Asian option which is why we call this contract the Parasian option. Note that in all cases that the Parasian contract knocks in/out, the equivalent Parisian will have done so too. This variation requires only a minor modification in the model, we just have to change the definition (3) of the dynamics of τ so that it does not reset at the boundary S = S: dτ a = { dt if S S, 0 if S < S. (7) where we have called the non-reset clock τ a. It is easy to imagine many other possible specifications of the clock τ. There could be another, lower barrier S and the time S spent below S would be subtracted from τ, or the speed with which τ changes could be proportional to the distance the share price S is beyond the barrier (thus weighting large deviations beyond the barrier more strongly). We invite the reader to find other variations himself and to find the appropriate specification of dτ. 4 Boundary Conditions At S = S we have to impose pathwise continuity of V, which will mean for Parisians (where τ is reset to zero at S = S) V (S, t, τ) = V (S, t, 0). (8) However, in the case of a Parasian option V does not jump at S, thus invalidating condition (8). The exact specification of the option enters our model via the boundary conditions that we specify. These conditions are to be applied at the boundaries of the state space as shown in exhibit 2: at expiry t = T, at trigger of the barrier τ = T and at the limit values for the share price S = 0, S. In its most general form a Parisian-type option is specified as follows: If the knock-in/knock-out has not been triggered by expiry T, then the option has the share price contingent payoff F (S) at expiry. This payoff might also depend on τ and would then be given by F (S, τ). For example, a boost option is an option whose payoff is proportional to the 7

8 time the share price spent beyond the barrier. This option would then have a payoff of F (S, τ) = cτ. In exhibit 2 the function F prescribes the option values on the rectangle A and the line D. If the knock-in/knock-out has been triggered during the lifetime of the option, the option pays off G(S) at expiry. In exhibit 2 the function G prescribes the option values on the rectangle B. All common variations of the option can be accommodated within this framework, e.g. 1. In Boundaries: One gets a European Call of maturity T and exercise price E as soon as the knock-in is triggered. Set F (S) = 0 and G(S) = (S E) Out Boundaries: One gets a European Call of maturity T and exercise price E unless the knock-out is triggered. Set F (S) = (S E) + and G(S) = 0. In this framework ins and outs are treated the same. The boundary conditions to specify are V (S, T, τ) = F (S, τ) 0 τ < T, (9) V (S, T, T ) = G(S). 5 An American in Paris The pricing of American-style options in the pde framework could not be simpler, either conceptually or from a numerical analysis point of view. We shall be very general in setting an American-style Parisian option, simply stating that the option gives its holder the additional right to exercise the option at any time prior to expiry and thereby receive an amount A(S, t, τ). From the simplest of arbitrage considerations we must have V (S, t, τ) A(S, t, τ). (10) Since we can exercise whenever we want, we should act to maximise the value of the contract to us. This optimality amounts to insisting that V/ S, the 8

9 option delta, is continuous. In solving the US-style contract numerically by an explicit finite-difference scheme, discussed below, all we need do is to add to our code a line representing (10). 6 Numerical Algorithm The numerical solution to equations (5) and (6) is implemented using an explicit finite difference scheme. For simplicity we choose to only discuss the case of an up barrier. The share price S, time t and barrier time τ are discretised as S, t and τ, respectively, where for convenience we set τ = t. For stability of the scheme t has to be chosen small enough. 1 We denote with Vi,j n the numerical approximation to the option value V (S, t, τ) at share price S = i S, time from expiry T t = n t, and time from the knockin/knockout T τ = j τ. We call ī the discrete barrier in the share price (i.e. ī S = S) and j is the value of the clock if τ is at zero(i.e. j τ = T ). The explicit finite difference approximation to the spacial part of the Black- Scholes equation (5) is defined for fixed times n, j by the operator L as L n i,j = i2 σ 2 2 ( ) V n i+1,j 2Vi,j n + Vi 1,j n i(r D) ( ) + V n 2 i+1,j Vi 1,j n rv n ij. (11) There are two regions to consider (see exhibit 2): The area where the clock τ is running (i.e. beyond the barrier) and the area where the clock τ stands still. For an up barrier the barrier domain is defined by i > ī and the time stepping is accomplished by the difference equation V n+1 i,j+1 = V n i,j + t L n i,j i > ī. (12) Note that in equation (12) we had to increase j since the clock τ is ticking. The system can be visualized as diffusively propagating in time along a diagonal plane, with normal vector (0, 2, 2) in (S, t, τ) space, as shown in 2 2 exhibit 1. 1 The stability condition t < 1/(i 2 maxσ 2 ) must be satisfied, where i max is the largest i index in the scheme. It is advisable to discretise x := ln S instead of S which will make the stability condition less stringent. See Wilmott et al. [5] for details. Here we chose the direct setup for its greater simplicity. 9

