Local volatility models are popular because they can be simply calibrated to the market of European options. For such models, we propose a modified Leland method which allows us to approximately replicate a European contingent claim when the market is under proportional transaction costs. The convergence of our scheme is shown by means of a new strategy of proof based on PDEs techniques allowing us to obtain appropriate Greeks estimations.
This thesis studies the problem of approximate hedging with constant proportional transaction costs in stochastic volatility models in different situations, using a simpler form for adjusted volatility in the Leland's algorithm. We show that asymptotic properties of hedging error are the same to those in deterministic volatility models and the rate of convergence can be impoved by controlling the model parameter. These can be extended to the case where transaction costs are defined by a general rule. We also show that jumps appear in asset price and/or in stochastic volatility do not affect asymptotic property of hedging error. In the next part, we consider the problem of approximate hedging in the presence of liquidity risks suggested by Cetin, Jarrow and Protter, of which proportional transaction costs models are a particular case. We show that liquidity costs due to smooth supply surves can be ignored using Leland's increasing volatility principle. In the third part, we study the case where the option is written on multiple risky assets. We demonstrate that approximately complete replication can be reached for exchange options using the same parameter suggested by Leland, but it is far from being obvious for other kinds of exotic options. Finally, we propose a simple method to reduce the option price which clearly approaches to the super hedging price in Leland's algorithm. whenever the seller accepts to take a risk defined by a given significance level.
We introduce a jump-diffusion model for asset returns with jumps drawn from a mixture of normal distributions and show that this model adequately fits the historical data of the Samp;P500 index. We consider delta-hedging strategy for vanilla options under the diffusion model (DM) and the proposed jump-diffusion model (JDM) assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profit-and-loss (Pamp;L) of the delta-hedging strategy under the both models. We find that, under the log-normal model by Black-Scholes-Merton, the actual PDF of the Pamp;L can be well approximated by the chi-squared distribution with specific parameters. We derive an approximation for the Pamp;L volatility in the DM and JDM. We show that, under the both DM and JDM, the expected loss due to transaction costs is inversely proportional to the square root of the hedging frequency. We apply the mean-variance analysis to find the optimal hedging frequency given the hedger's risk tolerance. Since under the JDM it is impossible to reduce the Pamp;L volatility by increasing the hedging frequency, we consider an alternative hedging strategy, following which the Pamp;L volatility can be reduced by increasing the hedging frequency.
In this paper we consider the problem of hedging contingent claims on a stock under transaction costs and stochastic volatility. Extensive research has clearly demonstrated that the volatility of most stocks is not constant over time. As small changes of the volatility can have a major impact on the value of contingent claims, hedging strategies should try to eliminate this volatility risk. We propose a stochastic optimization model for hedging contingent claims that takes into account the effects of stochastic volatility, transaction costs and trading restrictions. Simulation results show that our approach could improve performance considerably compared to traditional hedging strategies.
We derive a nonlinear parabolic partial differential equation for the value of portfolios of options in the presence of proportional transaction costs. This assumes a Leland world of transacting after each time interval, which is of fixed length. The equation reduces to the modified variance case described by Leland in the case of a single option. We demonstrate the nonlinear nature of option portfolios and give results for several simple combinations of options.
We develop a dynamic model in which traders have differential information about the true value of the risky asset and trade the risky asset with proportional transaction costs. We show that without additional assumption, trading volume can not totally remove the noise in the pricing equation. However, because trading volume increases in the absolute value of noisy per capita supply change, it provides useful information on the asset fundamental value which cannot be inferred from the equilibrium price.We further investigate the relation between trading volume, price autocorrelation, return volatility and proportional transaction costs. Firstly, trading volume decreases in proportional transaction costs and the influence of proportional transaction costs decreases at the margin. Secondly, price autocorrelation can be generated by proportional transaction costs: under no transaction costs, the equilibrium prices at date 1 and 2 are not correlated; however under proportional transaction costs, they are correlated - the higher (lower) the equilibrium price at date 1, the lower (higher) the equilibrium price at date 2. Thirdly, we show that return volatility may be increasing in proportional transaction costs, which is contrary to Stiglitz 1989, Summers amp; Summers 1989's reasoning but is consistent with Umlauf 1993 and Jones amp; Seguin 1997's empirical results.
Although the reconstructed local volatility function can give an accurate approximation to the market volatility which leads to more efficient hedging, using the implied volatility calculated from a carefully chosen model is more ready to be accepted for the sake of easy implementation. We compare the dynamic hedging performance of the constant elasticity of variance model family with constant volatility Black-Scholes model. By assuming that the underlying stock follows a specific diffusion process, we use hypothetic examples to illustrate that Black-Scholes model yields the largest hedging parameter and the greater standard error of relative hedging error. Our observation shows that the constant volatility model can result in significant hedging error, and by allowing for varying volatility, the constant elasticity of variance model is more adaptive.
The behaviour of a smile model when applied to hedging should be consistent with market evidence that asset prices and market smiles move in the same direction (Hagan et al. 2002). Local volatility models are criticized because not consistent with this desired behaviour, and this has been an important driver towards the use of stochastic volatility models.In this work we perform a simple analysis showing that, if we take into account explicitly the correlation between stochastic volatility and underlying asset which is typical of the most common stochastic volatility models, the hedging behaviour of stochastic volatility models does not always conform with the desired behaviour of a smile model in hedging.With further simple tests we show that the behaviour of local volatility and stochastic volatility models calibrated to market skew is less different than assumed in current market wisdom. Both approaches, when used consistently with model assumptions, do not show the desired behaviour in hedging, while for both models the desired behaviour is obtained in market practice by hedging techniques which are not fully consistent with rigorous model assumptions.
