Review of Quantitative Finance and Accounting

, 34:23

The valuation of multivariate contingent claims under transformed trinomial approaches

Authors

  • Chuang-Chang Chang
    • Department of FinanceNational Central University
    • Department of Money and BankingNational Kaohsiung First University of Science and Technology
Original Research

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 options

JEL Classification

C52G12

1 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

Prior to setting out to construct our transformed-trinomial model for the valuation of options written on a single underlying asset, we must introduce some basic concepts of functional transformations to normality. Here, we follow the spirit of the Johnson (1949a) model, considering a transformation of variable x with following type:
$$ \varepsilon = \gamma + \delta g\left( {\frac{x - \mu }{\lambda }} \right), $$
(1)
where g is a strictly monotonic function and μ, λ, γ and δ are parameters. We must choose the parameters and the function, g, such that ε is a unit normal variable, or approximately so.
Without any loss of generality, we define a new variable, y, as follows:
$$ y = \frac{x - \mu }{\lambda }, $$
(2)
then,
$$ \varepsilon = \gamma + \delta g(y). $$
Hence, we have the density function of y:
$$ \begin{aligned} p(y) = & \delta g^{\prime}(y)p(\varepsilon )\left| {_{\varepsilon = \gamma + \delta g(y)} } \right. \\ = & \frac{\delta }{{\sqrt {(2\pi )} }}g^{\prime}(y)\exp \left\{ { - \frac{1}{2}[\gamma + \delta g(y)]^{2} } \right\}, \\ \end{aligned} $$
(3)
where \( g^{\prime}(y) = dg(y)/dy. \)

2.1.1 Definition of the transformed-binomial process

An asset price, Sij, follows a transformed-binomial process over discrete periods if g(Sij) has an arithmetic-binomial process over such discrete periods, where g is a strictly monotonic differentiable function with \( g(S_{ij} ) \in R\)https://static-content.springer.com/image/art%3A10.1007%2Fs11156-009-0121-3/MediaObjects/11156_2009_121_Figa_HTML.gifwhere 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

A random variable X has a four-parameter transformed-normal distribution, if:
$$ \begin{aligned} X & \, = \beta g^{ - 1} (Z\sigma + \mu ) +\alpha \quad {\text{where}}\quad Z\sim N(0,1) \\ & \Rightarrow g\left( {\frac{X - \alpha }{\beta }} \right)\sim N(\mu ,\sigma )\\ & \Rightarrow g(Y)\sim N(\mu ,\sigma )\quad{\text{let}}\quad Y = \frac{X - \alpha }{\beta }. \\ \end{aligned}$$
The selection of U and D should be workable for any member of the transformed-binomial family. Without any loss of generality, we assume that the stock price at Date t has a two-parameter transformed-normal distribution. For the case of multiple periods, we assume that there are n-periods until Date t. Here, μt, σ2t are the mean and variance of the transformed-normal distribution, g(St), with S denoting the current price of the asset. We use the following two equations to match the mean and variance of the stock return.
$$ \begin{aligned} & qU + (1 - q)D = \mu \Updelta t \\ & q(1 - q)(U - D)^{2} = \sigma^{2} \Updelta t. \\ \end{aligned} $$
(8)
Let U = −D, we then have:
$$ \begin{aligned} U & = \sqrt {\sigma^{2} \Updelta t + (\mu \Updelta t)^{2} } \\ D & = - \sqrt {\sigma^{2} \Updelta t + (\mu \Updelta t)^{2} } . \\ \end{aligned} $$
(9)

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(μt,σ2t).

