The valuation of multivariate contingent claims under transformed trinomial approaches
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DOI: 10.1007/s11156-009-0121-3
- Cite this article as:
- Chang, C. & Lin, J. Rev Quant Finan Acc (2010) 34: 23. doi:10.1007/s11156-009-0121-3
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Abstract
This study develops a transformed-trinomial approach for the valuation of contingent claims written on multiple underlying assets. Our model is characterized by an extension of the Camara and Chung (J Futur Mark 26: 759–787, 2006) transformed-binomial model for pricing options with one underlying asset, and a discrete-time version of the Schroder (J Finance 59(5): 2375–2401, 2004) model. However, unlike the Schroder model, our model can facilitate straightforward valuation of American-style multivariate contingent claims. The major advantage of our transformed-trinomial approach is that it can easily tackle the volatility skew observed within the markets. We go on to use numerical examples to demonstrate the way in which our transformed-trinomial approach can be utilized for the valuation of multivariate contingent claims, such as binary options.
Keywords
Transformed-trinomial approachesMultivariate contingent claimsBinary optionsJEL Classification
C52G121 Introduction
There are two main branches within the extant literature on the derivation of option pricing formulae, with the first of these advocating the adoption of an arbitrage-free replication approach, or a risk-neutral approach, as the means of deriving closed-form solutions for pricing European options. Representatives of this branch of the literature are Black and Scholes (1973), Merton (1973) and Cox and Ross (1976), with the subsequent examples of extensions including those undertaken by Merton (1976), Leland (1985), Hull and White (1987) and Rabinovitch (1989).
The second branch of the literature in this field posits that equilibrium option prices are dependent on the marginal utility function of investors. Rubinstein (1976) and Brennan (1979) derived preference-free option pricing models within this framework, under the assumptions of a discrete-time trading model and asset returns with continuous distributions. Camara (2003, 2005) and Schroder (2004) have also recently gone on to generalize the Brennan–Rubinstein preference-free option valuation formula.
The most widely used option pricing model—apart from the famous option pricing formula of Black and Scholes (1973)—is the multiplicative-binomial model of Cox et al. (1979) (hereafter referred to as the CRR model). The beauty of the CRR model is not only its simplicity, but also its limit convergence to the Black–Scholes model. As pointed out by Camara and Chung (2006), the CRR model is of particular interest to both academicians and practitioners alike, because it facilitates the derivation of equations for hedging portfolios, the assessment of the risk-neutral multiplicative binomial probability of the risky asset, and the derivation of pricing equations for call options and other derivatives.
The current literature contains several extensions of the CRR model, which include: (1) attempts at improving the convergence speed of the model;^{1} (2) methods of dealing with the valuation of options written on multiple underlying assets;^{2} (3) methods of examining the convergence properties for binomial option pricing models;^{3} (4) approaches to dealing with option pricing with stochastic interest rates or stochastic volatility;^{4} and (5) the design of various lattice approaches for the valuation of exotic options.^{5}
More recently, Camara and Chung (2006) developed a transformed binomial model which was capable of pricing options when the underlying asset price followed a transformed-binomial process. They also demonstrated that the European-style transformed-binomial option price converges to the European transformed-normal option price of Camara (2003) and Schroder (2004). They used several examples to demonstrate the accuracy and convergence of the transformed-binomial model, and also showed that their model was capable of dealing with the smile effects observed within the markets. Indeed, it could be argued that Camara and Chung (2006) have made a significant contribution to the literature through their provision of a more flexible lattice framework for the valuation of options.
In this study, we extend the Camara and Chung (2006) transformed-binomial model by developing a transformed-trinomial model capable of pricing options written on multiple underlying assets. Hence we contribute to the literature in two significant ways; the first relates to our provision of a more general transformed-lattice model for the valuation of options, whilst the second is our ability to demonstrate the way in which the European-style transformed-trinomial price converges to the European-style transformed-normal digital option price of Camara (2005).
