Abstract. Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated. * Presented at the first annual conference: Computational and Quantitative Finance '98, New York. This research was conducted using resources of the Cornell Theory Center, which is supported by Cornell University, New York State, the National Center for Research Resources at the National Institutes of Health, and members of the Corporate Partnership Program. 0 1. Introduction. An option pricing model establishes a relationship between the traded derivatives, the underlying asset and the market variables, e.g., volatility of the underlying asset [4,24]. Option pricing models are used in practice to price derivative securities given knowledge of the volatility and other market variables.The celebrated constant-volatility Black-Scholes model [4,24] is the most often used option pricing model in financial practice. This classical model assumes constant volatility; however, much recent evidence suggests that a constant volatility model is not adequate [27,26]. Indeed, numerically inverting the Black-Scholes formula on real data sets supports the notion of asymmetry with stock price (volatility skew), as well as dependence on time to expiration (volatility term structure). Collectively this dependence is often referred to as the volatility smile. The challenge is to accurately (and efficiently) model this volatility smile.In practice, the constant-volatility Black-Scholes model is often applied by simply using different volatility values for options with different strikes and maturities. In this paper, we refer to this approach as the constant implied volatility approach. Although this method works well for pricing European options, it is unsuitable for more complicated exotic options and options with ea...

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Abstract. Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated. * Presented at the first annual conference: Computational and Quantitative Finance '98, New York. This research was conducted using resources of the Cornell Theory Center, which is supported by Cornell University, New York State, the National Center for Research Resources at the National Institutes of Health, and members of the Corporate Partnership Program. 0 1. Introduction. An option pricing model establishes a relationship between the traded derivatives, the underlying asset and the market variables, e.g., volatility of the underlying asset [4,24]. Option pricing models are used in practice to price derivative securities given knowledge of the volatility and other market variables.The celebrated constant-volatility Black-Scholes model [4,24] is the most often used option pricing model in financial practice. This classical model assumes constant volatility; however, much recent evidence suggests that a constant volatility model is not adequate [27,26]. Indeed, numerically inverting the Black-Scholes formula on real data sets supports the notion of asymmetry with stock price (volatility skew), as well as dependence on time to expiration (volatility term structure). Collectively this dependence is often referred to as the volatility smile. The challenge is to accurately (and efficiently) model this volatility smile.In practice, the constant-volatility Black-Scholes model is often applied by simply using different volatility values for options with different strikes and maturities. In this paper, we refer to this approach as the constant implied volatility approach. Although this method works well for pricing European options, it is unsuitable for more complicated exotic options and options with ea...

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We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion.In order to compute a better hedge parameter, accurate estimation of the local volatility function in a region surrounding the current asset price is crucial. We illustrate that a suitably implemented volatility function method can estimate this local volatility function sufficiently accurately to generate more accurate hedge parameters. Hedging using this volatility function for the absolute diffusion example leads to a smaller average absolute hedging error when compared with using the implied/constant volatility rate.When comparing the hedging performance in the S&P 500 index option market as well as the S&P 500 futures option market, we similarly observe that the delta hedge parameter from the implied/constant volatility method is typically greater than that using the volatility function approach. Examination of the hedging error reveals that using a larger delta factor greater than that of the true volatility yields more positive 1

Background and Aims:Preemptive analgesia is an antinociceptive treatment that prevents establishment of altered processing of afferent input. Pregabalin has been claimed to be more effective in preventing neuropathic component of acute nociceptive pain of surgery. We conducted a study to compare the effect of oral gabapentin and pregabalin with control group for post-operative analgesiaMaterials and Methods:A total of 90 ASA grade I and II patients posted for elective gynecological surgeries were randomized into 3 groups (group A, B and C of 30 patients each). One hour before entering into the operation theatre the blinded drug selected for the study was given with a sip of water. Group A- received identical placebo capsule, Group B- received 600mg of gabapentin capsule and Group C — received 150 mg of pregabalin capsule. Spinal anesthesia was performed at L3-L4 interspace and a volume of 3.5 ml of 0.5% bupivacaine heavy injected over 30sec through a 25 G spinal needle. VAS score at first rescue analgesia, mean time of onset of analgesia, level of sensory block at 5min and 10 min interval, onset of motor block, total duration of analgesia and total requirement of rescue analgesia were observed as primary outcome. Hemodynamics and side effects were recorded as secondary outcome in all patients.Results:A significantly longer mean duration of effective analgesia in group C was observed compared with other groups (P < 0.001). The mean duration of effective analgesia in group C was 535.16 ± 32.86 min versus 151.83 ± 16.21 minutes in group A and 302.00 ± 24.26 minutes in group B. The mean numbers of doses of rescue analgesia in the first 24 hours in group A, B and C was 4.7 ± 0.65, 4.1 ±0.66 and 3.9±0.614. (P value <0.001).Conclusion:We conclude that preemptive use of gabapentin 600mg and pregabalin 150 mg orally significantly reduces the postoperative rescue analgesic requirement and increases the duration of postoperative analgesia in patients undergoing elective gynecological surgeries under spinal anesthesia

Abstract. Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.