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J Futures Markets. 2019;1 2019 Wiley Periodicals, Inc. | 1 Received: 4 June 2019 | Accepted: 22 November 2019 DOI: 10.1002/fut.22080 RESEARCH ARTICLE Efficient trinomial trees for localvolatility models in pricing doublebarrier options U Hou Lok1|YuhDauh Lyuu2 1College of Business, National Taipei University of Business, Taipei, Taiwan 2Department of Finance and Department of Computer Science 1973) prices options by assuming constant volatility for the stock price. Closedform pricing formulas are then available for vanilla European options. However, the constantvolatility assumption is not supported by empirical data (Rubinstein, 1994). Instead, options with different times to maturity or strike prices have different volatilities when inverted by the BS formula. These volatilities are called implied volatilities, which form the implied volatility surface. This surface typically exhibits complex dependency on both the maturity and strike price of an option, a phenomenon known as the volatility smile or simply the smile. Models that can reproduce the smile are desirable because exotic options can then be priced and hedged consistently against the benchmark options. The localvolatility (LV) model is one such model. The LV model makes the local volatility (also called the instantaneous volatility or the deterministic volatility) depend on the stock price and time (Derman Dupire, 1994; Rubinstein, 1994). This assumption makes the LV model preferencefree, like the BS model. As a result, the market is complete and options can be valued by the no arbitrage argument without the need to estimate the market price of risk, which is difficult (Fengler, 2005; Rebonato, 2004). Indeed, the LV model is the only smileconsistent model that is complete (Bennett, 2014). An option pricing model like the LV model can be used in two ways (Hull, 2012; Lyuu, 2002). (a) A stochastic process for the underlying asset is assumed and the options are priced thereof. (b) Option prices are used to infer the stochastic process that reproduces the smile as closely as possible (a step called calibration), after which other options can be priced consistently. Empirical studies of LV models can be found in Crpey (2004), Dumas, Fleming, and Whaley (1998), Lim and Zhi (2002), Linaras and Skiadopoulos (2005). Applications of the LV model can be found in Derman, Miller, and Park (2016). Options without closedform pricing formulas require numerical methods, such as trees. A calibrated tree converges to the underlying continuoustime model as the number of time steps increases (Duffie, 2001). Trees have been widely used to price options because (a) trees can handle the earlyexercise feature of American options and (b) trees are reasonably efficient if the options are not strongly pathdependent. The bestknown tree is the CoxRossRubinstein (CRR) tree of Cox, Ross, and Rubinstein (1979). Many alternative trees have also been proposed (Chance, 2008). The continuous local volatilities form the LV surface. A tree that fits the LV surface is called an LV tree. Common LV trees are either binomial or trinomial. Their up, middle (if applicable), and down moves as well as the associated transition probabilities are such that the local volatilities on the tree match the LV surfaces. A tree is valid if (a) its transition probabilities lie between 0 and 1, and (b) the stock prices are all positive. A few LV trees have been studied. Amin (1993) presents trees for LV surfaces that depend only on time. The LV surface of the constant elasticity of variance (CEV) model, on the other hand, depends only on the stock price in the power form (Cox, 1975). Its trees are given by Lu and Hsu (2005) and Nelson and Ramaswamy (1990). Kamp (2009) proposes trees for LV surfaces which depend on both stock price and time, but they are not guaranteed to be valid. Guthries (2011) tree maintains constant up and down moves as well as constant transition probabilities by varying the durations of the time steps. However, the tree may not match the LV surface. Lok and Lyuus (2017) provably valid binomial tree matches only separable LV surfaces. An implied tree aims to