The remainder of this paper is organized as follows. Section 1 introduces the transformed-trinomial model with one underlying asset, followed in Sect. 2 by our extension of the transformed-trinomial model for the valuation of American-style options written on multiple underlying assets. We go on to provide numerical examples in Sect. 3 to demonstrate the accuracy and convergence of our transformed-trinomial model. Finally, the conclusions drawn from this study are presented in Sect. 4.
2 The transformed-trinomial model settings
2.1 The model for valuing options written on a single underlying asset
2.1.1 Definition of the transformed-binomial process
An asset price, S_{ij}, follows a transformed-binomial process over discrete periods if g(S_{ij}) has an arithmetic-binomial process over such discrete periods, where g is a strictly monotonic differentiable function with \( g(S_{ij} ) \in R\)where U: = ln u and D: = ln d are the additive shocks with U > D; and q and 1 − q are the respective physical and subjective probabilities of a U shock and a D shock. Based upon the above definition, we know that in order to completely specify the transformed-binomial option pricing model, we have to select values for U and D.
As noted by Camara and Chung (2006), in order to select U and D, we have to fit in the skewness and the kurtosis from the transformed-normal distribution implied by the market prices.^{6}
2.1.2 Definition of four-parameter transformed-normal distribution
So far, we have described the basic concepts for the transformed-binomial tree method; however, in this paper, we will go on to use a transformed-trinomial tree method. The main reason for this is that we know from the literature that a trinomial process has a better convergence property when used to evaluate option prices.
Given that g(S) follows normal distribution, \( \varepsilon = \left[ {\hat{g}(S_{(0,0)} ) - g(S_{(0,0)} )} \right] \) also has normal distribution. The term ĝ(S_{(0,0)}) indicates that after a single time step, an asset with a starting price of g(S_{(0,0)}) will have a possible value represented by ε ~ N(μ∆t,σ^{2}∆t).
We can now complete our construction of the transformed-trinomial model for the valuation of options written on a single underlying asset.
2.2 The model for pricing options written on two underlying assets
In this subsection, we begin by constructing a transformed-trinomial model for pricing options written on two underlying assets. Let S^{1}, S^{2} denote the asset price pair. Again, in order to have a trinomial process, we have to know the magnitudes of the up and down movements of the underlying asset price.
\( \varepsilon_{1}^{a} \) | \( \varepsilon_{2}^{a} \) | Prob |
---|---|---|
v_{1} | v_{2} | p_{1} |
v_{1} | −v_{2} | p_{2} |
−v_{1} | −v_{2} | p_{3} |
−v_{1} | v_{2} | p_{4} |
0 | 0 | p_{5} |
We can now complete our construction of the transformed-trinomial model for the valuation of options written on two underlying assets.
2.3 The model for pricing options written on K underlying assets
In this subsection, we set out to construct a transformed-trinomial model for pricing options written on K underlying assets. As in the above sub-sections, we assume that the joint density of the asset prices follows multivariate normal distribution.
We can now complete the construction of a general transformed-trinomial model for the valuation of options written on K underlying assets.
3 Numerical examples and convergence
In this section, we demonstrate the way in which our model can be used for the valuation of options, beginning with examples for computing the values of European options when the option is written on only one underlying asset. We compare the option values calculated by the ‘original trinomial model’ (hereafter denoted as O-trinomial models) price process with those computed by our ‘iterated logarithm transformed trinomial model’ (hereafter denoted as I-trinomial models) price process. Under these two price processes, the asset price has the same value over the first period, but thereafter has different price paths. We take the values of the O-trinomial price processes at Dates 0 and 1 as a comparison benchmark.
