Tree Methods Jérôme Lelong, Antonino Zanette

Tree methods Jérôme Lelong, Antonino Zanette

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Jérôme Lelong, Antonino Zanette. Tree methods. Rama Cont. Encyclopedia of Quantitative Finance, John Wiley & Sons, Ltd., 7 p., 2010, 10.1002/9780470061602.eqf12017. hal-00776713

HAL Id: hal-00776713 https://hal.archives-ouvertes.fr/hal-00776713 Submitted on 16 Jan 2013

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Tree methods in finance complex models. Binomial (see eqf05-006) and trinomial trees may also be constructed to ap- Tree methods are amongst the most popular proximate the stochastic differential equation numerical methods to price financial derivatives. governing the short rate [21, 28] or the inten- They are mathematically speaking easy to un- sity of default [33] permitting hereby to obtain derstand and they do not require severe impl- the price of respectively interest rate derivatives mentation skills to obtain algorithms to price fi- or credit derivatives. Implied binomial trees are nancial derivatives. Tree methods basically con- generalizations of the standard tree methods, sist in approximating the diffusion process mod- which enable us to construct trees consistent eling the underlying asset price by a discrete ran- with the market price of plain vanilla options dom walk. In fact, the price of a European op- used to price more exotic options (see Derman tion of maturity T can always be written as an and Kani [11] and Dupire [14]). expectation either of the form For the sake of simplicity, consider a market model where the evolution of the risky asset is E rT (e− ψ(ST )) driven by the Black-Scholes stochastic differential equation in the case of vanilla options or of the form

rT dS S rdt σdW S s E(e− ψ(S 0 t T )) t = t( + t) 0 = 0 (1) t ≤ ≤ in the case of exotic options, where (St t 0) is in which (Wt)0 t T is a standard Brownian mo- ≤ ≤ a stochastic process describing the evolution≥ of tion (under the so called risk neutral probabil- the stock price, ψ is the payoff function and r is ity measure) and the positive constant σ is the the instantaneous interest rate. The basic idea volatility of the risky asset. of tree methods is to approximate the stochastic The seminal work of Cox-Ross-Rubinstein process (St t 0) by a discrete Markov chain [10] (denoted CRR hereafter) paves the road to ¯N ≥ (Sn n 0), such that the use of tree methods in financial applications ≥ and many variants of the CRR model have been rT E(e− ψ(S 0 t T )) introduced to improve the quality of the approx- t ≤ ≤ ≈ rT N imation when pricing plain vanilla or exotic op- E(e− ψ(S¯ 0 n N)) n ≤ ≤ tions. for N large enough ( is used to remind the reader that the equality≈ is only guarantied for N = ). To ensure the quality of the approxi- Plain Vanilla options mation,∞ we are interested in a particular notion of convergence called convergence in distribution The multiplicative binomial CRR model [10] (weak convergence) of discrete Markov chains to is interesting on its own as a basic discrete- continuous stochastic processes. It is interesting time model for the underlying asset of a finan- to note that tree methods can be also regarded cial derivative, since it converges to a log-normal as a particular case of explicit finite difference diffusion process under appropriate conditions. algorithms. One of its most attractive features is the ease of Tree methods provide natural algorithms to implementation to price plain vanilla options by price both European and American options backward induction. when the risky asset is modeled by a geometric Let N denote the number of steps of the T Brownian motion (see [26] for an introduction tree and ∆T = N the corresponding time step. on how to use tree methods in financial prob- The log-normal diffusion process (Sn∆T )0 n N lems). When considering more complex mod- is approximated by the CRR binomial process≤ ≤ ¯N n els — such as models with jumps or stochastic (Sn = s0 Yj)0 n N where the random j=1 ≤ ≤ volatility models — the use of tree methods is variables Y1 YN are independent and iden- much more difficult; analytic approaches like fi- tically distributed with values in d u (u is nite difference (see eqf12-003) or finite element called the up factor and d the down{ factor)} with (see eqf12-007) methods are usually preferred, pu = P(Yn = u) and pd = P(Yn = d). The Monte Carlo methods are also widely used for dynamics of the binomial tree (see Figure 1) is