recover the unknown LV surface from option prices (equivalently, the implied volatility surface). The option prices are supposed to be generated by some LV model. This tree is thus the result of model calibration, an inverse problem, which is in general illposed (Atkinson, 1989). Once the implied tree is in place, it can be used in pricing. To avoid ambiguities, an implied tree will not be called an (implied) LV tree in this paper. Many implied trees have been proposed. Rubinstein (1994) suggests a binomial implied tree from options with the same maturity. The probabilities are determined by nonlinear optimization. But this tree can only deal with options with the same maturity, and it may not match the implied volatilities at maturity. Jackwerth (1997) relaxes Rubinsteins pathindependence assumption. This greatly increases the degree of freedom and allows the tree to tackle the whole implied volatility surface. Still, the tree may not match the implied volatility surface. Derman and Kanis (1994) alternative impliedtree methodology is widely adopted. Option prices with different strike prices and times to maturity are used to determine both the geometry and the transition probabilities of the binomial implied tree. But their binomial tree (DK henceforth) contains invalid transition probabilities and may not fit the implied volatility surface exactly. To address this issue, stock prices that violate the noarbitrage principle are replaced via ad hoc procedures. But the resulting tree may still fail to fit the surface exactly. Derman, Kani, and Chriss (1996) propose a trinomial implied tree which has more degrees of freedom. However, negative probabilities remain an issue. Barle and Cakici (1999) improve the DK tree with a better nodeplacement strategy to reduce, but not eliminate, the occurrences of invalid probabilities. Exotic options are used in hedging, speculation, or, much less often, model calibration (Ayache, Henrotte, Nassar, Rubinstein Dai, Liu, Tavella thuss = 00 . The numbers u m, and d denote the step sizes of an up move, a middle move, and a down move of the logarithmic return, respectively, whereumd. In one time step, the logarithmic returnsican make an up move tosu+ i with probabilitypn i , (u), a middle move to sm+ i with probabilitypn i , (m), and a down move to sd+ i with probabilitypn i , (d). These probabilities must lie between 0 and 1 besides satisfying ppp+= 1. n in in i, (u) , (m) , (d) (2) The trinomial LV tree for the logarithmic return can now be described. Choose a flat movement for the middle move (hencem = 0) and imposeud+= 0(henceud 0 ). The magnitude of u will be set tot UB . (A similar scheme appears in Haahtela, 2010 albeit in a different context.) In summary, ut= , UB(3) m = 0,(4) dt= . UB(5) Note that u is the grid spacing between adjacent horizontal grid lines. Place the root node at the grid point with time t = 0 0 and logarithmic returns0. The root nodes three successor nodes after one time step have logarithmic returns ssu s=+, 100, ands sd=+ 10 . The procedure is repeated for those successor nodes and so on until the maturity is reached. The trinomial LV tree is illustrated in Figure 1. We now derive the transition probabilities. The following equations match the mean and variance of the trinomial tree to those of the continuoustime process (1): FIGURE 1A trinomial localvolatility tree 4 | LOKANDLYUU pupdr S t t+= ( ,) 2 , n in i in , (u) , (d) 2 (6) pupdpupdS tt+ (+) =( ,) . n in in i u n i in , (u)2 , (d)2 , ( ) , (d)22 (7) The unique transition probabilities that solve Equations (2), (6), and (7) are p S t rS t t rS t t= ( ,) 2 + ( ,)/2 2 + ( ,)/2 2 , n i ininin , (u) 2 UB 2 2 UB 22 UB 2 (8) p S t rS t t rS t t= ( ,) 2 ( ,)/2 2 + ( ,)/2 2 , n i ininin , (d) 2 UB 2 2 UB 22 UB 2 (9) p S t rS t t= 1 ( ,) ( ,)/2 . n i inin , (m) 2 UB 2 22 UB 2 (10) Let 0 LB be a lower bound on S t( ,) in. As S t/ 0 for= 1, 2, . All the grid points having been determined, it only remains to lay the remaining tree nodes on them. We start by connecting the root node A to select three grid points in the next time step. They are to be called nodes B C, and D. We proceed to determinet0and the placements of nodesB C