Valuing Time-Dependent CEV Barrier Options
We have derived the analytical kernels of the pricing formulae of the CEV knockout options with time-dependent parameters for a parametric class of moving barriers. By a series of similarity transformations and changing variables, we are able to reduce the pricing equation to one which is reducible to the Bessel equation with constant parameters. These results enable us to develop a simple and efficient method for computing accurate estimates of the CEV single-barrier option prices as well as their upper and lower bounds when the model parameters are time-dependent. By means of the multistage approximation scheme, the upper and lower bounds for the exact barrier option prices can be efficiently improved in a systematic manner. It is also natural that this new approach can be easily applied to capture the valuation of other standard CEV options with specified moving knockout barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, more comparative pricing and precise risk management in equity options can be achieved by incorporating term structures of interest rates, volatility, and dividend into the CEV option valuation model.
In recent years European barrier options have become extremely popular in world markets. Unlike standard options, a barrier option is a path dependent option in which the existence of the option depends upon whether the underlying asset price has touched a critical value, called the barrier, during the option's lifetime. Should the price of the underlying asset breach this barrier before option expiration, the option will be extinguished immediately. An advantage of trading barrier options is that they provide more flexibility in tailoring the portfolio returns while lowering the cost of option premiums. The pricing of barrier options has been studied in many literatures assuming the underlying asset price to follow a lognormal diffusion process, that is, the Black-Scholes environment . Merton  was the first to derive a closed-form solution for a down-and-out European call option. Other closed-form pricing formulae of single-barrier options were published in [3–8]. The analytical valuation of double-barrier options was discussed in [9–14].
The Black-Scholes option pricing model is a member of the class of constant elasticity of variance (CEV) option pricing models. The diffusion process of stock price in a CEV model can be expressed as where is the instantaneous mean, is the instantaneous variance of the stock price, is a Weiner process, and is the elasticity factor. The equation shows that the instantaneous variance of the percentage price change is equal to and is a direct inverse function of the stock price. In the limiting case , the CEV model returns to the conventional Black-Scholes model in which the variance rate is independent of the stock price. In another case , it is the Ornstein-Uhlenbeck model. Several theoretical arguments imply an association between stock price and volatility. Black  and Christie  consider the effects of financial leverage on the variance of the stock. A fall in the stock price increases the debt-equity ratio of the firm; therefore, both the risk and the variance of the stock increase. Black also proposes that a downturn in the business cycle might lead to an increase in the stock price volatility and hence to a fall in the stock prices.
Empirical evidence has shown that the CEV process may be a better description of stock behavior than the more commonly used lognormal model because the CEV process allows for a nonzero elasticity of return variance with respect to prices. Schmalensee and Trippi  find a strong negative relationship between stock price changes and changes in implied volatility after examining over a year of weekly data on six stocks. By applying the trading profits approach on 19000 daily warrant price observations, Hauser and Lauterbach  find that the CEV model roughly doubles the trading excess returns of the Black-Scholes model. The superiority of the CEV model is strongest in out-of-the-money and longer time to expiration warrants. The results are consistent with the findings in Lauterbach and Schultz . If the relationship between the variance and the stock price is deduced from the empirical data, an option pricing formula based on the CEV model could fit the actual market option prices better than the Black-Scholes model. Beckers  finds thirty-seven out of forty-seven stocks in a year daily data set to have estimated to be less than two and concludes that the CEV diffusion process could be a better candidate of describing the actual stock price behavior than the Black-Scholes model.
In addition to providing a better description of stock behavior, the CEV process can be employed in the contingent-claims approach to valuing defaultable bonds. For example, in a valuation model of defaultable bonds proposed by Cathcart and El-Jahel  recently, default occurs when some signaling process hits some constant default barrier (i.e., the option to default can be considered as a barrier option). The model assumes the signaling process for each firm that determines the occurrence of default rather than the value of the assets of the firm. The signaling process can capture factors that can affect the probability of default. The use of the signaling process is also appropriate for entities such as sovereign issuers that issue defaultable debts but do not have an identifiable collection of assets. The signaling process could follow diffusion processes such as lognormal, Ornstein-Uhlenbeck, or CEV processes.