1 given by the following Markov chain programming equations

+ S¯N u with probability p vN (N x) = (K x) S¯N = n u − n+1 S¯N d with probability p v (n x) = max ψ(x) n d  N  r∆T  e− puvN (n + 1 xu) Kushner [23] proved that the local consistency  conditions given by Equation (2) — that is the vN (n + 1 xd)  matching at the first order in ∆T of the first  where ψ(x) = (K x)+. Note that the algo- and second moments of the logarithmic incre- − ments of the approximating chain with those of rithm requires the comparison between the in- the continuous-time limit — grant the conver- trinsic value and the continuation value. When gence in distribution. considering European options, ψ 0. The initial price of a Put option≡ in the ¯N Black-Scholes model can be approximated by E Sn+1 E S(n+1)∆T log ¯N = log + o (∆T ) Sn Sn∆T vN (0 s0). The initial delta, which is the quan- ¯N (2)  2 Sn+1 2 S(n+1)∆T E log N = E log + o (∆T ) tity of risky asset in the replicating portfo- S¯ Sn T  n ∆ lio on the first time step in the CRR model, vN (1s u) vN (1s d)  is approximated by 0 − 0 . Note This first order matching condition rewrites s0(u d) that in order to obtain the− approximated 2 price and delta, one only needs to compute p log u + (1 p ) log d = (r σ )∆T u u 2 v (n s ukdn k) 0 k n by backward in- 2 − 2 −2 (3) N 0 − pu log u + (1 pu) log d = σ ∆T duction on n from N≤ to≤ 0. Figure 2 gives an − example of backward computation of the price The usual CRR tree corresponds to the choice of an American Put option using N = 4 time 1 σ√∆T u = d = e , which leads to pu = steps. The complexity of the algorithm is of or- r∆T σ√∆T 2 2 e e− 1 r σ 2 32 der N , more precisely the function v has to − √ T T N σ√∆T− σ√∆T = 2 + 2σ ∆ + (∆ ). (N+1)(N+2) e e− O be evaluated at nodes. When− ∆T is small enough (i.e. for N large), 2 the above value of pu belongs to ]0 1[. For this choice of u, d and p , the difference between u 0 both sides of each equality in (3) is of order 0 2 0 (∆T ) . This is sufficient to ensure the conver- 0 1438 0 gence to the Black-Scholes model when N tends 3700 0 4543 to infinity. 9516 0516 18127 18127 4 s0u 3 25918 s0u 32968

2 s0u 3 s0u d s0u s0u

pu s0 2 2 s0 s0u d s0d Figure 2: Backward induction for a CRR tree pd

s0d with N = 4 for an American Put option with 3 s0ud 2 s0d parameters s0 = K = 100, r = 01, σ = 02,

3 S0d T = 1. 4 s0d For the computation of the delta, Pelsser and Figure 1: CRR tree. Vorst [29] suggested to enlarge the original tree by adding two new initial nodes generated by an As (S¯n)n deﬁned by the CRR model is Marko- extended two period back tree (dashed lines in vian, the price at time n 0 N of an Figure 2). To achieve the convergence in distri- ∈ { } American Put option (see eqf05-007) in the CRR bution, many other choices for u, d, pu and pd model with maturity T and strike K can be writ- may be done, leading to as many other Markov ten as v(n S¯n) where the function v(n x) can be Chains. We may choose other 2-point schemes computed by the following backward dynamic such as a random walk approximation of the

2 Brownian motion, or 3-point schemes (trinomial Discussion on the conver- trees) or more general p-point schemes. The ran- gence dom walk approximation of the Brownian motion (Zn+1 = Zn + Un+1 with (Ui)i independent Over the last years, signiﬁcant advances have P and identically distributed with (Ui = 1) = been made in understanding the convergence be- P(U = 1) = 1 ) can be used as long as S is i − 2 T havior of tree methods for option pricing and given by hedging (see [24, 25, 27, 13]). As noticed in