D,so that the transition probabilities are valid while matching the mean and variance of the stock price process (1). Define r St t Stt = (,) 2 , =(,) , A A 0 2 0 0 0 22 00 which are, respectively, the mean and variance of the logarithmic return from the root node A. Fixt0to be the duration from timet0to the first vertical grid line that lies within St t St t 1 4(,) , 3 4(,) . AA UB 0 2 UB 0 2 (13) Such a vertical grid line exists because the above interval has a width of St t St tt 3 4 1 4(,) = 1 2(,) . AA UB 0 2 UB 0 2 Name the timet1. As u is the spatial grid spacing, there must exist a unique grid point at timet1whose logarithmic return lies within the interval u u 2 ,+ 2 . 00 FIGURE 2A trinomial localvolatility tree for doublebarrier options with high barrier h and low barrier 6 | LOKANDLYUU Designate that grid point as node C, the destination of the middle move from node A. The numbermAdenotes the step size of this middle move; so the middle move of the logarithmic returnsAlands atsm+ AA. By design,mAis nearest to0among all the logarithmic returns of grid points at timet1. Place node B (node D) on the grid point that is, one grid line above (below, respectively) node C. The logarithmic returns at nodes B and D are thusmumu+and AA , respectively. We now derive the transition probabilities in the first time step. Define m u u =, =+, =. A 0 Note thatuu(1/2), (1/2) and. Letpp, AA0, (u) 0, (m), and p A0, (d) be the transition probabilities to nodesB C, and D, respectively. They can be solved by ppp+= 0, AAA0, (u) 0, (m) 0, (d) (14) ppp+= , AAA0, (u)2 0, (m)2 0, (d)2 0 2 (15) ppp+= 1. AAA0, (u) 0, (m) 0, (d) (16) Equations (14) and (15) match the first two moments of the trinomial LV tree to those of the stock price process (1), whereas Equation (16) ensures that the probabilities sum to one. The above equations have the following solution: p = + ( ) ( )( )( ) , A0, (u)0 2 (17) p = + () ( )( )( ) , A0, (m)0 2 (18) p = + ( ) ( )( )( ) . A0, (d)0 2 (19) The above transition probabilities are valid (see Appendix A). Beyond the tree nodesB C, and D, all tree nodes are placed by the procedure described in Section 3. Because S t sup( , ) StT UB ,0 +by inequality (11), the transition probabilities calculated by Equations (8)(10) are valid for sufficiently smallt. Thus the trinomial LV tree is valid fornlarge enough to maketsufficiently small. Moreover, the tree is compact with onlyN2nodes. For doublebarrier knockout options, the size of the tree is even smaller as the tree is truncated by the two barriers. 5|TRINOMIAL IMPLIED TREE FROM IMPLIED VOLATILITY SURFACE Section 3 constructed a trinomial LV tree from an LV surface. In this section, the process is reversed: The implied volatility surface is used to construct a trinomial implied tree. This implied tree should approximate the LV surface that gives rise to the implied volatility surface. The procedure is based on Derman et al. (1996). An alternative to obtaining the LV surface is via Dupires formula (1994). 5.1|Trinomial implied tree by forward induction The nodes of the trinomial implied tree are placed on the grid with the nodeplacement strategy in Section 3. A numerical value for S t sup( , ) StT UB ,0 +is selected. Again, it is not important to have S tsup( , ) StT,0 +as long as its finitude is assured: One can always determine an appropriateUBby the bisection method. LOKANDLYUU | 7 Call prices of various strikes and maturities are used to calculate the transition probabilities in the upper half of the tree, while put prices are used to calculate the transition probabilities in the lower half of the tree. This is adopted merely to be consistent with market practices (Derman thus interpolation, extrapolation, and even smoothing are needed to form the implied volatility surface. As a result, Equation (28) remains unstable (Crpey, 2003; Marco, Friz, those of the middlemove probabilities range from 54.8% to 45.3%; those of the down move probabilities range from 28.9% to 29.1%. Although the ranges seem expansive, extreme values account for only a small portion. Indeed, the averages of the absolute relative differences for the upmove, middlemove, and downmove probabilities are 8.8%, 9.3%, and 