Let εa be a discrete random variable, which has the approximated distribution for ε under the following distribution:
$$ \varepsilon^{a} = \left\{ {\begin{array}{*{20}c} {v\quad {\text{with}}\,{\text{probability}}\,p_{1} } \\ {0\quad {\text{with}}\,{\text{probability}}\,p_{2} } \\ { - v\quad {\text{with}}\,{\text{probability}}\,p_{3} } \\ \end{array} } \right \} $$
(10)
where \( v = \lambda \sqrt {\sigma^{2} \Updelta t + (\mu \Updelta t)^{2}}.\) It is obvious, when Δt is small, \( v \approx \sqrt {\sigma^{2} \Updelta t}.\)
The mean and variance of the approximated distribution are chosen to equal the mean and variance of ε; in other words, we have:
$$ E(\varepsilon^{a} ) = v(p_{1} - p_{3} ) = \mu \Updelta t $$
(11)
$$ {\text{Var}}(\varepsilon^{a} ) = v^{2} (p_{1} + p_{3} ) = \sigma^{2} \Updelta t + O(\Updelta t) $$
(12)
and
$$ \sum\limits_{j = 1}^{3} {pj = 1} . $$
(13)
By solving the above three equations, we then have:
$$ p_{1} = \frac{{v\mu \Updelta t + \sigma^{2} \Updelta t}}{{2v^{2} }} $$
(14)
$$ p_{2} = 1 - \frac{{\sigma^{2} \Updelta t}}{{v^{2} }} $$
(15)
$$ p_{3} = \frac{{ - v\mu \Updelta t + \sigma^{2} \Updelta t}}{{2v^{2} }}. $$
(16)

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 S1, S2 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.

Assuming that g1(S1), g2(S2) follow normal distributions for the underlying assets (Assets 1 and 2), we have:
$$ (\varepsilon_{1} ,\varepsilon_{2} ) = \left[ {\hat{g}_{1} (S_{{1,_{(0,0)} }} ) - g_{1} (S_{{1,_{(0,0)} }} ),\hat{g}_{2} (S_{{2,_{(0,0)} }} ) - g_{2} (S_{{2,_{(0,0)} }} )} \right], $$
(17)
which also has normal distribution with mean and variance, \( \mu_{i} t,\sigma_{i}^{2} t. \) The correlation between (ε2, ε2) is denotes as ρ.
The joint normal random variable (ε2, ε2) can be approximated by a pair of multi-normal discrete random variables with the following distribution:

\( \varepsilon_{1}^{a} \)

\( \varepsilon_{2}^{a} \)