The derivation of the CEV option pricing formula with (commonly known as the “square-root process”) was first presented by Cox and Ross  as an alternative diffusion process for valuation of options. Cox  also derived the option pricing formula for . All these derivations assume the model parameters such as volatility, interest rate, and dividend yield are constant. However, the model parameters are actually time-dependent in market. The time-dependent term structures of interest rates and volatility which can be implied from the money market and the option market, respectively, are expressed as time-dependent stepwise functions. The term structures can also be expressed as analytical functions to reflect expectation and dynamics of market factors. Only recently Lo et al.  succeeded in introducing time-dependent parameters into the CEV process, and obtaining the closed-form option pricing formula explicitly.
The valuation of European CEV barrier options with time-dependent model parameters is not a trivial extension. So far as we know, no simple and accurate approximation scheme is available yet. In a recent paper Lo and Hui  generalize the Lie-algebraic technique of Lo et al.  to derive the analytical kernels of the pricing formulae of the CEV knockout options with time-dependent parameters for a parametric class of moving barriers. By best fitting the fixed barrier by these parametric moving barriers, they also provide a simple method for computing tight upper and lower bounds of the prices of the single-barrier options (both call and put options). In this paper we try to provide an alternative derivation of the analytical pricing kernels via a more systematic approach, which involves a series of similarity transformations and changing variables so as to convert the pricing equation into one reducible to the Bessel equation with constant parameters. The solution of the resultant equation subject to absorbing boundary conditions is well known and discussed in most standard textbooks on partial differential equations. Then very tight upper and lower bounds of the exact barrier option prices can be calculated efficiently using the multistage approximation, and these bounds can also be systematically impoved in a straightforward manner.
The remainder of this paper is structured as follows. In the next section we present the derivation of the analytical kernels of the pricing formulae of the CEV knockout options with time-dependent parameters for a parametric class of moving barriers, and describe our formulation for evaluating accurate approximation of the value of a single-barrier European CEV option with time-dependent parameters. Section 3 presents some illustrative examples and examines the accuracy and efficiency of our approximate approach. Tight upper and lower bounds of the exact barrier option prices are also calculated. In Section 4 the multistage approximation scheme is proposed to systematically tighten the upper and lower bounds. It is found that even a rather low-order approximation can yield very tight bounds of the exact barrier option prices. Numerical results are discussed in detail. In Section 5, we apply the multistage approximation scheme to valuate barrier options with time-dependent volatilities. In Section 6 we summarize our investigation and provide suggestions for future research.
2. CEV Single-Barrier Options
The CEV model with time-dependent model parameters for a standard European option is described by the partial differential equation  for . Here is the option value, is the underlying asset price, is the time to maturity, is the volatility, is the risk-free interest rate, and is the dividend. Introducing a simple change of variables: , (2.1) can be recast in the following form: where and This equation represents a generalization of the Fokker-Planck equation associated with the well-known Rayleigh process . It is not difficult to show that the operator can be rewritten as follows: where The operators , and are the generators of the Lie algebra su(1,1) :
Without loss of generality we first assume the solution of (2.2) takes the form Substituting (2.6) into (2.2) yields Then we set the coefficient associated with in (2.7) equal to zero and obtain where is an arbitrary real constant and the are defined by Next, we perform the time-dependent similarity transformation to (2.7) so that it becomes where and Assuming that , where , (2.11) is reduced to whose formal solution is given by It should be noted that (2.13) is reducible to the Bessel equation of order by the separation of variables, and detailed analysis of its solutions for different boundary conditions are well documented in most standard textbooks on partial differential equations [28–30].
2.1. Up-and-Out Moving Barrier Options
Now we try to solve (2.13) for and with absorbing boundary conditions. Without loss of generality, we assume that is defined in terms of the Fourier-Bessel integral : for . Here denotes the zero of the Bessel function of the first kind of order . Then, making use of the fact that it is not difficult to show that As a result, the price of the corresponding up-and-out option is given by where In the above derivation we have made use of the well-known relation It can be easily seen that at time the kernel vanishes at . That is, we have derived the kernel of the pricing equation in (2.2) with an up-and-out barrier belonging to the class of trajectories: parametrized by the real adjustable parameter .
2.2. Down-and-Out Moving Barrier Options
On the other hand, for a down-and-out option with the barrier following the trajectory defined by (2.21), we would suppose that is defined in terms of the Weber transform : for . Then, it is straightforward to show that is simply given by where is the kernel of the pricing equation in (2.2) associated with a down-and-out barrier moving along the trajectory given in (2.21). Here denotes the Bessel function of the second kind of order . It should also be noted that the Gaussian decaying factor of the integrand ensures the rapid convergence of the integration over . Accordingly, the price of the corresponding down-and-out moving barrier option is found to be Furthermore, it is not difficult to see that in the special case of , that is, no barrier, the kernel in (2.24) is reduced to the one obtained by Lo et al.  which has a Gaussian decaying factor in the variable . By the maximum principle for the parabolic partial differential equation , we can thus conclude that the kernel in (2.24) must have a decaying factor in the variable , which decays at least as fast as the Gaussian decaying factor in the special case of .