10 In the second region, i < ī, the clock τ does not change. Hence the system evolves along the vertical plane, with normal vector (0, 0, 1), (the rectangle C in exhibit 2) in accordance to the time-stepping scheme V n+1 i,j = V n i,j + t L n i,j i < ī. (13) Until now we have neglected the dynamics at the the boundary i = ī which link the upper and lower regions of the state space. In fact, it is this condition that differentiates between the Parisian and Parasian option. A Parisian option requires the resetting of the clock at i = ī. We first use (13) to calculate the values of V n+1 for i ī of the option at or below the i, j barrier S = ī S and for barrier time zero (τ = 0 or j = j). V n+1 i, j = V n i, j + t Ln i, j i ī. (14) Next we set the remaining values V n+1 ī,j of the option at the boundary S = ī S. These values are set for all positive barrier times (j < j) to the values at barrier time zero (j = j) i.e. V n+1 i,j = V n+1 i, j j < j. (15) Then we proceed to calculate V n+1 i,j for i > ī according to the scheme (12). This stepping means that all values of V at or below the barrier are given by the values of V for τ = 0, the values of V on the rectangle C in exhibit 2. In contrast, for a Parasian option we simply apply (13) over the domain i ī, and then use (12) for i > ī without resetting the values at the boundary. This means that we need to solve and calculate values of V for all possible values of τ a even if S is below the barrier S, in figure 2 the whole box with l.h.s. C is relevant, not only the rectangle C. The payoff at knockin or knockout enters the scheme through the specification of the value of the payoff at j = 0 τ = T by way of boundary condition Vi,0 n = Ĝ(i S, T n t) i, (16) where Ĝ(S, t) is the (time t)-value of receiving G(S) at time T. The final payoff is included by starting the scheme off with the values V 0 i,j = F (i S, T j t) i, j > 0. (17) Finally, for the case of an early exercise feature we have to check before updating V n+1 i,j, if the newly determined value is larger than the early exercise 10

11 Inputs Outputs Underlying Barrier type Spot In/Out Option value Volatility Call/Put Delta Dividend yield Strike Gamma Interest rate Expiry Theta Reset (Y/N) Trigger time Exhibit 3: Inputs and outputs of the algorithm payoff A(i S, T (n + 1) t, T j t). If not, we simply set the two equal by way of the conditional relation V n+1 i,j = A(i S, T (n + 1) t, T j t). The program, which may be downloaded from solves for the price and hedging variables by way of the aforementioned explicit finite-difference scheme. Although explicit finite-difference schemes are similar in spirit to the binomial numerical method they are more general and thus more flexible. The downloadable program is fast but certainly not optimal. Had speed had been our prime concern we would have used an implicit method such as Crank-Nicolson. This, along with other methods, is discussed by Dewynne and Wilmott [3] and in great depth (with sample code) in Wilmott et al. [5]. The finite-difference solution of financial partial differential equations is becoming accepted as the most time-efficient method of pricing and hedging certain types of contract. The method is time efficient because it is extremely easy to program and the programs run very quickly. It is suitable for many types of contract including most common path-dependent derivatives and is trivially with one extra line of code extended to American-style early exercise. It easily outperforms Monte Carlo simulation, the other popular choice for path-dependent contracts. The downloadable program has the inputs and outputs as given in exhibit 3. The outputs are arrays (against the underlying). 7 Results and Discussion All presented results are for options on a dividend-free share S with volatility σ = 20% and a risk-free instantaneous interest rate r = 5%. The numerical scheme is identical to the one presented in the previous section, with the only exception that the share price was discretised using equal steps in ln S and 11

