Complexity of the Ritchken-Trevor-Cakici-Topyan
GARCH Option Pricing Algorithm
Yuh-Dauh Lyuu
Computer Science & Information Engineering
National Taiwan University
No. 1, Sec. 4, Roosevelt Rd.
Taipei, Taiwan
Chi-Ning Wu
Department of Finance
National Taiwan University
No. 1, Sec. 4, Roosevelt Rd.
Taipei, Taiwan
Abstract
The trinomial-tree GARCH option pricing algo-rithm of Ritchken and Trevor (1999) is claimed to be efficient. That algorithm is subsequently mod-ified by Cakici and Topyan (2000). However, this paper proves that both algorithms explode expo-nentially when the number of partitions per day, n, exceeds a typically small number determined by the GARCH parameters. Worse, when explo-sion happens, the tree cannot grow beyond a cer-tain maturity date and the usual tradeoff between accuracy and efficiency ceases to exist. Hence the algorithms must be limited to using small n’s, which may have accuracy problems. Numerical data confirm the theoretical results. The problem of designing efficient tree-based GARCH option pricing algorithms therefore remains open.
Keywords: GARCH, trinomial tree, path de-pendency, option
1
Introduction
Efficient numerical algorithms are paramount to derivatives pricing. Because the exponential function grows so fast, exponential-time algo-rithms present the difficult choice between accu-racy and reasonable running time much earlier than polynomial-time algorithms. Exponential-time algorithms are therefore said to suffer from combinatorial explosion.
In pricing derivatives numerically, the diffusion process of the asset price can be discretized to
yield a tree structure. Derivatives can then be priced on the tree. The lognormal diffusion, for instance, gives rise to the well-known CRR bino-mial tree of Cox, Ross, and Rubinstein (1979). Two critical features of the CRR tree, as well as its many variations, are that it recombines and that an m-period tree contains only O(m2)
nodes. As a consequence, vanilla options can be efficiently priced. Efficient as the tree may be, a pricing algorithm based on it may still explode if the derivative itself is complex. The Asian option fits this characterization because of the vast amount of extra states added by its path-dependent feature. To tackle this problem, ap-proximations are necessary as surveyed in Lyuu (2002).
A qualitatively more serious problem emerges when the explosion arises from the model itself. If the model generates unrecombining trees, pric-ing can be expensive even for simple derivatives like the vanilla option. When the volatility is not a constant, such as the interest rate model of Cox, Ingersoll, and Ross (1985), a brute-force discretization leads to trees that do not recom-bine: An m-period binomial tree now contains 2m+1− 1 nodes. The situation may be rectified
by a technique of Nelson and Ramaswamy (1990) to transform the diffusion process into one with a constant volatility. But the methodology, even where applicable, does not guarantee to reduce the tree size to subexponential. The complexity issue is particularly relevant when the diffusion process is bivariate. Two bivariate models are the
interest rate model of Ritchken and Sankarasub-ramanian (1995) and the GARCH option pricing model of Duan (1995), the focus of this paper.
Bollerslev (1986) and Taylor (1986) indepen-dently propose the GARCH process popular in modeling the stochastic volatility of asset re-turns. Duan (1995) later applies the model to options pricing. Because of the massive path de-pendency of the model, simulation has been the algorithm of choice. The situation changes with the appearance of the tree algorithm of Ritchken and Trevor (1999). Their algorithm is claimed to be efficient; furthermore, it is general enough to work beyond GARCH models.
This paper investigates the performance of the Ritchken-Trevor algorithm and its modified ver-sion by Cakici and Topyan (2000). The re-sults are discouraging, both theoretically and nu-merically. It will be shown that the Ritchken-Trevor-Cakici-Topyan (RTCT) algorithm creates exponential-sized trees. The condition for this combinatorial explosion mirrors that for GARCH to be nonstationary. It is satisfied when the num-ber of partitions per day, n, exceeds a typically small number. The algorithm is hence not effi-cient unless the tree is restricted to small n’s.