2.3. Illustrative Applications
If we take a closer look at the trajectory of the moving barrier defined in (2.21), we would immediately realize that the special case of a fixed barrier does not belong to the class of parametric barriers. In order to simulate a fixed barrier, we will thus choose an optimal value of the adjustable parameter in such a way that the integral is minimum. In other words, we try to minimize the deviation from the fixed barrier by varying the parameter . Here denotes the time at which the option price is evaluated. An illustrative example of such an optimal fluctuating barrier (represented by the dashed line) is shown in Figure 1. It is clear that the approximation is indeed very good. Within the framework of the new approach, we can also determine the upper and lower bounds for the exact barrier option prices. It is not difficult to show that for an up-and-out option the upper bound can be provided by the option price associated with a moving barrier whose is greater than or equal to for the duration of interest. (The proof is based upon the maximum principle for the parabolic partial differential equation .) Similarly, the option price associated with a moving barrier whose is less than or equal to for the duration of interest can serve as the desired lower bound. In this example, the best lower bound can be obtained by choosing an appropriate value of such that . That is, at time the moving barrier will return to its initial position and merge with the fixed barrier. In Figure 1 an example of such a barrier movement is denoted by the long-dashed line. On the other hand, the best upper bound can be obtained by choosing a value which satisfies the requirement that at . That is, the instantaneous rate of change of is required to be zero at time . An example of such a barrier movement is represented by the solid line in Figure 1. On the contrary, for a down-and-out option we can simply switch the above two choices of barrier movement in order to determine the upper and lower bounds of the option price.
3. Illustrative Examples
For illustration, we apply the approximate method to a “”-CEV up-and-out call option with constant model parameters: , , . (Note that the value of to be used for the CEV model is adjusted to be .) The strike price and the knockout barrier are set equal to and , respectively. We now try to evaluate the barrier option price associated with the current underlying asset price at time .
First of all, we determine the optimal value of the adjustable parameter : Then an estimate of the exact up-and-out barrier option price can be evaluated by numerically computing the integral in (2.18) (with, e.g., Mathematica): Since the exact value of the barrier option price is found to be  the approximate estimate is indeed very close to the exact result with a percentage error of only. The numerical results for the corresponding upper and lower bounds are determined as follows: The barrier tracks for the estimate and bounds of the option price are shown in Figure 1. Clearly, the new approach is able to yield very tight upper and lower bounds for the exact barrier option price. To further illustrate the accuracy of the new approach, we also calculate the estimates and bounds of the option prices corresponding to some other underlying asset prices, namely, , , and . The numerical results are listed in Tables 1(a) and 1(b). It is clear that the estimates and bounds are remarkedly good; all of them have a percentage error of less than percent above or below the exact option value.
In order to assess the efficiency of the new approach, we also perform Monte Carlo simulations to evaluate the option prices. As shown in Tables 1(a) and 1(b), using a time-step of and a sample of random paths of the underlying asset price, the Monte Carlo method gives much poorer estimates in comparison with the new approach. Furthermore, to examine the robustness of the new approach, we carry out the same kind of investigations for up-and-out call options in different CEV environments too. The numerical results corresponding to different values, namely, and , are tabulated in Tables 2 and 3, respectively. Beyond question, the advantages of the new approach are clearly demonstrated by these data.
4. Systematic Multistage Approximation
As the time to maturity increases beyond one year, that is, , or the model parameters have more dramatic term structures, the accuracy of the estimates and bounds of the approximate method decreases. To obtain the same accuracy as before, we can approximate the fixed barrier by a continuous and piecewise smooth barrier, leading to the multistage approximation scheme. For simplicity, we will concentrate on the up-and-out options to demonstrate the multistage approximation in the following. Generalization to the down-and-out options should be very straightforward. First of all, we consider the estimate of the lower bound and perform the evaluation in two stages.
Stage (the time interval ). Following the same procedure as that discussed in Section 3, we choose an appropriate value of the parameter , denoted by , such that . This determines the movement of the barrier within the time interval . The corresponding price function is given by the integral in (2.18) with the kernel associated with . One can efficiently calculate the integral numerically using either the Mathematica or the Gauss quadrature method.
