Numerical Valuation Of Discrete Barrier Options With The Adaptive Mesh .

by nodes just outside the true barrier. Fig. 2.1 shows the inner and outerbarrier for a binomial and trinomial tree when the true barrier is horizontaland constantly monitored. The usual tree calculations implicitly assume theouter barrier is the true barrier because the barrier condition is first met onthe outer barrier.Outer barrier?Outer barrierTrue barrier?True barrier?Inner barrier66Inner barrier(b)(a)Figure 2.1: Barrier assumed by tree lattice(a) Barriers assumed by binomial trees. (b) Barriers assumed by trinomialtrees.Bolye and Lau [6] describe this condition and propose a method to constrain the time steps that make the true barrier coincide with or occur justabove the underlying asset price level in trees. Nevertheless, the time stepconstraint makes the lattice impracticable to compute because of the incredible large number of time steps when the initial asset price is too close tothe barrier. On the other hand, the constraint of time step number is alsoannoying. In 1995, Derman et al. propose an adjusting for nodes not lyingon barriers by assuming the barrier calculated by the tree is incorrect[8].Ritchken (1995) [9] offers another approach under trinomial framework introducing a ”stretch” parameter into the lattice, which changes the pricestep just enough to place nodes on the barrier. Cheuk and Vorst [10] alsointroduce a deformation of the trinomial tree permitting one to adjust thelocation of nodes differently in each time period, and allows great flexibilityin matching a time-varying barrier. Although those methods have been proposed, a quite slow convergence rate still occur when they are used to price8

discretely monitored barrier options.For pricing discrete barrier options, Wei (1998) [12] offers an approximation approach based on interpolating between the formula for a barrieroption with the highest number of monitored points that can be handledwith the analytic formula and the continuous case (infinite monitored dates).Broadie, Glasserman and Kou (1999) develop the enhanced trinomial modelfrom Ritchken’s lattice framework. Like their earlier paper in 1997 [5], they 0.5λ σ hshift the discrete barrier at level H to a new barrier at level H 0 Hep(with for an up option and - for a down option), where λ , 3/2 andh is the size of one time step [11]. Both these techniques, however, can beused only for European options, and in Broadie et al.’s model, the ”barriertoo-close” problem still exists.Figlewski and Gao (1999) [13] present the adaptive mesh model (AMM)as an efficient trinomial lattice approach to deal with ”barrier-too-close”problem in continuous barrier options. Furthermore, in the same year, ananother kind of adaptive mesh model is proposed for pricing discrete barrieroptions by Ahn, Flglewski, and Gao [14]. The AMM model is very powerfulin both efficiency and flexibility and is going to be discussed further in thisthesis.Besides, there is the quadrature method presented by Andricopoulos etal. (2003) [15] using somewhat multinomial-like integral method to pricediscrete barrier options with speed and accuracy which can also deal withbarrier-too-close problem. We will numerically compare it with the AMMmodel later.9

Chapter 3The Adaptive Mesh Model3.1Approximation Error in Lattice ModelsAlthough lattice models provide powerful, intuitive and asymptotically exactapproximations to the theoretical option values under Black-Scholes assumptions, there are essentially two related but distinct kinds of approximationerrors in any pricing techniques of lattice framework, which we refer to asdistribution error and nonlinearity error, where the latter can be minimizedby the adaptive mesh model with slight computation increase.1. Distribution error: The lattice model approximates the true assetprice distribution with continuous lognormal density by a finite setof nodes with probabilities. Even though the mean and variance ofthe continuous distribution are matched by the discrete distribution oflattice model, the discrepancy between discrete and continuous distribution still produces distribution error in option value.2. Nonlinearity error: The finite set of nodes with probabilities used bylattice model can be thought as a set of probability weighted averageoption price over a range of the continuous price space around thenode. If the option payoff function is highly nonlinear, evaluating thenonlinear region with only one or several nodes would gives a poorapproximation to the average value over the whole interval.Fig. 3.1 illustrates the two sources of error graphically around at themoney nodes of a one year European put at expiration date with the initialasset price S0 100, the exercise price X 100, riskless rate r 0.1 andvolatility σ 0.25. The solid line represents the option payoff. The grayshaded bars represent the nodes in the trinomial lattice, corresponding to10

