1. Trinomial model. This chapter discusses the implementation of trinomial probability trees for pricing

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1 TRINOMIAL TREES AND FINITE-DIFFERENCE SCHEMES 1. Trinomial model This chapter discusses the implementation of trinomial probability trees for pricing derivative securities. These models have a lot more exibility than binomial trees. As shown (early on) by Cox, Ross Rubinstein, binomial models for stock equity derivatives are dynamically complete. This may be one of the reasons for their popularity among practitioners: not only can they be used for pricingbutthe price actually represents the cost of a replicating strategy. However, a drawback of binomial trees is that they cannot be used to model processes with local means variances which dependon the value of the underlying index. The reason is that there are not enough parameters to adjust or degrees of freedom. Therefore, binomial models are only useful to model stochastic processes with constant or, at best, with time-dependent parameters that do not vary with the underlying index or other factors. In certain situations, we would prefer to describe the underlying index by an Ito process with parameters that depend on the index itself or other factors. This is where trinomial trees become a useful numerical tool. Trinomial trees permit us to specify dierent local volatilities drifts at each node of a recombining tree. This means that we can model Ito processes of the form ds S = (S t) dz + (S t) dt (1) in which the local parameters are functions of the asset price time, with a structure that has O(N )nodes if the number of periods is N. Theelementary structure of the trinomial tree is shown below. S U S S M 1 S D

2 This structure is reproduced at each time to generate a \tree" in which each vertex has three ospring. A necessary sucient condition for the tree to recombine, i.e to have 3n nodes at time n, isthat the parameters U M D satisfy U D = M () This constraint the stochastic dierential equation (1) lead us to parametrize the model as follows: U = e p dt + dt M = e dt D = e p dt + dt : () Here are constants that will be determined dt represents a (small) interval of time between successive shocks, measured in years. We shall assign conditional probabilities to eachofthe three outcomes, P U P M P D. Let us denote by dt dt the mean variance of the logarithm of the price shock over a period. We have dt = p dt ( P U P D ) + dt (3) dt = dt ( P U + P D ) ( dt) (4) We will always assume that dt 1 that the parameters are O(1). Under these conditions, we are only interested in keeping terms of order dt in the calculation of the mean variance. 1 In particular, we will replace equation (4) by the simpler equation dt = dt ( P U + P D ) (5) Equations (4) (5) can be used to calculate the probabilities P U P M P D in terms of the mean the variance of the logarithm of the shock over a period. In fact, we can easily deduce from (3) (5) that 1 Lower-order terms of order dt 3= give anegligible contribution to the sumofthe localvariances.

3 P U = 1 + (6) P D = 1 (7) P M = 1 : (8) Financial considerations typically require adjusting the drift(s t) ofthe process in (1) to satisfy no-arbitrage conditions. For instance, in the world of currency derivatives, = r d r f =the dierence between the domestic foreign interest rates. Thus, in practice, we would like to treat (i.e. the annualized expected return) as the input parameter rather than. Of course, we know from Ito calculus that = 1 : (9) Let us rewrite equations (6) (7) in terms of. Theresultis: P U = 1 p! 1 dt + (10) P D = 1 p! 1 + dt (11). Stability analysis We would like to derive conditions that must be satised so that the three probabilities calculated above are non-negative. Introducing the parameters p q (1) 3

4 the equations for the probabilities become P U = p 1 p dt! + q p dt (13) P D = p 1 + p dt! q p dt (14) P M = 1 p: (15) These expressions are useful to determine whether P U P M P D are are nonnegative less than one. In fact, we conclude from (15) that wemust have Moreover, (from (13) with q = 0), 0 p 1 : (16) < p dt (17) must hold as well. A further condition, which guarantees that P U P M P D are positive, is that p > jqj p dt p dt : (18) 3. Calibration To calibrate our tree to given volatility drift \surfaces" (S t) consider the four numbers (S t), we min inf (S t) max sup (S t) 4

5 min inf (S t) max sup (S t) : We shall assume that these numbers are uniformly bounded. Regarding the volatility parameter, we assume that min > 0. This last requirement is not necessary in fact, it is possible to model using the trinomial tree processes with a local volatility whichvanishes at certain levels of spot prices. We begin with the calibration of volatility. Since equation (5) tells us that = p (19) we must have in view of the restriction (16). We shall therefore make the choice = max (0) which corresponds to the minimal value of compatible with the volatility range that we are trying tomodel. Of course, any choice of greater than max is also possible. 3 It follows from equation (19) that, by varying the parameter p in the interval min max p 1 (1) we can achieve anyvalue of in the desired range. Let us turn next to the drift. Equation (1) suggests that should be chosen in the range This implies, in particular, that min max : jqj max max min : () Notice that the bound is tighter when we set = max + min (3) This property willprove useful later in modeling the impact of transaction costs onhedging. 3 We note, however, that the greater, the smaller we have tochoose dt in order to ensurethat (17) holds. 5

6 when it becomes jqj max min : (4) Combining equations (1) (4), we conclude that the stability condition (18) is satised if we have or, more concisely, if dt < min max 4 max > max p dt max p dt max min max min min + = max min : (5) This condition is, in general, more restrictive than the one implied by (17), namely dt < 4 max : (6) It reduces to the latter condition when the drift is constant ( = 0). (In the latter case, it is possible to implement a trinomial tree with volatility that vanishes at certain nodes without generating negative probabilities.) To construct a trinomial tree with transition probabilities corresponding to the diusion process (1), we calculate the parameters min max min max dene the parameters accordingly. We then select a time interval dt which satises the stability condition (5). Once this is done, we obtain a trinomial tree, or lattice, that describes the values of the index S in discrete increments. These values can be denoted by S j n,where n represents the time variable j the \height" on the tree. Next, we discretize the drift volatility surfaces, setting Finally, weset j n (Sj n t n) j n (Sj n t n) : p j n = j n q j n = j n dene the probabilities at the node(n j) according toequations (13), (14) (15), substituting p j n for p q j n for q. Inthis way, wehave specied a discrete approximation for the diusion process (1) on a trinomial tree. 6

7 4. Finite-dierence scheme for the Black-Scholes PDE with prescribed volatility drift surfaces Pricing contingent claims on the trinomial tree by discounting expected cash-ows leads to a recursion relation for the value of the claim at the dierent nodes of the tree. This relation is analogous to the Cox-Ross-Rubinstein \backward induction" method for the binomial model, but we can incorporate volatility drift \surfaces". The discrete pricing equation is V j n = F j n + e rj n dt f 1 pj n 1 p dt! V j p! n+1 pj n 1+ dt V j1 n+1 +(1 p j n ) V j n qj n V j+1 n+1 V j1 n+1 g (7) where F j n represents a cash-ow due at time t n if the index value is S j n. This equation can be viewed as a nite-dierence scheme for solving the Black-Scholes equation. The construction of the previous sections guarantees the stability convergence of the scheme as dt! 0. Of course, dynamic programming equations derived from (7) can be used for pricing American options other contingent claims which involve stopping times. In particular, the trinomial tree provides a more accurate alternative than the binomial model for the pricing of barrier options because the barrier can be made to coincide with a particular level in the tree. This eliminates to some extent numerical roundo error. Other applications involve dynamical programming equations that are used in worstcase scenario analyis of portfolios (Avellaneda, Levy Paras (1994) Avellaneda Paras (1995)), which will be discussed in other chapters. 7