Now suppose one is willing to trade efficiency for better accuracy with a large n. Such a trade-off usually exists for numerical algorithms. But this sensible practice turns out to be impossible: When explosion occurs, the RTCT tree cannot grow beyond a certain maturity. This extremely negative result obliterates the tradeoff between accuracy and efficiency taken for granted in the literature. It also throws into question some of the calculated prices in Ritchken and Trevor (1999). All the theoretical results are backed up by numerical data. The existence of efficient tree-based GARCH option pricing algorithms there-fore remains open.
The paper is organized as follows. The GARCH model is presented in Section 2. Sec-tion 3 reviews the essence of the RTCT tree. In Section 4, conditions for the tree to explode exponentially are proved. The nonexistence of the tradeoff between efficiency and accuracy is proved in Section 5. Section 6 provides numer-ical data to back up the theoretnumer-ical results and throw into question some of the calculated prices in Ritchken and Trevor (1999). Section 7 con-cludes.
2
The GARCH model
Let St denote the asset price at date t and ht
the conditional volatility of the return over the (t + 1)st day [ t, t + 1 ]. Here, “one day” is a convenient term for any elapsed time ∆t. The following risk-neutral process for the logarithmic price yt≡ ln St is due to Duan (1995):
yt+1= yt+ r − h2 t 2 + htt+1, where h2 t+1 = β0+ β1h2t+ β2h2t(t+1− c)2, (1)
t+1 ∼ N(0, 1) given information at date t,
r = daily riskless return, c ≥ 0.
It is postulated that β0, β1, β2 ≥ 0 to make the
conditional variance h2
t positive. Further impose
β1+ β2< 1 to make the model stationary. The
violation of a version of this inequality will be shown to make the RTCT tree explode. Pro-cess (1) for conditional variance, due to Engle and Ng (1993), is called the nonlinear asymmet-ric GARCH.
3
Building the Tree
The RTCT trinomial tree approximates the continuous-state GARCH process with discrete states. Partition a day into n periods. Three successor states will follow each state (yt, h2t)
af-ter a period. As the trinomial tree recombines, 2n + 1 states at date t + 1 follow each state at date t. We next pick the jump size and the branching probabilities to match the distribution of yt+1. Let γ = h0 and γn = γ/√n . (Our
re-sults will be seen to be independent of how γ is picked.) The jump size will be some integer mul-tiple η of γn. See Fig. 1 for illustration. Note
that the middle branch does not change the un-derlying asset’s price. The probabilities for the up, middle, and down branches are
pu = h2 t 2η2γ2 + r − (h2 t/2) 2ηγ√n , (2) pm = 1 − h2 t η2γ2, (3) pd = h2 t 2η2γ2 − r − (h2 t/2) 2ηγ√n . (4)
The intermediate nodes between dates are dis-pensed with to create a (2n + 1)-nomial tree as in Fig. 2 to reduce the node count. The resulting model is multinomial with 2n + 1 branches from any state (yt, h2t). From Eqs. (2)–
(4), valid branching probabilities exist (i.e., 0 ≤ pu, pm, pd≤ 1) if and only if | r − (h2 t/2) | 2ηγ√n ≤ h2 t 2η2γ2 ≤ min(1− | r − (h2 t/2) | 2ηγ√n , 1 2). (5) The updating rule (1) must be modified to account for the adoption of the discrete-state model. For −n ≤ ` ≤ n, state (yt, h2t) at date
t is followed by state (yt+ `ηγn, h2t+1) at date
t + 1, where h2 t+1 = β0+ β1h2t+ β2h2t(0t+1− c)2, (6) 0 t+1 = `ηγn− (r − h2t/2) ht .
This transition happens with probability
X ju,jm,jd n! ju! jm! jd! pju u p jm mp jd d , where ju, jm, jd≥ 0, n = ju+ jm+ jd, and ` = ju− jd.