σ2 [15, 12], there are two sources of error in tree T r 2 T +σW ST = s0e − methods for Call (see eqf07-001) or Put (see eqf07-002) options: the first one (the distribu- 1 This leads to pu = 2 and tion error) ensues from the approximation of a continuous distribution by a discrete one, while 2 r σ ∆T +σ√∆T u = e − 2 the second one (the non linearity error) stems σ2 from the interplay between the strike and the r ∆T σ√∆T d = e − 2 − grid nodes at the final time step. Because of the non linearity error, the convergence is slow ex- The most popular trinomial tree has been in- CRR cept for at-the-money options. Let PN and troduced by Kamrad and Ritchken [22] who P BS denote the initial price of the European have chosen to approximate (Sn∆T )0 n N by a Put option with maturity T and strike K re- ¯ ≤ ≤ symmetric 3-point Markov chain (Sn)0 n N spectively in the CRR tree (with N steps) and ≤ ≤ in the Black-Scholes model. Using the Call-Put ¯ Snu with probability pu parity relationship in both models and the re- ¯ ¯ Sn+1 = Sn with probability pm sults given for the Call option in [13], one finds  ¯  Snd with probability pd

The convergence is ensured as soon as the first rT d2 Ke− 2 2 S T CRR BS 2 two moment matching condition on log ∆ PN = P e− s0 − N π is satisfied. With u = eλσ√∆T and d = 1 , this 1 u κ (κ 1)σ√T + D + condition leads to N N − 1 O N 32 2 σ s σ2 r √∆T log( 0 )+(r )T 1 2 1 where d = K − 2 , κ denotes the p = + − p = 1 2 σ√T N u 2 m 2 K 2λ 2λσ − λ log( s ) 0 N N 2 fractional part of and D is a r σ √ T 2σ T − 2 1 1 2 ∆ p = − constant. For at-the-money options (i.e. K = d 2λ2 − 2λσ 1 s0), κN = 0 for N even and κ = 2 for N CRR BS odd, hence the difference (PN P )N is The parameter λ — the stretch parameter — 1 − appears as a free parameter of the geometry of an alternating sequence. Figure 3 shows that N P CRR the tree, which can be tuned to improve the con- for even N gives an upper estimate of P BS N P CRR vergence. The value λ 122474 corresponding , whereas for odd N gives a lower P to p = 1 is reported≈ to be a good choice for estimate. The monotonicity of ( 2N+1)N and m 3 P at-the-money plain vanilla options. Note that ( 2N )N for at-the-money options enables us to 1 use a Richardson extrapolation (i.e. consider choosing u = d is essential to avoid a complex- CRR CRR CRR CRR 2P4N P2N or 2P4N+1 P2N+1) to make ity explosion. In this case, the complexity is still − 1 − of order N 2, but this time (N + 1)2 evaluations the terms of order N disappear. For not at- of the function v are required. The value λ = 1 the-money options, Figure 4 shows that the se- N CRR CRR corresponds to the CRR tree. quences (P2N+1)N and (P2N )N are not mono- Note that complexity is intimately related to tonic and present an oscillating behavior. In this the quality of the approximation. Therefore, one context, a naive Richardson extrapolation per- should always try to balance the additional com- forms badly. putational cost with the improvement of the con- 1The graphics have been generated using PREMIA vergence it brings out. software [30].

lated (with Richardson extrapolation), intro-

3.78 + odd number of steps duced by Gaudenzi and Pressacco [19] tries to × even number of s teps Black Scholes price recover the regularity of the sequences giving the

3.77

× CRR price of the European at-the-money op-

× × × ×

× ×

× × × 3.76 × tions. The logic of the BI approach is to create a set of computational options, each one with

3.75 a computational strike lying exactly on a ﬁnal

3.74 +

+ node of the tree. The value of the option with

+ +

+ + the contractual strike is then obtained by in- 3.73 terpolating the values of the computational op- 100 110 120 130 140 150 160 170 180 190 200 tions. Let us finally remark that similar techniques have been developed in numerical analysis for Figure 3: Convergence for an at-the-money Put PDEs associated to option pricing problems (see option with parameters s0 = K = 100, r = 01, [17]). In the case of non-smooth initial condi- σ = 02, T = 1 tions, to get good convergence of finite element and finite difference methods, there should always be a node at the strike and the payoff may

1.45 + odd number of steps have to be smoothed (see [31] for a classical ref- × even number of s teps Black Scholes price erence).