Stage (the time interval ). We repeat the procedure in Stage 1 such that . This will give us another value of , denoted by , and determine the moving barrier's trajectory for the time interval . Then, the corresponding price function is given by
Again, the integration in (4.1) can be efficiently evaluated using either the Mathematica or the Gauss quadrature method.
In Figure 2 the long-dashed line gives an illustrative example of the moving barrier's trajectory within the two-stage approximation scheme. It is clear that the deviation from the fixed barrier is much reduced in this two-stage approximation. The corresponding numerical results for different CEV up-and-out call options in Tables 1–3 also demonstrate that the lower bounds are dramatically improved. Apparently, one can further improve the lower bounds by splitting the evaluation process into more stages instead.
Next, we discuss how to implement the multistage approximation scheme to improve the upper bound. For the two-stage approximation, the in the single-stage approximation is used for the time interval , that is, we set . At , another value of , denoted by , is selected so that the moving barrier will then start moving back to its initial position and merge with the fixed barrier at . As a result, the corresponding price function is given by (i) the integral in (2.18) with the kernel associated with the parameter for , and (ii) the following expression:
where for . All the convolution integrals can be numerically evaluated using either the Mathematica or the Gauss quadrature method. Further improvement in the estimation of the upper bound can be easily achieved by the approximation involving more stages. In Figure 2 the solid line gives an example of the moving barrier's trajectory for the two-stage approximation. The numerical results in Tables 1–3 show that the upper bounds are significantly improved.
As expected, the multistage approximation for both the upper and lower bounds becomes better and better as the number of stages increases; in fact, the gap between the bounds is asymptotically reduced to zero. In practice even a rather low-order approximation can yield very tight upper and lower bounds to the exact option price function. It should be pointed out that the above multistage approximation can be applied to an up-and-out put option in a similar manner as well.
5. Barrier Options with Time-Dependent Volatilities
Now, we apply the multistage approximation method to the case of a time-dependent volatility with the term structure. where , , and . This term structure can be interpreted as a pulse of surge or drop (depending upon the sign of ) in market volatility. The centre of the pulse is at time , and the width of the pulse is determined by . Other input parameters remain unchanged. In this example the three-stage approximation is used, and Figure 3 shows a couple of the typical barrier tracks. Because of the term structure of the volatility, the barrier movement is a little bit more complicated than those cases with constant volatilities. However, the spirit of the approximation scheme remains the same. Numerical results of the bounds of the barrier option prices for , and are shown in Table 4. For comparison, we also include the numerical data generated by Monte Carlo simulations using a time step of and a sample of random paths of the underlying asset price. It can be seen that the bounds are indeed very tight whereas the Monte Carlo simulations give rather poor estimates. Furthermore, even though the approximation involves three stages, the convergence of the calculations is still very rapid for it takes less than two minutes to give one estimate of the barrier option value.
Finally, we generalize the multistage approximation scheme to the CEV down-and-out options. Table 5 shows the results of a down-and-out put option with time-dependent volatility. It should be noted that although we need to evaluate a double integral in this case, our proposed method works very well in evaluating the option prices, in terms of both accuracy and efficiency.
By a series of similarity transformations and changing variables, we have derived the analytical kernels of the pricing formulae of the CEV knockout options with time-dependent parameters for a parametric class of moving barriers. These results enable us to develop a simple and efficient method for computing accurate estimates of the single-barrier option prices (both call and put options) as well as their upper and lower bounds in the CEV model environment when the model parameters are time-dependent. By means of the multistage approximation scheme, the upper and lower bounds for the exact barrier option prices can be efficiently improved in a systematic manner. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, more comparative pricing and precise risk management in equity options can be achieved by incorporating term structures of interest rates, volatility, and dividend into the CEV option valuation model. Extension to the CEV double-knockout options with time-dependent parameters can also be straightforwardly achieved by solving (2.13) for and with absorbing boundary conditions at both endpoints. Such a boundary value problem is well documented in most standard textbooks on partial differential equations, but the solution will involve explicit searching for eigenvalues numerically. Moreover, it is natural that this new approach can be easily applied to capture the valuation of standard CEV options with specified moving knockout barriers. Finally, we would like to point out that the results for the down-and-out CEV option can be generalized to price the CEV American put option and lookback options with time-dependent parameters too. This research is now in progress and results will be published elsewhere.
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