As volatility ht changes through time, we may
have to pick different η’s for different states so that pu, pm, and pd lie between 0 and 1. This
implies varying jump sizes. As the necessary re-quirement pm ≥ 0 implies η ≥ ht/γ, we go
through
η = d ht/γ e, d ht/γ e + 1, d ht/γ e + 2, . . . (7)
until valid probabilities are obtained or until their nonexistence is confirmed by inequalities (5). Obviously, the magnitude of η grows with ht. Backward induction starts after the tree has
been built.
Figure 3 depicts a 3-day tree with n = 1. Nodes A and B pick η = 2. Observe that dif-ferent states may pick difdif-ferent η’s. The number of possible values of ht at a node can be
expo-nential as each path leading to the node carries a different ht. To handle this complexity, only the
maximum and minimum ht are recorded at each
node. For example, the maximum and minimum ht at nodes C and D pick different jump sizes.
Ritchken and Trevor (1999) add extra volatilities
between the maximum and minimum h2 t.
Be-cause these interpolated volatilities can only in-crease the range of future volatilities, our analy-sis will stand without considering them. Hollow nodes in Fig. 3 are not occupied because they are unreachable. As will be shown later, their count is minuscule.
4
Sufficient Conditions
for
Explosion
One typically increases n for better accuracy. Unfortunately, the maximum value of ht grows
exponentially in t if n is large enough. When this happens, the tree explodes because it must pick an η that expands exponentially by virtue of Eq. (7). Hence the RTCT tree must be re-stricted to small n’s to have any hope of being efficient. We next provide the argument for the claimed exponential growth of ht.
Assume r = 0 and c = 0 first. The updating rule (6) is now h2 t+1= β0+ β1h 2 t+ β2 `ηγn+ (h2t/2) 2 , where ` = 0, ±1, ±2, . . . , ±n. To make h2 t+1 as
large as possible, set ` = n. The updating rule becomes h2 t+1 = β0+ β1h 2 t+ β2 √ n ηγ + (h2 t/2) 2 ≥ β0+ β1h 2 t+ β2 √ n ht+ (h2t/2) 2 ≥ β0+ β1h 2 t+ β2nh 2 t = β0+ (β1+ β2n) h2t. By induction, h2 t+1 ≥ β0 t X i=0 (β1+ β2n)i+ (β1+ β2n)t+1h20 = β0 1 − (β1+ β2n) +[ h2 0+ β0 (β1+ β2n) − 1 ](β1+ β2n)t+1.
The above expression grows exponentially if β1+
β2n > 1. This inequality is reminiscent of the
necessary condition β1 + β2 ≥ 1 for GARCH
to be nonstationary. When r 6= 0 or c 6= 0, the maximum value of ht still grows exponentially in
more tedious but essentially identical. We con-clude that the RTCT tree grows exponentially if n is large enough.
5
The Shallowness of an
Ex-ploding Tree
Can a large n be chosen to improve accuracy if we are willing to accept long running times? Un-fortunately, the RTCT tree does not admit such a tradeoff. The reason is that there is a ceiling on volatility ht for valid branching probabilities
to exist. With the maximum value of ht growing
exponentially, this ceiling will quickly be reached at some nodes and the tree can grow no further. The choice of n is thus capped even if infinite resources are available. We next derive the said upper bound. Inequalities (5) imply | (h2 t/2) − r | 2ηγ√n ≤ h2 t 2η2γ2, h2 t 2η2γ2 ≤ 1 2. Hence h2 t ≤ (ηγ) 2 ≤ h2 t √ n | (h2 t/2) − r | 2 ,
which can be simplified to yield
(h2 t/2) − r
2 ≤ nh2
t.
Finally, the above quadratic inequality (in h2 t) is equivalent to 2(r + n) − 2p2rn + n2≤ h2 t ≤ 2(r + n) + 2p2rn + n2. We conclude that h2 t ≤ 2(r + n) + 2 p 2rn + n2 (8)
is necessary for the existence of valid branching probabilities. This condition does not depend on the choice of γ because the identity γ = h0 did
not enter the analysis.