Transformed-trinomial price processes with a single underlying asset
Prices | Models | |
---|---|---|
O-trinomial | I-trinomial | |
u | 2 | 1.0214 |
d | 0.5 | 0.9790 |
S_{0,0} | 100 | 100 |
S_{1,0} | 50 | 50 |
S_{1,1} | 100 | 100 |
S_{1,2} | 200 | 200 |
S_{2,0} | 25 | 28.26 |
S_{2,1} | 50 | 50 |
S_{2,2} | 100 | 100 |
S_{2,3} | 200 | 200 |
S_{2,4} | 400 | 452.27 |
As noted earlier, the potential underlying asset prices at Date 1 are identical under both the O-trinomial and I-trinomial tree processes, whereas the potential underlying asset prices at Date 2 are totally different under each of these trinomial tree processes. This numerical example therefore demonstrates that, under the two diverse trinomial processes, the underlying asset price can assume quite a different price path in the future.
European put option values with displaced distribution
K/S | Maturity period^{a} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 Month | 3 Months | 9 Months | 3 Years | |||||||||
1.10 | 1.00 | 0.90 | 1.10 | 1.00 | 0.90 | 1.10 | 1.00 | 0.90 | 1.10 | 1.00 | 0.90 | |
Theoretical prices^{b} | 4.7971 | 0.8146 | 0.0055 | 4.6551 | 1.2894 | 0.1084 | 4.6162 | 1.8924 | 0.4850 | 4.2784 | 2.4414 | 1.1791 |
Trinomial tree prices^{c} | ||||||||||||
m = 101 | 4.7966 | 0.8138 | 0.0054 | 4.6548 | 1.2895 | 0.1077 | 4.6148 | 1.8956 | 0.4841 | 4.2790 | 2.4504 | 1.1851 |
m = 201 | 4.7970 | 0.8144 | 0.0055 | 4.6545 | 1.2901 | 0.1085 | 4.6186 | 1.8951 | 0.4846 | 4.2810 | 2.4443 | 1.1821 |
m = 301 | 4.7970 | 0.8146 | 0.0055 | 4.6538 | 1.2901 | 0.1081 | 4.6178 | 1.8944 | 0.4849 | 4.2756 | 2.4407 | 1.1792 |
m = 401 | 4.7970 | 0.8147 | 0.0055 | 4.6553 | 1.2901 | 0.1084 | 4.6170 | 1.8940 | 0.4854 | 4.2806 | 2.4384 | 1.1789 |
m = 501 | 4.7970 | 0.8147 | 0.0055 | 4.6546 | 1.2900 | 0.1085 | 4.6152 | 1.8936 | 0.4852 | 4.2795 | 2.4404 | 1.1803 |
We demonstrate, from Table 2, that under our transformed trinomial tree approach, the benchmark values are able to converge under reasonable time steps. For instance, when the time to maturity is 3 years and there are only 250 steps in the transformed-trinomial tree, the pricing error is, on average, <0.2%.
Valuation of American put options with a single underlying asset under displaced distribution
K/S | Maturity period^{a} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 Month | 3 Months | 9 Months | 3 Years | |||||||||
1.10 | 1.00 | 0.90 | 1.10 | 1.00 | 0.90 | 1.10 | 1.00 | 0.90 | 1.10 | 1.00 | 0.90 | |
Trinomial tree prices (10,000 steps)^{b} | 5.0000 | 0.8305 | 0.0056 | 5.0139 | 1.3448 | 0.1111 | 5.2808 | 2.0837 | 0.5193 | 6.1027 | 3.3048 | 1.5233 |
Trinomial tree prices^{c} | ||||||||||||
m = 101 | 5.000 | 0.8297 | 0.0054 | 5.0106 | 1.3442 | 0.1104 | 5.2749 | 2.0830 | 0.5173 | 6.0914 | 3.2886 | 1.5214 |
m = 201 | 5.000 | 0.8303 | 0.0055 | 5.0134 | 1.3450 | 0.1111 | 5.2778 | 2.0842 | 0.5188 | 6.0870 | 3.3013 | 1.5208 |
m = 301 | 5.000 | 0.8305 | 0.0055 | 5.0135 | 1.3452 | 0.1107 | 5.2805 | 2.0842 | 0.5189 | 6.0990 | 3.3023 | 1.5215 |
m = 401 | 5.000 | 0.8305 | 0.0055 | 5.0133 | 1.3452 | 0.1111 | 5.2794 | 2.0842 | 0.5193 | 6.0973 | 3.3014 | 1.5212 |
m = 501 | 5.000 | 0.8306 | 0.0055 | 5.0131 | 1.3452 | 0.1111 | 5.2799 | 2.0841 | 0.5193 | 6.1014 | 3.3013 | 1.5222 |
One-period transformed-trinomial price processes with two underlying assets
Price processes^{a} | Underlying Asset 1 | Underlying Asset 2 |
---|---|---|
S_{0,0} | 100 | 100 |
S_{1,0} | 41.1952 | 101.244 |
S_{1,1} | 41.1952 | 100.804 |
S_{1,2} | 100 | 100 |
S_{1,3} | 299.927 | 101.244 |
S_{1,4} | 299.927 | 100.804 |
Theoretical binary option value = 0.427184 |
Using the Camara (2005) closed-form solution as a benchmark, we report the price differences in the valuation of European binary call options under different time steps and under the different scale parameter, λ, as noted by Kamrad and Ritchken (1991), λ will affect the convergence speed; thus, when λ is equal to 1, this trinomial tree method will reduce to the binomial tree method.