1.44 In the American option case, a new source of er-

× × × ×

× +

× ×

+ + + +

+ +

+ ×

1.43 +

× +

+ +

+ ror arises compared to the European case: the +

× ×

+ +

+ ×

× +

+ ×

× ×

+ × × ×

× loss of opportunity to early exercise in any in- + ×

+ ×

1.42 × +

+ ×

+ + terval between two discrete times of the tree.

1.41

1.40 Exotic options

100 110 120 130 140 150 160 170 180 190 200 The classical CRR approach may be trouble- some when applied to barrier options (see eqf07- Figure 4: Convergence for a not at-the-money 003) because the convergence is very slow in Put option with parameters s0 = 100, K = 90, comparison with plain vanilla options. The rea- r = 01, σ = 02, T = 1 son is quite obvious: let L be the barrier and nL denote the index such that

nL nL+1 Several tree methods [6, 15, 19] try to deal s0d L > s0d with the non linearity error at maturity repro- ≥ ducing in some sense an at-the-money situation. Then, the binomial algorithm yields the same nL The BBS (Binomial Black-Scholes) method in- result for any value of the barrier between s0d nL+1 troduced by Broadie and Detemple [6] replaces and s0d , while the limiting value changes at each node of the last but one time step be- for every barrier L. fore maturity, the continuation value with the Several different approaches have been pro- Black-Scholes European one [3]. A two point posed to overcome this problem. Richardson extrapolation aiming at improving Boyle and Lau [5] choose the number of time the convergence leads to the BBSR method. steps in order to have a layer of nodes of the Adaptive Mesh Model (AMM) is a trinomial tree as close as possible to the barrier. Ritchken based method introduced by Figlewski and Gao [32] noted that the trinomial method is more [15]. By taking into account that the non lin- suitable than the binomial one. The main idea is earity error at maturity only affects the node to choose the stretch parameter λ such that the nearest to the strike, AMM resorts to thicken- barrier is exactly hit. Later, Cheuk and Vorst ing the trinomial grid only around the strike and [9] presented a modification of the trinomial tree only at maturity time. (based on a change of the geometry of the tree) The BI(R) method, that is Binomial Interpo- which enables to set a layer of the nodes exactly