The impossibility result may sound puzzling at first. Under the Black-Scholes model, valid branching probabilities always exist if n is large
enough. Why, one may ask, can’t the same prop-erty hold here? The answer lies in volatility. The daily volatility in the Black-Scholes model is a constant, which amounts to setting ht to some
fixed number. So every state solves the same Eqs. (2)–(4) for probabilities, and increasing n will eventually have inequality (8) satisfied for all states. In contrast, the volatility ht under
GARCH fluctuates. So each state (yt, h2t) faces
different Eqs. (2)–(4) in solving for probabilities. Increasing n makes inequality (8) harder to sat-isfy for those states with a large h2
t, whose
exis-tence under GARCH has been confirmed earlier.
6
Numerical Examples
The following parameters from Ritchken and Trevor (1999) and Cakici and Topyan (2000) will be assumed throughout the section: S0 = 100,
y0 = ln S0 = 4.60517, r = 0, h20 = 0.0001096,
γ = 0.010469, β0 = 0.000006575, β1 = 0.9,
β2 = 0.04, and c = 0. As r = c = 0,
combi-natorial explosion occurs when
n > 1 − β1 β2
=1 − 0.9 0.04 = 2.5.
Figure 4 picks n = 3, 4, 5 to demonstrate the ex-ponential growth of the RTCT tree. The rate of growth clearly increases with n. For comparison, the standard trinomial tree contains only 2t + 1 nodes at date t.
The number of nodes is a critical indicator because the running time is proportional to it. We mentioned earlier that there may be nodes which are not reachable (recall Fig. 3). In prin-ciple, if such nodes are numerous, the algorithm can potentially run more efficiently by skipping them. Figure 5 shows that the proportion of un-reachable nodes among all the nodes is small for n = 3, 4, 5. We will see shortly that the same conclusion also holds for larger n’s. As the over-whelming majority of nodes between the top and bottom nodes are reachable, no substantive bene-fits can be obtained by clever programming tech-niques to skip unreachable nodes.
Now suppose we pick n = 100 to seek very high accuracy at the expense of efficiency. The theory predicts a high risk of having the RTCT tree cut short. Indeed, with r = 0, inequality (8) imposes the upper bound h2
This means that a node with ht > 20 cannot
have valid branching probabilities and thus can-not grow further. As this ceiling is breached at date 9 because of the exponential growth of the maximum value of ht, the tree stops growing
then if not earlier. See Table 1 for the final dates under various n’s beyond the threshold of explo-sion. Observe that the tree’s longest maturity decreases rapidly as n increases. It is therefore important not to pick too large an n, for only trees of very short maturities will be generated otherwise. Table 1 also tabulates the total num-bers of nodes and unreachable ones. Again, the overwhelming majority of the nodes are occupied as mentioned earlier.
Some of the calculated option prices in Ritchken and Trevor (1999) use n as large as 25 and maturity dates as far as 200. These choices contradict our analysis and data. For example, Table 1 says that the RTCT tree with n = 25 should stop growing at date 18. These prices must therefore be viewed with caution. Cakici and Topyan (2000) use n = 1 throughout; thus explosion is avoided.
7
Conclusions
We proved that for n suitably large, the RTCT tree explodes. The Ritchken-Trevor-Cakici-Topyan GARCH option pricing algorithm is hence inefficient. Worse, a global upper bound on the volatility renders the tree shallow when ex-plosion occurs. Cakici and Topyan (2000) claim that their pricing algorithm is empirically accu-rate enough at n = 1 for vanilla options. But ac-curacy must remain a concern with under-refined trees. The problem of designing efficient tree-based GARCH option pricing algorithms there-fore remains open. Our results literally carry over to the BDT-GARCH interest rate model of Bali (1999).
References
[1] Bali, T.G. (1999) An Empirical Compar-ison of Continuous Time Models of the Short Term Interest Rate. Journal of Fu-tures Markets, 19(7) , pp. 777–797.