Prob

v1

v2

p1

v1

v2

p2

v1

v2

p3

v1

v2

p4

0

0

p5

where \( v_{i} = \lambda_{i} \sqrt {\sigma_{i}^{2} \Updelta t + (\mu_{i} \Updelta t)^{2}}.\) Again, when Δt is small, \( v_{i} \approx \lambda_{i} \sqrt {\sigma_{i}^{2} \Updelta t}.\)
In order to ensure the convergence of the approximated distribution to the true distribution, ∆t → 0, the first two moments of the approximation distribution are set as being equal to the true moments of the continuous one. Apart from the equality of the two moments, the covariance term must also be equal.
$$ E\left( {\varepsilon_{1}^{a} } \right) = v_{1} (p_{1} + p_{2} - p_{3} - p_{4} ) = \mu_{1} \Updelta t $$
(18)
$$ E\left( {\varepsilon_{2}^{a} } \right) = v_{2} (p_{1} - p_{2} - p_{3} + p_{4} ) = \mu_{2} \Updelta t $$
(19)
$$ {\text{Var}}\left( {\varepsilon_{1}^{a} } \right) = v_{1}^{2} (p_{1} + p_{2} + p_{3} + p_{4} ) = \sigma_{1}^{2} \Updelta t + O(\Updelta t) $$
(20)
$$ {\text{Var}}\left( {\varepsilon_{2}^{a} } \right) = v_{2}^{2} (p_{1} + p_{2} + p_{3} + p_{4} ) = \sigma_{2}^{2} \Updelta t + O(\Updelta t) $$
(21)
$$ E\left( {\varepsilon_{1}^{a} ,\varepsilon_{2}^{a} } \right) = v_{1} v_{2} (p_{1} - p_{2} + p_{3} - p_{4} ) = \sigma_{1} \sigma_{2} \rho \Updelta t + O(\Updelta t) $$
(22)
By solving the above equation, we then have,
$$ p_{1} = \frac{1}{4}\left( {\frac{{\mu_{1} }}{{v_{1} }}\Updelta t + \frac{{\mu_{2} }}{{v_{2} }}\Updelta t + \frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t + \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(23)
$$ p_{2} = \frac{1}{4}\left( {\frac{{\mu_{1} }}{{v_{1} }}\Updelta t - \frac{{\mu_{2} }}{{v_{2} }}\Updelta t + \frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t - \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(24)
$$ p_{3} = \frac{1}{4}\left( { - \frac{{\mu_{1} }}{{v_{1} }}\Updelta t - \frac{{\mu_{2} }}{{v_{2} }}\Updelta t + \frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t + \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(25)
$$ p_{4} = \frac{1}{4}\left( { - \frac{{\mu_{1} }}{{v_{1} }}\Updelta t + \frac{{\mu_{2} }}{{v_{2} }}\Updelta t + \frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t - \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(26)
$$ p_{5} = 1 - \frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t $$
(27)
From Eqs. 23 and 24, we can also determine the relationship between λ1 and λ2, which can be expressed as:
$$ \frac{{\lambda_{1}^{2} }}{{\lambda_{2}^{2} }} = \frac{{\sigma_{1}^{2} \left( {\sigma_{2}^{2} + \mu_{2} \Updelta t} \right)}}{{\sigma_{2}^{2} \left( {\sigma_{1}^{2} + \mu_{1} \Updelta t} \right)}} $$
(28)
Rearranging Eqs. 2327, we have:
$$ p_{1} = \frac{1}{4}\left( {\frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t + \left( {\frac{{\mu_{1} }}{{v_{1} }} + \frac{{\mu_{2} }}{{v_{2} }}} \right)\Updelta t + \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(29)
$$ p_{2} = \frac{1}{4}\left( {\frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t + \left( {\frac{{\mu_{1} }}{{v_{1} }} - \frac{{\mu_{2} }}{{v_{2} }}} \right)\Updelta t + \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(30)
$$ p_{3} = \frac{1}{4}\left( {\frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t + \left( { - \frac{{\mu_{1} }}{{v_{1} }} - \frac{{\mu_{2} }}{{v_{2} }}} \right)\Updelta t + \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(31)
$$ p_{4} = \frac{1}{4}\left( {\frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t + \left( { - \frac{{\mu_{1} }}{{v_{1} }} + \frac{{\mu_{2} }}{{v_{2} }}} \right)\Updelta t + \frac{{\sigma_{1} \sigma_{2} \rho \Updelta t}}{{v_{1} v_{2} }}} \right) $$
(32)
$$ p_{5} = 1 - \frac{{\sigma_{2}^{2} }}{{v_{2}^{2} }}\Updelta t $$
(33)
and
$$ \lambda_{i}^{2} = \frac{{\sigma_{i}^{2} \left( {\sigma_{j}^{2} + \mu_{j} \Updelta t} \right)}}{{\sigma_{j}^{2} \left( {\sigma_{i}^{2} + \mu_{i} \Updelta t} \right)}}\lambda_{j}^{2} $$
(34)

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.