4 on the barrier for any choice of the number of ternative and more efficient approach to evalu- time steps. Numerical interpolation techniques ate the pure binomial prices associated with the have also been provided by Derman et al. [12]. path-dependent options. Moreover, because the In the case of Asian options (see eqf05-008 and piecewise linear function representing the price eqf12-013) with arithmetic average, the CRR is convex it is easy to obtain upper and lower method is not efficient since the number of pos- bounds of the price. sible averages increases exponentially with the For the rainbow options (see eqf07-013), ex- number of the steps of the tree. For this reason, tensions of the binomial approach for pricing Hull and White [20] and in a similar way Bar- American options on two or more stocks have raquand and Pudet [2] proposed more feasible been made by Boyle, Evnine and Gibbs [4], and approaches. The main idea of their procedure is Kamrad and Ritchken [22]. In higher dimen- to restrict the possible arithmetic averages to a sional problems (say, dimension greater than 3) set of representative values. These values are se- the straightforward application of tree methods lected in order to cover all the possible values of fails because of the so-called “curse of dimen- the averages reachable at each node of the tree. sion”: the computational cost and the memory The price is then computed by a backward in- requirement increase exponentially with the di- duction procedure, whereas the prices associated mension of the problem. to the averages outside of representative value set are obtained by some suitable interpolation methods. References These techniques drastically reduce the computation time compared to the pure binomial [1] Babbs S. (2000). Binomial valuation of tree, however they present some drawbacks (con- lookback options. Journal of Economic Dy- vergence and numerical accuracy) as observed namics and Control 24, 1499-1525. by Forsyth et al. [16]. Chalasani et al. [7, 8] proposed a completely different approach to ob- [2] Barraquand, J. & Pudet, T. (1996). Pric- tain precise upper and lower bounds on the pure ing of American path-dependent contingent binomial price of Asian options. This algorithm claims. Mathematical Finance 6, 17-51. significantly increases the precision of the estimates but induces a different problem: the im- [3] Black, F. & Scholes, M. (1973). The pricing plementation requires a lot of memory compared of options and corporate liabilities. Journal to the previous methods. of Political Economy 81, 637-654 . In the case of lookback options (see eqf07- 007), the complexity of the pure binomial algo- [4] Boyle, P.P., Evnine, J. & Gibbs, S. (1989). rithm is of order O(N 3) and the methods pro- Numerical evaluation of multivariate con- posed in [20, 2] do not improve the efficiency. tingent claims. Review of Financial Studies, Babbs [1] gave a very efficient and accurate solu- 2, 241-250. tion to the problem for American floating strike lookback options by using a procedure of com- [5] Boyle, P.P. & Lau, S.H. (1994). Bumping plexity of order O(N 2). He proposed a change of up against the barrier with the binomial “numeraire” approach, which cannot be applied method. Journal of Derivatives 1, 6-14. in the fixed strike case. Gaudenzi et al [18] introduced the singular [6] Broadie, M. & Detemple, J. (1996). Ameri- point method to price American path-dependent can option valuation: new bounds, approxi- options. The main idea is to give a continuous mations and a comparison of existing meth- representation of the option price function as ods. The Review of Financial Studies 9-4, a piecewise linear convex function of the path- 1221-1250. dependent variable. These functions are char- acterized only by a set of points named “sin- [7] Chalasani, P., Jha, S., Egriboyun, F. & gular points”. Such functions can be evalu- Varikooty, A. (1999). A refined binomial ated by backward induction in a straightforward lattice for pricing American Asian options. manner. Hence, this method provides an al- Review of Derivatives Research 3, 85-105.

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[16] Forsyth, P.A., Vetzal, K.R., & Zvan, R. [27] Walsh, J.B. (2003). The rate of convergence (2002). Convergence of numerical meth- of the binomial tree scheme. Finance and ods for valuing path-dependent options us- Stochastics 7, 337-361. ing interpolation. Review of Derivatives Re- search 5, 273-314. [28] Li, A., Ritchken, P. & Sankarasubrama- nian, L. (1995). Lattice methods for pricing [17] Giles, M. B. & Carter, R. (2006) Conver- American interest rate claims. The Journal gence analysis of Crank-Nicolson and Ran- of Finance 2, 719-737. nacher time-marching. Journal of Compu- tational Finance 4-9, 89-112. [29] Pelsser, A. & Vorst, T. (1994). The binomial model and the greeks. Journal of [18] Gaudenzi, M., Lepellere, M.A. & Zanette Derivatives 1-3, No. 3, 45-49. A. (2007). The singular points binomial method for pricing American path- [30] PREMIA (2008). An Option Pricer dependent options. Working paper Finance Project. MathFi, INRIA - ENPC. Department University of Udine 1, 1-17. http://www.premia.fr [19] Gaudenzi, M. & Pressacco, F. (2003). An [31] Rannacher, A. (1984). Finite element so- eﬃcient binomial method for pricing Amer- lution of diﬀusion problems with irregular ican put options. Decisions in Economics data. Numerische Mathematik, 43-2, 309- and Finance 4-1, 1-17. 327.

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[33] Schonbucher, P.J. (2002). A tree implementation of a credit spread model for credit derivatives. Journal of Computational Fi- nance 6-2, 1-38.

J´erˆome Lelong, Projet MathFi, INRIA Rocquencourt, Domaine de Voluceau, France. [email protected].

Antonino Zanette, Dipartimento di Finanza dell’Impresa e dei Mercati Finanziari, Udine University, Italy. [email protected].