12 not in S. We will first consider the case of a Parisian up-and-out European Call option with exercise price E = 10, barrier level S = 12, barrier time T = 0.1 and time to expiry T = 1. Exhibit 4 depicts the option value V versus share price S and time τ beyond the barrier, exhibit 5 depicts the associated delta of the option. The option value shows the typical up-and-out knockout shape: first (for low S) the Call option feature dominates and the option value increases with S but from a certain level of S onwards (here this is around S = 10.18) the approaching knockout barrier becomes stronger and the option price decreases with increasing S. Because the option is of the Parisian type, the option price for share prices below the barrier (S < S = 12) is constant: Here the barrier time τ will immediately be re-set to τ = 0 as described in the numerical algorithm. For share prices larger than the barrier the option prices start to differ depending on how long the share has spent beyond the barrier so far. If the share price has just crossed the barrier from below (i.e. at τ = 0) the smoothing effect of the Parisian specification can be seen, while for τ close to the barrier activation time (τ = 0.1) the option price approaches zero. Looking at the delta of the option in this area shows that the Parisian specification only provides a smooth transition for small τ (or τ = 0 which is the case when the barrier is first crossed from below). Crossing backwards from above will result in large changes in the delta of the option because τ is reset to zero when the barrier is touched. Consequently, the hedging of Parisian options will provide difficulties if the share price returns close to the reset barrier S after having spent some time beyond it. Exhibit 6 highlights this effect: for τ = 0 the smooth transition at S = 12 can be seen, while for τ = (i.e. close to knockout) crossing back below the activation barrier S = 12 will have a significant effect. The numerical values for exhibit 6 are given in exhibit 7. Exhibits 8 and 9 show the value and delta of the equivalent Parasian upand-out European Call option, again with the same parameter values as for the previous option. The Parasian option exhibits the same up-and-out shape (increasing, then decreasing) as the Parisian but because the clock τ for the Parasian option is not reset after crossing back below S, the option price depends on τ even for S < S. This can be seen clearly in the price plot in exhibit 8. For large 12

13 0,35 0,3 0,25 0,2 V 0,15 0,1 0,05 0 6,95 7,31 7,68 8,07 8,48 8,92 9,37 S 9,85 10,35 10,88 11,43 12,02 1 2,63 13,27 13,95 14,66 tau= 0 tau=.03 tau=.06 tau=.09 Exhibit 4: Value V versus barrier time τ and asset price S for a Parisian up and-out European Call option with exercise price E = 10, barrier level S = 12, barrier time T = 0.1 and time to expiry T = 1. The underlying asset S has volatility σ = 20% and no dividends, the risk-free instantaneous interest rate is r = 5%. 13

14 0,2 0,1 0-0,1 Delta -0,2-0,3-0,4-0,5-0,6 tau= 0 tau=.02 tau=.04 tau=.06 tau=.08 7,07 7,31 7,55 7,81 8,07 8,34 8,63 8,92 9,22 9,53 9,85 10,18 S 10,52 10,88 11,24 11,62 12,02 12,42 12,84 13,27 13,72 14,18 14,66 Exhibit 5: Delta = V versus barrier time τ and asset price S for a Parisian S up and-out European Call option with exercise price E = 10, barrier level S = 12, barrier time T = 0.1 and time to expiry T = 1. The underlying asset S has volatility σ = 20% and no dividends, the risk-free instantaneous interest rate is r = 5%. 14

15 0,3000 0,2500 0,2000 V 0,1500 tau= 0 tau=.05 tau=.095 tau= 0 tau=.05 tau=.095 0,1000 0,0500 0, ,00 11,50 12,00 12,50 13,00 1 3,50 14,00 S Exhibit 6: The value V of a Parisian and a Parasian up and-out European Call option with exercise price E = 10, barrier level S = 12, barrier time T = 0.1 and time to expiry T = 1 at different levels of τ. The Parisian option values are with straight lines, the Parasian option with dashed lines. The underlying asset S has volatility σ = 20% and no dividends, the risk-free instantaneous interest rate is r = 5%. 15

