Dynamic hedging with a deterministic local volatility function model (original) (raw)
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An improved framework for approximating option prices with application to option portfolio hedging
Economic Modelling, 2016
As the price of the underlying asset changes over time, delta of the option changes and a gamma hedge is required along with delta hedge to reduce risk. This paper develops an improved framework to compute delta and gamma values with the average of a range of underlying prices rather than at the conventional fixed 'one point'. We find that models with time-varying volatility price options satisfactorily, and perform remarkably well in combination with the delta and delta-gamma approximations. Significant improvements are achieved for the GARCH model followed by stochastic volatility models. The new approach can ensure significant improvement in modelling option prices leading to better risk-management decision-making.
Finance Research Letters, 2007
We pointed out a mathematical flaw in Leland's (1985) paper, whose results were earlier questioned by Kabanov and Safarian (1997). This means that one cannot modify volatility for transactions costs in the Black-Scholes (1973) option pricing model to obtain exact pricing and hedging results. However, there is the question of how large the pricing and hedging errors are if Leland's technique is used. We presented a simulation study where the volatility is adjusted by the length of the trading interval and the associated transaction costs. Unfortunately, our Table 2 (but not the other tables) contained a computational error. A value of k = 0.001 was used in the calculation of options premiums, when in fact 0.002 is the round trip transaction cost. So k should really be 0.002. Hence the hedging errors in Table 2 in our article overstate the true errors. Furthermore, our original simulations include a trading cost for establishing the initial hedging position, on the presumption that investors initially hold a cash-only portfolio. The corrected Table 2 is shown here. If one considers the initial transaction costs, the errors are not negligible; see the new Table 2. Hedging on a daily basis introduces mean errors ranging from −0.0148 to −0.0706 with standard deviations from 0.3240 to 0.5635. As rebalancing becomes finer, the hedging errors generally increase with higher strike prices. For example, when t = 1/8320, mean hedging errors range from −0.0379 to −0.0904. As expected, the standard deviations of the hedging errors shrink substantially. The original Fig. 2 should be replaced by Fig. 2 shown here.
Exploring Option Pricing and Hedging via Volatility Asymmetry
SSRN Electronic Journal
This paper evaluates the application of two well-known asymmetric stochastic volatility (ASV) models to option price forecasting and dynamic delta hedging. They are specified in discrete time in contrast to the classical stochastic volatility (SV) models used in option pricing. There is some related literature, but little is known about the empirical implications of volatility asymmetry on option pricing. The objectives of this paper are to estimate ASV option pricing models using a Bayesian approach unknown in this type of literature, and to investigate the effect of volatility asymmetry on option pricing for different size equity sectors and periods of volatility. Using the S&P MidCap 400 and S&P 500 European call option quotes, results show that volatility asymmetry benefits the accuracy of option price forecasting and hedging cost effectiveness in the large-cap equity sector. However, asymmetric SV models do not improve the option price forecasting and dynamic hedging in the mid-cap equity sector.
OPTION HEDGING AND IMPLIED VOLATILITIES IN A STOCHASTIC VOLATILITY MODEL
Mathematical Finance, 1996
In the stochastic volatility framework of Hull and White (1987), we characterize the so-called Black and Scholes implied volatility as a function of two arguments: the ratio of the strike to the underlying asset price and the instantaneous value of the volatility. By studying the variations in the first argument, we show that the usual hedging methods, through the Black and Scholes model, lead to an underhedged (resp. overhedged) position for in-the-money (resp. out-of-the-money) options, and a perfect partial hedged position for at-the-money options. These results are shown to be closely related to the smile efect, which is proved to be a natural consequence ofthe stochastic volatility feature. The deterministic dependence of the implied volatility on the underlying volatility process suggests the use of implied volatility data for the estimation of the parameters of interest. A statistical procedure of filtering (of the latent volatility process) and estimation (of its parameters) is shown to be strongly consistent and asymptotically normal.
Does model fit matter for hedging? Evidence from FTSE 100 options
Journal of Futures Markets, 2012
This study implements a variety of different calibration methods applied to the Heston model and examines their effect on the performance of standard and minimum-variance hedging of vanilla options on the FTSE 100 index. Simple adjustments to the Black-Scholes-Merton model are used as a benchmark. Our empirical findings apply to delta, delta-gamma, or delta-vega hedging and they are robust to varying the option maturities and moneyness, and to different market regimes. On the methodological side, an efficient technique for simultaneous calibration to option price and implied volatility index data is introduced.
On Leland’s Option Hedging Strategy with Transaction Costs
2003
Nonzero transaction costs invalidate the Black-Scholes (1973) arbitrage argument based on continuous trading. Leland (1985) developed a hedging strategy which modifies the Black-Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Leland claimed that the exact hedge could be achieved in the limit as the length of rebalance intervals approaches zero. Unfortunately, the main theorem (Leland 1985, P1290) is in error. Simulation results also confirm opposite findings to those in Leland (1985). Since standard delta hedging fails to exactly replicate the option in the presence of transaction costs, we study a pricing and hedging model which is similar to the delta hedging strategy with an endogenous parameter, namely the volatility, for the calculation of delta over time. With transaction costs, the optimally adjusted volatility is substantially different from the stock’s volatility under the criteri...
Journal of Futures Markets, 2007
The Black-Scholes (BS; F. option pricing model, and modern parametric option pricing models in general, assume that a single unique price for the underlying instrument exists, and that it is the mid-(the average of the ask and the bid) price. In this article the authors consider the Financial Times and London Stock Exchange (FTSE) 100 Index Options for the time period 1992-1997. They estimate the We would like to thank Mike Dixon, Bryan Read, and Alexandros Kontonikas for useful comments and suggestions. The comments of an anonymous referee were proven invaluable in improving the paper. The usual disclaimer applies. ask and bid prices for the index, and show that, when substituted for the mid-price in the BS formula, they provide superior option price predictors, for call and put options, respectively. This result is reinforced further when they fit a non-parametric neural network model to market prices of liquid options. The empirical findings in this article suggest that the ask and bid prices of the underlying asset provide a superior fit to the mid/ closing price because they include market maker's, compensation for providing liquidity in the market for constituent stocks of the FTSE 100 index.
A Brief Analysis of Option Implied Volatility and Strategies
With the implementation of reform of financial system and the opening-up of financial market in China, knowing and properly utilizing financial derivatives becomes an inevitable road. The phenomenon of B-S-M option pricing model underpricing deep-in/out option prices is called volatility smile. The substantial reasons are conflicts between model's presumptions and reality; moreover, the market trading mechanism brings extra uncertainties and risks to option writers when doing delta hedging. Implied volatility research and random volatility research have been modifying B-S-M model. Giving a practical case may let reader have an intuitive and in-depth understanding.