T. F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2:77--102, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....# = 2 to price these instruments and charge the fair value. This defines a market consistent with the jump di#usion model parameters given above. In order to facilitate comparisons between the various models of the underlying asset, we calibrated a local volatility function as described in [6], and a constant implied These parameters are approximately the values reported in [1] which the authors found were implied in a certain set of S P options market prices. Source: the local volatility function was computed using the Calcvol volatility surface calibration program developed ....

T. F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2(3):77--102, 1999.

....derivatives of the solution for the fair value. It is common knowledge that the constant volatility Black Scholes model is not consistent with market prices. In order to match observed market prices for options, traders use a matrix of implied volatilities [28] or generate a volatility surface [5]. However, as discussed in [2] volatility surfaces tend not to be very stable as a function of time. In particular, the surface obtained by matching today s prices, tends to become very flat as one looks out farther in time. This is a significant problem if this surface is used to price and hedge ....

T.F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2:77--102, 1999.

.... recently (see [4] for an overview and some empirical evidence) Using a fully numerical approach enables us to study the effect of deviations from this common assumption by employing alternative models such as the constant elasticity of variance (CEV) model [12] and implied volatility surfaces [2, 11]. We will find that different models for the volatility can lead to dramatically different prices and hedging strategies. ffl Best worst case uncertain parameter (e.g. volatility, interest rate, dividend yield) models [3, 25] may be particularly suited to this type of application, because the ....

....Dimensional Valuation So far we have concentrated on contracts and modeling assumptions which admit a similarity solution. Although these are important special cases, we also wish to compute the solution for alternative underlying asset price models (e.g. CEV models [12] or volatility surfaces [2, 11]) In such cases we must solve the full three dimensional time dependent numerical problem. We are then faced with two additional possible sources of error: i) interpolation of the numerical PDE solution to obtain the constraint; and ii) meshing complications arising from locally refined meshes ....

[Article contains additional citation context not shown here]

T.F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2(3):77--102, 1999.

.... been subjected to increasing scrutiny recently (see [2] for an overview and some empirical evidence) Using a fully numerical approach enables us to study the effect of deviations from this common assumption such as constant elasticity of variance (CEV) models [9] and implied volatility surfaces [1, 8]. Numerical schemes frequently used in the finance industry are not well suited for valuation of these contracts. At present, standard Monte Carlo approximation techniques, which work forward in time, cannot effectively handle the optimization component of these contracts. However, a backward ....

....Dimensional Valuation So far we have concentrated on contracts and modelling assumptions which admit a similarity solution. Although these are important special cases, we also wish to compute the solution for alternative underlying asset price models (e.g. CEV models [9] or volatility surfaces [1, 8]) In such cases we must solve the full three dimensional time dependent numerical problem. We are then faced with two additional possible sources of error: i) interpolation of the numerical PDE solution to obtain the constraint; and ii) meshing complications arising from locally refined meshes ....

[Article contains additional citation context not shown here]

T.F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2(3):77--102, 1999.

....coupon semi annually. We have only considered constant parameters in this case, but parameters dependent on time or the underlying factors do not introduce any new numerical issues and thus can be easily accommodated. For example, volatility can be made a function of time and the stock price (see [13, 24]) Although there are regions where the convection terms are large relative to the diffusion terms, we did not find this to cause any difficulties. Consequently, central weighting with = 1 2 was used to obtain the solutions. Table 4 demonstrates that a solution that is no more than 0:10 away ....

T. F. Coleman, Yuying Li, and A. Verma. Reconstructing the Unknown Local Volatility Function. In Proceedings of Computational and Quantitative Finance '98, New York, 1998.

....in specifying the model for the underlying price movement (model specification error) can lead to poor hedge performance. In this article, we compare the e#ectiveness of dynamic hedging using the constant volatility method, the implied volatility method, and the recent volatility function method [3]. We provide evidence that dynamic hedging using the volatility function method [3] produces smaller hedge error. We assume that there are no transaction costs, and both the risk free interest rate r and the dividend rate q are constant. Many studies have shown that the classical Black Scholes ....

....can lead to poor hedge performance. In this article, we compare the e#ectiveness of dynamic hedging using the constant volatility method, the implied volatility method, and the recent volatility function method [3] We provide evidence that dynamic hedging using the volatility function method [3] produces smaller hedge error. We assume that there are no transaction costs, and both the risk free interest rate r and the dividend rate q are constant. Many studies have shown that the classical Black Scholes constant volatility model does not adequately describe the stock price dynamics, see ....

[Article contains additional citation context not shown here]

Thomas F. Coleman, Yuying Li, and Arun Verma. Reconstructing the unknown local volatility function. The Journal of Computational Finance, 2(3):77--102, 1999.

....employ a numerical PDE approach. This offers several potential benefits: ffl Using a fully numerical approach allows for more general specifications of volatility than the basic geometric Brownian motion assumption of Black Scholes. Examples include CEV models [8] and implied volatility surfaces [1, 7]. ffl Best worst case uncertain parameter (e.g. volatility, interest rate, dividend yield) models [2, 15] can be used. Such models may be particularly suited to this type of application, because the contracts are typically quite long term and have complicated provisions. Uncertain parameter ....

....an initial strike is set become equivalent to the case of no initial strike (Figure 3(b) Since the delta of these contracts is non decreasing in S (Figure 9(a) we have S Delta Gamma V 0. 21 If even further generality is desired, it is also easy to incorporate an implied volatility surface [1, 7] into our algorithm. In the case of a simple shout option the contract received upon shouting is a vanilla option. The volatility surface can be viewed as a method of interpolating the prices of traded options to increase the consistency of the model with presently observed market prices. 6 ....

T.F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2(3):77--102, 1999.

....in specifying the model for the underlying price movement (model specification error) can lead to poor hedge performance. In this article, we compare the e#ectiveness of dynamic hedging using the constant volatility method, the implied volatility method, and the recent volatility function method [3]. We provide evidence that dynamic hedging using the volatility function method [3] produces smaller hedge error. We assume that there are no transaction costs, and both the risk free interest rate r and the dividend rate q are constant. Many studies have shown that the classical Black Scholes ....

....can lead to poor hedge performance. In this article, we compare the e#ectiveness of dynamic hedging using the constant volatility method, the implied volatility method, and the recent volatility function method [3] We provide evidence that dynamic hedging using the volatility function method [3] produces smaller hedge error. We assume that there are no transaction costs, and both the risk free interest rate r and the dividend rate q are constant. Many studies have shown that the classical Black Scholes constant volatility model does not adequately describe the stock price dynamics, see ....

[Article contains additional citation context not shown here]

Thomas F. Coleman, Yuying Li, and Arun Verma. Reconstructing the unknown local volatility function. The Journal of Computational Finance, 2(3):77--102, 1999.

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T. F. Coleman, Y. Li, and A. Verma. Reconstructing the unknown local volatility function. Journal of Computational Finance, 2:77--102, 1999.

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