9.4%, respectively. That the transition probabilities of implied trees can witness significant departures from the LV trees has been observed by Lok and Lyuu (2017) for the more restrictive separable LV models. To the best of our knowledge, no other paper brings up this issue. This phenomenon arises mainly because the transition probabilities are highly sensitive to the ArrowDebreu prices of Equations (23) and (26). Appearing in the denominators, small variations in them produce huge swings in FIGURE 6The local volatilities of the trinomial implied tree minus the true localvolatility surface TABLE 2Call prices from the trinomial LV tree vs those from the trinomial implied tree, both with 20 time steps StrikeLV treeImplied treeRelative difference (%) 2080.79680.7850.01 4061.83561.8220.02 6044.78244.7570.06 8031.41031.3720.12 10021.51021.4230.40 12014.92914.8840.30 14010.35110.3200.30 1607.1937.1600.46 1805.2015.1590.81 2003.7323.6931.05 2202.5992.5721.04 2401.9681.9481.02 2601.3751.3630.87 2801.0591.0481.04 3000.7420.7331.21 Note: The current stock price is 100. Abbreviation: LV, localvolatility. 12 | LOKANDLYUU the transition probabilities. The ArrowDebreu pricesthus the transition probabilities of the trinomial implied treein turn can deviate from those of the trinomial LV tree because the ArrowDebreu prices are derived from option values which are obtained in two steps. First, option prices are generated by trinomial LV trees with a much finer mesh (100 time steps for all maturities). These values are then mediated by the socalled contravariate trees. Consequently, option prices used in the trinomial implied treeC S t( ,) in+1 in Equation (23) andP S t( ,) in+1 in Equation (26)do not equal the abovementioned option prices as calculated by the trinomial LV trees. Instead, they are calculated by the CRR tree with maturitytnand+ 1 n+1 time steps (thus sharing the sametas the trinomial implied tree) with a volatility equal to the options implied volatility. This CRR tree is called the contravariate tree, employed by all DKbased implied trees (Barle Li, 2000). The contravariate tree makes the nonlinearity error less severe but contributes to the discrepancies in option prices. This is because it employsn + 1time steps whereas the trinomial LV tree generating the implied volatilities uses 100 time steps for all maturitiestn. These discrepancies in option prices are particularly acute for those early nodes of the implied tree as n is too small then. TABLE 3Call prices from the trinomial LV tree vs those from the trinomial implied tree, both with 30 time steps StrikeLV treeImplied treeRelative difference (%) 2080.79280.7850.01 4061.82561.8160.01 6044.73544.7220.03 8031.34131.3220.06 10021.52321.4600.29 12014.97914.9540.17 14010.42210.4090.12 1607.3197.3030.22 1805.2025.1770.48 2003.7233.6930.81 2202.6532.6231.13 2401.9541.9350.97 2601.4441.4300.90 2801.0311.0220.87 3000.7910.7831.01 Abbreviation: LV, localvolatility. FIGURE 7The local volatilities via Dupires formula (marked with) and the true localvolatility surface (plotted as the mesh) LOKANDLYUU | 13 Although the ArrowDebreu prices and the transition probabilities of the trinomial implied tree differ, sometimes significantly, from those of the trinomial LV tree, their impact on option prices is small. Indeed, the option prices obtained by both trees are very close to each other. Table 2 shows that the relative differences of the call prices over a broad range of strike prices lie within 1.2%. The implied tree with 30 time steps has even lower relative differences compared with the one with 20 time steps except in theX = 220, 260cases, where its relative differences remain very small (see Table 3). Interestingly, if the implied tree has the sametas the trinomial LV tree that generates the implied volatility surface (thus the same number of time steps), the trinomial implied tree recovers the local volatilities and transition probabilities exactly. And this is achieved without the contravariate tree. Figures 7, 8, and 9 illustrates the near perfect match. We conclude that the primary reason for the numerical instability of implied t

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