True probability densityLattice approximation ofprobability distribution100 Strike Put Payoff0.381.40.371.20.36S 100.5 NODES 100 NODE0.32S 99.5 NODE0.33S 99 NODEProbability0.80.340.6Put Option 100.00100.25100.50Asset PriceFigure 3.1: Distribution error and nonlinearity error around the atthe-money nodes at maturity date.asset prices of 99.0, 99.5, 100.0, and 100.5. The heavy dashed line representsthe lognormal density over this region of the price space. The light dashedbars indicate how the probability density is discretized over this price range.The contribution of a particular node to the option value equals the value ofthe node probability multiplies the option payoff at the asset price for thatnode. The distribution error arises from the difference between the heavydashed line and the light dashed line. At the S 100 node, the nonlinearityerror is caused by undervaluing the probability weighted average payoff tozero in this interval [13].The adaptive mesh model presented in this thesis can significantly reducethe nonlinear error over a given region of the tree.11

3.2Building the ModelNow we start to build a lattice model to price plain vanilla options usingadaptive mesh mechanism around the nonlinear payoff region of exerciseprice at maturity. The essence of the AMM is to use a relatively coarselattice throughout the option life and insert meshes with higher resolutioninto the tree where the nonlinear error is contributed. It is important forthe fine mesh structure (higher resolution mesh) to be isomorphic so thatadditional, still finer mesh can be added using the same procedure. Thisallows increasing resolution in a given region of the lattice as much as onewishes without requiring the step size changes elsewhere.Here we introduce an isomorphic AMM structure that can be easily applied to each region of the lattice. Trinomial tree is chose as the base latticeto approximate the risk neutral distribution because it has more degrees offreedom and has proven to be more useful and adaptable for many derivativeapplications. Because the asset price is assumed to be lognormal, the tree isbased on the log of asset price S. Define X ln(S), which implies that X isnormally distributed. Under risk neutral assumption, X follows the standarddiffusion process:dX(t) αdt σdzwhere α r q σ 2 /2, σ denotes volatility, dz is standard Brownian motion,and r and q are the riskless interest rate and dividend yield.In trinomial tree, there are three different branches for any node to moveto next time state, which are called up (u), down (d), and middle (m). Fordeduction’s convenience, we change the variable X by X 0 X αt. X 0 isthe mean-adjusted value of the log of asset price and the mean of X 0 wouldbe 0 at any time state. Hence, The trinomial tree of X 0 is symmetric. Let kdenote the length of a time step (decided by the option’s maturity T , and thenumber of time steps N to be used for the tree with k T /N ) and h be thesize of an up and down move. Thus over one time period X goes to X hwith probability pu , to X h with probability pd , and remain unchangedwith probability pm .Matching the mean, variance, and summing up all probabilities to be one,there are three constraints must be obeyed by the three next state prices andthree probabilities at each node of tree.1 pu pm pd ,E[X 0 (t k) X 0 (t)] 0 pu h pm 0 pd ( h),E[(X 0 (t k) X 0 (t))2 ] σ 2 k pu h2 pm 0 pd ( h)2 .12(3.1)

By solving Eq. (3.1) we can get the following relations:σ2k,2h2σ2k 1 2 ,hσ2k. 2h2pu pmpu(3.2)Besides, because the tree of X 0 is symmetric distributed, all oddnumbered moments of the trinomial will be zero, as they are for the normal.Therefore, we can set the kurtosis in the tree equal to that of the normal.E[(X 0 (t k) X 0 (t))4 ] 3σ 4 k 2 pu h4 pm 0 pd ( h)4 .(3.3)Applying the relations in Eq. (3.2) into Eq. (3.3) for the probabilitiesyields: h σ 3k,pm 2/3,pu pd 1/6.(3.4)This is the trinomial process approximating the asset price distribution: with probability pu 1/6 h,000,with probability pm 2/3Xt k Xt h,with probability pd 1/6.which implies the process of X αk h,αk,Xt k Xt αk h,with probability pu 1/6with probability pm 2/3with probability pd 1/6.(3.5)The option value at a given asset price and time, V (X, t) is computedfrom the values at the three successor nodes as:V (X, t) exp( rk)[pu V (X αk h, t k) pm V (X αk, t k) pd V (X αk h, t k)].(3.6)Note that for generality Eq. (3.6) allows that the probabilities may varywith h and k, even though in the current case of Eq. (3.5) they are fixed.13