When the approximated process includes the opportunity for horizontal jumps, there will exist 2k + 1 probability, such that:
$$ \sum\limits_{m = 1}^{{2^{k} + 1}} {p_{m} } = 1. $$
where m denotes as the state of economy and m is allocated between 1 and 2k + 1 in our framework.
Here we set \( v_{i} = \lambda_{i} \sqrt {\sigma_{i}^{2} \Updelta t + (\mu_{i} \Updelta t)^{2} } \) so as to ensure the existence of horizontal jumps. The necessary conditions for convergence require the equality of the means, variance, and the pair-wise covariance terms. For the 2k probability, we have:
$$ p_{m} = \frac{1}{{2^{k} }}\left\{ {\frac{{\sigma_{k}^{2} }}{{v_{k}^{2} }}\Updelta t + \Updelta t\sum\limits_{i = 1}^{k} {x_{im} \left( {\frac{{\mu_{i} }}{{v_{i} }}} \right) + \pi_{i}^{k} \left( {\frac{{\sigma_{i} }}{{v_{i} }}} \right)\sum\limits_{i = 1}^{k - 1} {\sum\limits_{j = i + 1}^{k} {x_{ij}^{m} (\rho_{ij} )} } } } \right\} $$
(35)
where
$$ x_{im} = \left\{ {\begin{array}{*{20}c} { 1\quad{\text{if}}\,{\text{asset}}\,i\,{\text{has}}\,{\text{an}}\,{\text{up}}\,{\text{jump}}\,{\text{in}}\,{\text{state}}\,m} \\ { - 1\quad{\text{if}}\,{\text{asset}}\,i\,{\text{has}}\,{\text{a}}\,{\text{down}}\,{\text{jump}}\,{\text{in}}\,{\text{state}}\,m} \\ \end{array} } \right. $$
$$ x_{ij}^{m} = \left\{ {\begin{array}{*{20}c} { 1\quad{\text{if}}\,{\text{assets}}\,i\,{\text{and}}\,j\,{\text{have}}\,{\text{jumps}}\,{\text{in}}\,{\text{the}}\,{\text{same}}\,{\text{direction}}\,{\text{in}}\,{\text{state}}\,m} \\ { - 1\quad{\text{if}}\,{\text{asset}}\,i\,{\text{and}}\,j\,{\text{have}}\,{\text{jumps}}\,{\text{in}}\,{\text{the}}\,{\text{opposite}}\,{\text{direction}}\,{\text{in}}\,{\text{state}}\,m} \\ \end{array} } \right. $$
In addition, the probability for the horizontal jump, \( p_{{2^{k} + 1}} \) is:
$$ p_{{2^{k} + 1}} = 1 - \frac{{\sigma_{k}^{2} }}{{v_{k}^{2} }}\Updelta t $$

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.

Suppose that a call option written on an asset with strike price, K = 100, expires at Date 2. The possible underlying asset prices for the O-trinomial and I-trinomial tree processes at Dates 1 and 2, under the given parameters, are provided in Table 1.
Table 1

Transformed-trinomial price processes with a single underlying asset

Prices

Models

O-trinomial

I-trinomial

u

2

1.0214

d

0.5

0.9790

S0,0

100

100

S1,0

50

50

S1,1

100

100

S1,2

200

200

S2,0

25

28.26

S2,1

50

50

S2,2

100

100

S2,3

200

200

S2,4

400

452.27

The table shows the price processes for the O-trinomial and I-trinomial models; we assume that there are two periods, and that the initial price (S0,0) is 100, the exercise price (K) = 100, r = 0.1, and Δt = 1 year

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.

The convergence property of our transformed-trinomial model for the valuation of a European put option written on one underlying asset is shown in Table 2, under the assumption that the underlying asset follows displaced distribution. The option maturity is arranged from 1 month to 3 years and we assume that the current stock price is equal to 50, the annualized risk-free interest rate is equal to 0.05, the annualized return volatility of the underlying asset is equal to 0.2, and the parameter, α, is equal to 10. We use three strike prices, 45, 50 and 55, to compare the in-the-money, at-the-money and out-of-money results for the options. In addition, we use the closed-form solution derived by Camara (2005) as the benchmark for the convergence test.
Table 2

European put option values with displaced distribution

K/S

Maturity perioda

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 pricesb

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 pricesc

    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

aThe option maturity periods are arranged from 1 month to 3 years

bThe theoretical values are denoted as the option values computed by Camara (2005). We assume that the current stock price S(0) = 50, the annualized risk-free interest rate r = 0.05, the annualized return volatility of the underlying asset σ = 0.2, and the parameter α = 10

cThe trinomial tree prices are obtained from: (\( \frac{m - 1}{2} \) steps)

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%.