16 Parisian Parasian S tau= 0 tau= 0.05 tau= tau= 0 tau=.05 tau= Exhibit 7: The data to exhibit 6. Value V versus asset price S for a Parisian and a Parasian up and-out European Call options for different barrier times τ. (Parameters: Exercise price E = 10, barrier level S = 12, knockout barrier time T = 0.1, time to expiry T = 1. Underlying asset volatility σ = 20% zero dividends, risk-free instantaneous interest rate r = 5%.) 16

17 0,3 0,25 0,2 V 0,15 0,1 0,05 0 6,95 7,31 7,68 8,07 8,48 8,92 9,37 S 9,85 10,35 10,88 11,43 12,02 12,63 13,27 13,95 14,66 tau= 0 tau=.03 tau=.06 tau=.09 Exhibit 8: Value V versus barrier time τ and asset price S for a Parasian up and-out European Call option with exercise price E = 10, barrier level S = 12, barrier time T = 0.1 and time to expiry T = 1. The underlying asset S has volatility σ = 20% and no dividends, the risk-free instantaneous interest rate is r = 5%. 17

18 0,15 0,1 0,05 Delta 0-0,05-0,1-0,15 tau= 0 tau=.015 tau=.03 tau=.045 tau=.06 tau=.075 tau=.09 7,07 7,31 7,55 7,81 8,07 8,34 8,63 8,92 9,22 9,53 9,85 10,18 S 10,52 10,88 11,24 11,62 12,02 12,42 12,84 13,27 13,72 14,18 14,66 Exhibit 9: Delta = V versus barrier time τ and asset price S for a S Parasian up and-out European Call option with exercise price E = 10, barrier level S = 12, barrier time T = 0.1 and time to expiry T = 1. The underlying asset S has volatility σ = 20% and no dividends, the risk-free instantaneous interest rate is r = 5%. 18

19 barrier times τ T the Parasian option price approaches the price of a plain vanilla knockout option. The problem with the excessive deltas of Parisian options when S leaves the barrier region does not exist for the Parasian, as can be seen from exhibit 9. The deltas of the Parasian option remain within a much smaller range. Exhibit 6 and table 7 directly contrast the values of the Parisian and the Parasian version of the option. At any given τ, the Parisian option dominates its Parasian counterpart for all share prices S. Intuitively this makes goods sense since the cumulative effect on τ of the Parasian amplifies the out feature of the option, the Parasian is knocked out more often than the Parisian. In conclusion, out method provides an extremely flexible, fast, easy to program method for evaluating more sophisticated variations of traditional barrier options such as the Parisian and Parasian options under various exercise and payoff structures. This work clearly demonstrates the versatility and ease of utilizing the partial differential equation framework for the practical implementation of exotic option pricing. 19

20 References [1] Chesney, M, Cornwall, J, Jeanblanc-Picqué, M, Kentwell, G & Yor, M 1997 Parisian Pricing, Risk Magazine, January [2] Chesney, M, Jeanblanc-Picqué, M, Kentwell, G & Yor, M 1995: Brownian Excursions and Parisian Barrier Options, Adv. Appl. Prob. March [3] Dewynne, JN & Wilmott, P 1993 Partial to the Exotic, Risk Magazine, March [4] Taleb, N 1997 Dynamic Hedging, John Wiley [5] Wilmott, P, Dewynne, JN & Howison, SD 1993 Option Pricing: Mathematical Models and Computation, Oxford Financial Press. 20

21 8 About the Authors Richard Haber is a postdoctoral researcher in the Mathematical Institute, Oxford. Address: Mathematical Institute, St Giles, Oxford OX1 3LB, United Kingdom. Philipp Schönbucher is a research assistant in the Financial Markets Group at the London School of Economics and the Department of Statistics at Bonn University. Philipp would like to thank the DAAD and the SFB 303 at the University of Bonn for financial support. Address: University of Bonn, Department of Statistics,Adenaueralle 24-42, Bonn, Germany schonbuc@addi.finasto.uni-bonn.de Paul Wilmott is a Royal Society University Research Fellow in the Mathematical Institute, Oxford, and the Department of Mathematics, Imperial College, London. Paul would like to thank the Royal Society for support. Address: Mathematical Institute, St Giles, Oxford OX1 3LB, United Kingdom. Paul@Wilmott.com 21