3.3Application of the Adaptive Mesh Modelto Plain Vanilla OptionsFor a European option, nonlinearity error is around the exercise price atexpiration. It turns out that an American option’s nonlinearity error is alsolargely accounted for by the error in the last time step, for the prices thatbracket the strike price. Besides, for an American option there is also anapproximation error with regard to where the early exercise occur. However,by ”smooth pasting” property [16] of the American option value, this kind ofapproximation error does not translate into significant error in valuing optionbecause the values of option price nodes is not highly nonlinear around theearly exercise boundary.A7A6A5A4X XA3XA2hh/4A1kk/4Figure 3.2: An AMM for a put option around exercise price at expiration.14

While there is already a well-known analytic solution by Black and Sholesfor pricing European option, we do not only take an European put optionbut also take an American put option with AMM mechanism applied aroundexercise price at maturity date as examples. Fig. 3.2 illustrate the criticalregion of Adaptive Mesh trinomial tree that we wish to construct to valuea put option. The base coarse lattice, with price and time steps h and k,is represented by heavy lines, and light lines represent the finer mesh withprice step size h/2 and time step size k/4. The finer mesh covers all coarsenodes at time state T k, from which there are both fine-mesh paths thatend up in-the-money and out of-the money at expiration, i.e. A2 , A3 , A4 ,and A5 in this figure. X is the strike price, and X and X are the two dateT coarse-mesh node asset price that bracket the strike price. In finer mesh,X is the highest out of-the money node that branches from A2 whereas X is the lowest in-the-money price node from A2 . Since all branches startingfrom nodes below A1 all end up in-the-money and paths start from nodesabove A6 are all expired at the end, there is no need to fine the mesh.The finer mesh is set up with one-half price size of the previous coarsermesh. To cut the price step size in half with maintaining the relationship inEq. (3.5), the time step price must be set one-quarter of the size of the coarserone. By the isomorphism of AMM, the trinomial tree lattice introduced inFig. 3.2 can cut into any finer level as one wish in the same manner. Thus,if we set the base mesh as level 0, then the finer mesh of level M has pricestep size hM h/2M and time step size kM k/2M .In the traditional trinomial tree model, there are (N 1)2 nodes of pricecomputation in total, where N is the number of price steps. Therefore,cutting the price step in half to reduce the nonlinear error makes N becomequadrupled (h is proportional to k) which implies 16 times computationamount than before. On the other hand, as we can see in Fig. 3.2 adding onelevel of adaptive mesh model to cut the nonlinearity error down only needs40 more nodes of price computation in critical region (9, 11, 13, and 7 fortime states T 3/4k, T 2/4k, T 1/4k, and T ).Fig. 3.3 shows the convergence behavior of an in-the-money American putwhich is priced by Adaptive Mesh Model. The Label of AMM-M means theAMM model of level M. The yellow line represents the Traditional TrinomialApproach, while the blue line is the convergence behavior of Adaptive MeshModel of level 2. Although there is the approximation error contributed byearly exercise, AMM model still can improve the convergence behavior with alittle more calculations in American put options. If we rule out the influenceof early exercise, it comes to the European put option whose convergencebehavior is presented in Fig. 3.4 where we can see that a higher level ofAMM model gives rise to a better convergence rate.15

Put PriceNumber of Time StepsFigure 3.3: The AMM model convergence for at-the-money American put.S 100, X 100, σ 20%, r 10%, dividend yield q 0%, and T 0.5year.16

Put PriceNumber of Time StepsFigure 3.4: The AMM model convergence for at-the-money European put.S 100, X 100, σ 20%, r 10%, dividend yield q 0%, and T 0.5year.17