Table 3 reports the results for the convergence property for the valuation of an American put option, again under the assumption of displaced distribution for the underlying asset price. However, for this case we use the trinomial tree for 10,000 steps as the benchmark since we have no closed-form solution as a benchmark. Again, we find that the transformed-trinomial tree quickly converges to the benchmark value.
Table 3

Valuation of American put options with a single underlying asset under displaced distribution

K/S

Maturity perioda

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 pricesc

    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

aThe option maturity periods are arranged from 1 month to 3 years

bThe theoretical values are obtained by using our transformed-trinomial model with 10,000 time steps. We assume that the current stock price S(0) = 50; the annualized risk-free interest rate r = 0.05; the annualized return volatility of the underlying asset σ = 0.2; and the parameter α = 10

cThe trinomial tree prices are obtained from: (\( \frac{m - 1}{2} \) steps)

Table 4 uses a binary option as an example to demonstrate the way in which the price paths of the underlying assets evolve when the binary call option has one period to go. Following Camara (2005), we assume that Asset 1 follows log-normal distribution, and Asset 2 follows normal distribution. We then calculate the value of the binary option written on these two underlying assets with different distributions, and find that the binary option would have a payoff of $1 if S1 > K1 and S2 > K2.
Table 4

One-period transformed-trinomial price processes with two underlying assets

Price processesa

Underlying Asset 1

Underlying Asset 2

S0,0

100

100

S1,0

41.1952

101.244

S1,1

41.1952

100.804

S1,2

100

100

S1,3

299.927

101.244

S1,4

299.927

100.804

Theoretical binary option value = 0.427184

The table shows the results of the binary asset process within which Asset 1 is assumed to follow log-normal distribution, whilst Asset 2 is assumed to follow normal distribution

aWe assume that there is one period and that the initial prices of the two assets are 100, the exercise price k = 100, r = 0.1, and Δt = 1 year

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.

We can see from Table 5 that the transformed-trinomial tree method has a better convergence property than the binomial tree method; that is λ = 1 does not have the best convergence speed, which is consistent with the results of Kamrad and Ritchken.
Table 5

Price differences between the transformed-trinomial model and the benchmark values

Strike pricea

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

The table shows the differences between the true and computed European binary options with different jump sizes; the payoff for the binary option is $1 if S1 > K1 and S2 > K2

aWe assume that the initial prices of the two assets are 100, k1 = k2 = 100, r = 0.1, Δt = 1 year, σ1 = σ2 = 0.3, and the correlation coefficient is 0.5

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 \( \lambda = \sqrt 2 . \)

The convergence property of the transformed-trinomial model is shown in Fig. 1. In the spirit, for the case where λ = 1, the transformed-trinomial tree model reduces to the transformed-binomial tree model.7
https://static-content.springer.com/image/art%3A10.1007%2Fs11156-009-0121-3/MediaObjects/11156_2009_121_Fig1_HTML.gif
Fig. 1

Convergence of the five-jump models for the binary option, by jump size. a The figure shows the convergence properties of the five-jump models for the binary option with λ = 1.00 and 1.29100; the payoff for the binary option is $1 if S1 > K1 and S2 > K2. b We assume that the initial prices of the two assets are 100, k1 = k2 = 100, r = 0.1, Δt = 1 year, σ1 = σ2 = 0.3, and the correlation coefficient is 0.5

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.

Footnotes
1

Hull and White (1988), for example, used a control variable technique as the means of enhancing convergence.

 
2

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.

 
3

Nelson and Ramaswamy (1990), for instance, derived the conditions for a sequence of binomial processes to converge weakly to a diffusion process.

 
4

See, for example, Hilliard et al. (1996) and Ritchken and Trevor (1999).

 
5

Hull and White (1993) developed lattice approaches for the valuation of both look-back options and Asian options.

 
6

This is because: “μ and σ are first determined from skewness and kurtosis; the other two parameters, α and β, are then determined so that agreement in mean and standard deviation is obtained” (Johnson 1949a: 155).

 
7

We do not mean that our transformed-trinomial model will exactly reduce to Camara and Chung (2006) transformed-binomial model since they use replication approaches to construct their transformed-binomial model, whereas we use moment-matching method to build up our transformed-trinomial model.

 

Acknowledgments

We are especially grateful for Professors A.H.W. Wang constructive comments.

Copyright information

© Springer Science+Business Media, LLC 2009