Price differences between the transformed-trinomial model and the benchmark values
Strike price^{a} | λ | No. of Steps | |||||
---|---|---|---|---|---|---|---|
10 | 20 | 50 | 100 | 150 | 200 | ||
100 | 1.4142 | −0.1654 | −0.0867 | −0.0399 | −0.0574 | −0.0459 | 0.0119 |
1.2910 | −0.0066 | −0.1163 | −0.0223 | 0.0121 | 0.0120 | −0.0100 | |
1.2247 | −0.1477 | −0.1106 | −0.0709 | −0.0489 | 0.0120 | −0.0327 | |
1.1180 | 0.0016 | −0.1012 | −0.0086 | 0.0121 | −0.0348 | 0.0116 | |
1.0541 | −0.0842 | −0.0966 | −0.0603 | −0.0058 | 0.0118 | −0.0166 | |
1.0000 | −0.1905 | −0.1977 | −0.1257 | 0.0139 | −0.0188 | −0.0264 |
We also find that the parameter, λ, may change for different contracts. In the cases examined here, we find that the best performance amongst different values of λ occurs when λ is equal to 1.2900; however, Kamrad and Ritchken (1991) noted that better results are reported when \( \lambda = \sqrt 2 . \)
We again find from Fig. 1 that the transformed-trinomial tree model has better convergence property than that of the transformed-binomial tree. This finding is not at all surprising since the traditional trinomial tree model also has better convergence property than the traditional binomial tree model (Appendices 1, 2).
4 Conclusions
This study develops a transformed-trinomial approach to the valuation of multivariate contingent claims. Our model is an extension of the Camara and Chung (2006) transformed-binomial model for pricing options with one underlying asset, and a discrete-time version of the Schroder (2004) model. However, unlike the Schroder model, our model can facilitate straightforward valuation of American-style multivariate contingent claims. The advantage of our transformed-trinomial approach is that it can easily tackle the volatility skew observed within the markets.
The results from our numerical examples can be summarized as follows. Firstly, under the assumptions of traditional trinomial and transformed-trinomial price processes, we demonstrate that the future price paths of the underlying assets are quite different. Secondly, we show that our transformed-trinomial model has excellent convergence property for the valuation of options written on either one or two underlying assets. Hence, we contribute to the literature on lattice modeling by constructing a more flexible lattice framework which is easily capable of capturing the volatility skew observed within the markets.
Hull and White (1988), for example, used a control variable technique as the means of enhancing convergence.
See, for example, Boyle et al. (1989) in which a binomial option pricing model was developed for the valuation of options written on multiple underlying assets.
Nelson and Ramaswamy (1990), for instance, derived the conditions for a sequence of binomial processes to converge weakly to a diffusion process.
Hull and White (1993) developed lattice approaches for the valuation of both look-back options and Asian options.
Acknowledgments
We are especially grateful for Professors A.H.W. Wang constructive comments.