3.4Extending the AMM Model to DiscreteSingle Barrier OptionsWith only a little modification and extension, the model of Fig. 3.2 can beextended to price discretely monitored barrier options. Fig. 3.5 shows howthis is done in a discretely monitored down-and-out barrier call by an AMMstructure with one level of fine mesh around the barrier. The coarse meshnodes are labelled as A and the finer mesh nodes are labelled as B. As to thesubscript, Aj,k node means the k-th coarse mesh price node at time state j.The lattice before time state j 1 are of the same structure as Fig. 3.2 withonly the exercise price is replaced by the barrier price at time j 1. Latticebetween time states j 1 and j 2 connect the fine mesh lattice back to thecoarse lattice.There are two kinds of B-level nodes at time state j 1. Some are atthe same positions as A-level nodes, and the other are between two coarsenodes. If we directly connect all B-level nodes at time j 1 to all A-levelnodes at j 2. The former can intuitively use trinomial method and thelatter may use quadrinomial branching mechanism such as in [13]. But whenwe want to cut the mesh finer (i.e. add more level to the tree), it seems liketoo complicated under this kind of lattice structure. Hence, the mechanismin Fig. 3.5 is presented with isomorphism of adding any finer meshes. Thecoarse time step is divided into two subperiods. The first subperiod is onefine mesh time step ,and the second is three-quarters of a coarse time step.That is, the first of length k/4 and the second of length 3k/4 in the examplelattice in Fig. 3.5.Branching for the first subperiod is the same as at other B-level nodes.However, it also leads to two kinds of B5, nodes. For those nodes lyingat the same price level of coarse nodes, the trinomial branching method isstraightforward. The node values can be obtained from Eq. (3.6) with a pricestep h and a time step 3k/4. And the branch probabilities of pu pd 1/8and pm 3/4 can be derived from Eq. (3.2) with replacing k with 3k/4in probabilityequations of pu , pm , and pd , and maintaining the relationship h σ 3k (because it has been fixed by coarse mesh). Let k 0 3k/4 Hence,the new trinomial process for those nodes is as follows:Xt k Xt k/4 αk 0 h,αk 0 , αk 0 h,with probability pu 0 1/8with probability pm 0 3/4with probability pd 0 1/8.(3.7)Notice that the kurtosis is no longer matched by the process in Eq. (3.7)because matching the mean, variance, constraining the all probabilities to be18

A j 2,10B5,17A j 1,7B5,16A j 2,9B5,15A j,6B5,14A j 2,8B5,13A j,5B4,11A j 2,7B4,10A j,4A j 2,6A j,3A j 2,5A j,2A j 2,4A j,1A j 2,3BarrierA j 2,2time statej -1jj 1A j 2,1j 2Figure 3.5: An AMM for discrete down-and-out barrier call options.19

one, and maintaining the relationship of h σ 3k have used four degree offreedom while there are only five variables pu , pm , pd , h, and k.For the nodes lying between two coarse node price levels, a quadrinomialbranching mechanism should be applied. For example, we should connectB5,13 to four A-level nodes at time t 2, i.e. Aj 2,6 , Aj 2,7 , Aj 2,8 ,and Aj 2,9with price increments of 3h/2, h/2, h/2, and 3h/2. Matching the mean,variance, and adding four branch probabilities to be one give us the followingthree equations under the condition of h σ 3k.puu pu pd pdd 1,puu (αk 0 3h/2) pu (αk 0 h/2) pd (αk 0 h/2) pdd (αk 0 3h/2) αk 0 ,puu (3h/2)2 pu (h/2)2 pd ( h/2)2 pdd ( 3h/2)2 σ 2 k 0 .(3.8)which can be solved as:puu pdd 0,pu pd 1/2.(3.9)Surprisingly the solution in Eq. (3.9) collapses the quadrinomial branching into binomial one, as follows:00 Xt 1/4 Xt k αk 0 3σh/2,αk 0 σh/2,αk 0 σh/2,αk 0 3σh/2,with probability puu 0with probability pu 1/2with probability pd 1/2with probability pdd 0.The isomorphic structure of the fine mesh allows us to add the nextlayer, with price and time steps hC h/4 and kC k/16, using exactlythe same procedure as described above. Fig. 3.6 illustrates the resultinglattice structure. As we can see from this figure, we don’t care where thebarrier is and merely cut the lattice finer by adding more levels into thestructure around where the payoff function value is significantly nonlinear.Hence, the ”barrier-too-close” problem does not exists under the AMM latticemechanism.We can see the co

Barrier Options 2.1 Barrier Option Basics A barrier option is a kind of path-dependent options that comes into exis-tence or is terminated depending on whether the underlying asset's price S reaches a certain price level H called "barrier". A knock-out option ceases to exist if the underlying asset reaches the barrier, whereas a knock-in .