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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

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

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 [

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 [

The derivation of the CEV option pricing formula with

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 [

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

The CEV model with time-dependent model parameters for a standard European option is described by the partial differential equation [

Without loss of generality we first assume the solution of (

Now we try to solve (

On the other hand, for a down-and-out option with the barrier following the trajectory defined by (

If we take a closer look at the trajectory of the moving barrier defined in (

Barrier tracks for the bounds and barrier option price estimate within the single-stage approximation scheme. Upper bound’s track (solid line), lower bound’s track (long dashed line) and barrer option price estimate’s track (dashed line). Time to maturity is equal to one. Other input parameters are

For illustration, we apply the approximate method to a “

First of all, we determine the optimal value of the adjustable parameter

Comparison of estimates and bounds of option prices with the analytical results from Lo et al., “Pricing barrier options with square root process,”

CEV up-and-out call option with

% error of estimate | Analytical result of | Monte-Carlo method | % error of MC | Lower bound of | % error of lower bound | ||||
---|---|---|---|---|---|---|---|---|---|

CEV up-and-out call option with

Upper bound of | % error of upper bound | Improved lower bound of | % error of improved lower bound of | Improved upper bound of | % error of improved upper bound of | |
---|---|---|---|---|---|---|

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

CEV up-and-out call option with

Monte-Carlo result | Lower boun of | Upper boun of | Improved lower bound of | Improved upper bound of | |||||
---|---|---|---|---|---|---|---|---|---|

CEV up-and-out call option with

Monte Carlo result | Lower bound of | Upper bound of | Improved lower bound of | Improved upper bound of | |||||
---|---|---|---|---|---|---|---|---|---|

As the time to maturity increases beyond one year, that is,

Following the same procedure as that discussed in Section

We repeat the procedure in Stage

In Figure

Next, we discuss how to implement the multistage approximation scheme to improve the upper bound. For the two-stage approximation, the

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

Barrier tracks for the bounds improved by a two-stage approximation scheme.Upper bound’s track under a two-stage scheme (solid line) and lower bound’s track under a two-stage scheme (long dashed line). Time to maturity is equal to one. Other input parameters are

Now, we apply the multistage approximation method to the case of a time-dependent volatility with the term structure.

CEV up-and-out call option with time-dependent volatility. We extend our model to the time-dependent case with the volatility term structure expressed as:

Lower bound of | Upper bound of | Monte-Carlo result | ||
---|---|---|---|---|

16 | 0.27326 | 0.27458 | 0.27240 | |

18 | 0.51414 | 0.51468 | ||

20 | 0.63327 | 0.63608 | ||

22 | 0.53918 | 0.53809 | ||

24 | 0.27824 | 0.28684 | 0.29050 | |

16 | 0.28462 | 0.28602 | 0.28539 | |

18 | 0.55640 | 0.56046 | 0.56073 | |

20 | 0.67566 | 0.68348 | 0.68740 | |

22 | 0.55016 | 0.56068 | 0.56887 | |

24 | 0.27571 | 0.28432 | 0.28782 | |

16 | 0.29076 | 0.29228 | 0.29136 | |

18 | 0.60248 | 0.60683 | 0.60574 | |

20 | 0.72862 | 0.73700 | 0.73813 | |

22 | 0.57272 | 0.58380 | 0.58778 | |

24 | 0.27361 | 0.28224 | 0.28672 |

Barrier tracks for the bounds improved by a three-stage approximation scheme. Upper bound’s track (solid line) and lower bound’s track (long dashed line). Time to maturity is equal to one. Other input parameters are

Finally, we generalize the multistage approximation scheme to the CEV

CEV down-and-out put option with time-dependent volatility. We extend our model to the time-dependent case with the volatility term structure expressed as

Lower bound of | Upper bound of | Monte-Carlo result | ||
---|---|---|---|---|

16 | 0.80290 | 0.81341 | 0.82993 | |

18 | 0.93028 | 0.93662 | 0.95328 | |

20 | 0.73071 | 0.73371 | 0.74182 | |

22 | 0.47491 | 0.47632 | 0.47829 | |

24 | 0.27759 | 0.27822 | 0.27923 | |

16 | 0.79012 | 0.80025 | 0.81260 | |

18 | 0.87756 | 0.88452 | 0.89777 | |

20 | 0.67388 | 0.67754 | 0.68690 | |

22 | 0.44145 | 0.44324 | 0.44091 | |

24 | 0.26678 | 0.26764 | 0.26970 | |

16 | 0.77761 | 0.78771 | 0.79748 | |

18 | 0.82329 | 0.83014 | 0.84160 | |

20 | 0.61817 | 0.62192 | 0.62888 | |

22 | 0.40516 | 0.40728 | 0.40804 | |

24 | 0.25092 | 0.25193 | 0.25603 |

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 (