گزینه های توقف در زیر هزینه های معامله و نوسانات تصادفی

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.

In this paper, we consider the problem of hedging contingent claims under transaction costs and stochastic volatility. Extensive research during the last two decades has demonstrated that the volatility of stocks is not constant over time (Bollerslev et al., 1992). Engle (1982) and Bollerslev (1986) introduced the family of ARCH and GARCH models to describe the evolution of the volatility of the asset price in discrete time. Econometric tests of these model clearly reject the hypothesis of constant volatility and find evidence of volatility clustering over time. In the financial literature stochastic volatility models have been proposed to model these effects in a continuous-time setting (Hull and White, 1987; Scott, 1987; Wiggins, 1987). Pricing methods for options on a stock with a stochastic volatility process are now widely available, both in the discrete-time and the continuous-time framework (Heston, 1993; Finucane and Tomas, 1997; Ritchken and Trevor, 1999). Practicable methods for hedging options under stochastic volatility are rare however. Schweizer 1991 and Schweizer 1995 has proposed methods to minimize the replication error of contingent claims in general incomplete markets, including stochastic volatility as a special case. Schweizer (1995) only considers trading strategies involving the riskless bond and the underlying stock itself. As the bond and the underlying stock price are insensitive to changes of the volatility, these hedging schemes are deemed to be inefficient compared to strategies involving traded option contracts on the underlying stock (Frey and Sin, 1999). Traded options on the underlying stock are sensitive to changes in the stock price volatility. This observation is used in the simple delta–vega hedging scheme: traded options are added to the portfolio of the investor in order to eliminate the exposure to small changes of the volatility. Unfortunately, an effective delta–vega hedge has to be rebalanced frequently. As the bid–ask spreads on exchange-traded options are considerable, frequent updating of a delta–vega hedge could result in losses due to transaction costs. Static hedging methods try to compose a buy-and-hold portfolio of exchange-traded options that replicate the payoff of the contingent claim under consideration (Derman et al., 1995; Carr et al., 1998). The static hedging strategy does not require any rebalancing and is therefore quite efficient in avoiding transaction costs. Unfortunately, the odds of coming up with a perfect static hedge for a particular over-the-counter product are small, as the number of (liquid) traded option contracts is limited. Avellaneda and Paras (1996) proposes an algorithm to construct a static portfolio of options that matches the desired payoff as closely as possible, while the residual is priced and hedged with a trading strategy involving the underlying stock. A disadvantage of this approach is that the static hedge can only be efficient if traded options are available with sufficiently similar maturity and moneyness as the over-the-counter product that has to be hedged. In this paper, we propose a stochastic optimization model to extend the simple delta–vega hedging scheme. The hedge portfolio in our model consists of the underlying stock and exchange-traded options with sufficient liquidity. The model has a limited number of trading dates on which the hedge portfolio can be rebalanced (e.g. weekly), while transaction costs and trading restrictions are taken into account. The goal of the model is to minimize hedging errors by following an appropriate dynamic trading strategy. An important feature is that we only minimize the hedging error at the first few trading dates and not until the final maturity of the contingent claim. We think that our specification of the hedging model is useful because: (1) The planning horizon of traders is shorter than the maturity of their contingent claims and they are usually more interested in overnight profits and losses. (2) The portfolio of liabilities of a trader might change frequently due to additional buying and selling. (3) Risk limits like ‘Value At Risk’ are often imposed with a relatively short horizon. The stochastic optimization hedging (SOH) model requires a set of scenarios as input. The scenarios are distinct paths for the prices of the assets and liabilities at each trading date. It is clear that the performance of a hedge constructed with the stochastic optimization model will crucially depend on the quality of the price scenarios. Without a good scheme for generating scenarios the stochastic optimization model is merely a theoretical concept, not a practicable hedging tool. Our contribution is that we propose reliable methods to construct scenarios for stochastic volatility models. Moreover, results of simulations with an illustrative example show that the SOH strategy can really outperform a delta–vega hedging scheme in the presence of transaction costs. The strategy of the stochastic optimization model makes sense intuitively: it stays close to a delta–vega neutralized position but with some additional slack to avoid needless transaction costs. We now shortly outline the contents of the paper. In Section 2, we first describe traditional hedging strategies like delta–vega hedging and static replication. Next we introduce the SOH model for markets with stochastic volatility and transaction costs. The SOH model requires a set of asset price scenarios as input. In Section 3, we propose methods to construct accurate scenarios of stock and option prices for a one-factor stochastic volatility process. Given the fine-grained approximation of the underlying distribution of the prices in Section 3, the stochastic optimization model might become huge and hard to solve however. We propose an aggregation algorithm that reduces a set of price scenarios to a smaller size, while preserving important properties like the absence of arbitrage. In Section 4, we discuss methods to solve the stochastic optimization model. Finally, we use simulations to investigate the relative performance of our approach for a specific hedging problem.

In this paper, we considered hedging of options under transaction costs and stochastic volatility.As traditional methods like delta–vega hedging and static hedging are not fully appropriate in this context, we introduced the stochastic optimization hedging (SOH) model.The SOH model takes account of transaction costs, stochastic volatility and trading restrictions.The model has a number of trading dates on which the hedge portfolio of stocks and traded options can be rebalanced.The goal of the model is to minimize the hedging errors at the >rst few trading dates by following an appropriate dynamic trading strategy. The SOH model requires an event tree of asset prices as input.The performance of the SOH hedge depends crucially on the quality of this event tree, which is an approximation of the underlying price process.Our contribution is that we propose reliable methods to construct event trees for stochastic volatility models.Without loss of generality we focus on the asymmetric N-GARCH model as an underlying process. First, we represent the N-GARCH model on a grid of stock price versus volatility using a trinomial process.Next, we propose methods to avoid arbitrage opportunities in the set of stock and option prices on the grid.Finally, we construct sparse event trees for the SOH model by applying an aggregation algorithm which preserves the no-arbitrage property. We investigated the performance of the SOH model with a simulated hedging problem. Even though the SOH model for this example consists of just three trading dates, the solutions make sense both intuitively and economically.On average the hedging policy of the SOH model is quite close to a delta–vega neutral strategy, but with some slack to avoid needless transaction costs.The SOH hedging policy reduced transaction costs considerably compared to traditional delta hedging and delta–vega hedging strategies. Moreover, the SOH strategy could easily incorporate restrictions on short selling and borrowing and outperformed the optimal static hedge portfolio.