[2] Bollerslev, T. (1986) Generalized Au-toregressive Conditional Heteroskedastic-ity. Journal of Econometrics, 31, pp. 307– 327.
[3] Cakici, N., and Topyan, K. (2000) The GARCH Option Pricing Model: A Lat-tice Approach. Journal of Computational Finance, 3(4) , pp. 71–85.
[4] Cox, J.C., Ingersoll, J.E., and Ross, S.A.(1985) A Theory of the Term Struc-ture of Interest Rates. Econometrica, 53(2), pp. 385–407.
[5] Cox, J.C., Ross, S.A., and Rubin-stein, M. (1979) Option Pricing: a Sim-plified Approach. Journal of Financial Eco-nomics, 7(3), pp. 229–263.
[6] Duan, J.-C. (1995) The GARCH Op-tion Pricing Model. Mathematical Finance, 5(1), pp. 13–32.
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Figure 1: Trinomial tree for logarithmic price yt for the duration of one day.
(yt, h2t) 6 ? ηγn - 1 day
A day is partitioned into n = 3 periods, and the jump size is ηγn. The 7 values on the right should
approximate the distribution of yt+1 given yt
and h2 t.
Figure 2: Multinomial tree for daily loga-rithmic prices. (yt, h2t) 6 ? ηγn - 1 day
This heptanomial tree is the outcome of the tri-nomial tree in Fig. 1 after its interior nodes are removed. Recall that n = 3.
Figure 3: Possible geometry of a 3-day RTCT tree. (y0, h20) A B C D 6 ?γn= γ1 - 3 days
A day is partitioned into n = 1 period. Nodes A and B have a jump size of 2γ1. Nodes C and D
have two jump sizes: γ1 and 2γ1. All other nodes
have a jump size of γ1. Two nodes between the
top and bottom nodes are not reachable. They are shown as hollow nodes.
Figure 4: Exponential growth of the RTCT tree. 25 50 75 100 125 150 175 Date 5000 10000 15000 20000 25000
The parameters are S0 = 100, y0 = ln S0 =
4.60517, r = 0, h2
0 = 0.0001096, γ = 0.010469,
β0 = 0.000006575, β1 = 0.9, β2 = 0.04, and
c = 0. The dotted line is based on n = 3, the dashed line on n = 4, and the solid line on n = 5. The standard trinomial tree, in contrast, has only 2t + 1 nodes at date t.
Figure 5: The percent of unreachable nodes.
25 50 75 100 125 150 175 Date 0.5
1 1.5
The parameters are S0 = 100, y0 = ln S0 =
4.60517, r = 0, h2
0 = 0.0001096, γ = 0.010469,
β0 = 0.000006575, β1 = 0.9, β2 = 0.04, and
c = 0. The plots show the percent of unreach-able nodes among all nodes at each date. The dotted line is based on n = 3, the dashed line on n = 4, and the solid line on n = 5. The number of unreachable nodes is insignificant in all 3 lines.
Table 1: Final maturity dates and sizes of exploding trees.
Total number of Total number of n Final date (t) nodes unreachable nodes 3 182 1,017,327 5,565 4 100 499,205 3,028 5 72 368,523 947 10 34 222,935 42 25 18 286,844 6,925 50 12 305,113 448 100 9 578,710 3,961 150 8 795,309 2,011 200 7 652,808 1,596 250 7 1,747,758 20,291 300 7 2,929,508 11,510 350 6 1,179,157 3,151
The parameters are S0 = 100, y0 = ln S0 =
4.60517, r = 0, h2
0 = 0.0001096, γ = 0.010469,
β0 = 0.000006575, β1 = 0.9, β2 = 0.04, and
c = 0. With n > 2.5, all RTCT trees in the ta-ble explode. The final maturity date of the tree shortens quickly as n increases. The total num-ber of nodes in each tree far exceeds the (t + 1)2
of the standard trinomial tree. The overwhelm-ing majority of nodes are reachable in all trees.
