Computing the implied volatility in stochastic volatility models
Computing the implied volatility in stochastic volatility models
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DOI：
10.1002/cpa.20039
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发表时间：
2004-10-01
影响因子：
3
通讯作者：
Florent, I
中科院分区：
文献类型：
--
作者：
Berestycki, H;Busca, J;Florent, I
Black-Scholes模型[6,23]在金融市场上得到了广泛认可。然而，它的一个缺点是，它与大多数观察到的期权价格不一致。尽管该模型仍然可以非常有效地使用，但有人提出要放宽其假设，例如，考虑标的资产S的波动率不再是常数，而是一个随机过程。有两种众所周知的方法可以实现这一目标。在第一类模型中，假定波动率依赖于变量t（时间）和S，从而产生所谓的局部波动率模型。第二种理论在概念上更具雄心，认为波动性本身具有随机成分。在后者中，进入波动率模型的随机因素的数量增加了因素的数量。这两种模型都具有实际意义。在这种情况下，用隐含波动率来表示最终价格是相关的。给定价格，布莱克-斯科尔斯隐含波动率是确定的，对于每个给定的产品（即每个给定的执行和到期日，例如，看涨期权），作为波动率参数的唯一值，布莱克-斯科尔斯定价公式与该给定价格一致。实际上，以这种方式报价和观察价格是交易大厅的常见做法。用这种无量纲单位表示价格的一大优点是便于在不同特性的产品之间进行比较。原则上，隐含波动率可以通过反转Black-Scholes公式从计算期权价格中推断出来。然而，直接分析隐含波动率更为方便。事实上，这种方法使我们能够阐明定性性质，否则将更加难以建立。特别地，我们在这里导出了几个有实际意义的渐近公式，例如在校准问题中。后者是一个反问题，包含
The Black-Scholes model [6, 23] has gained wide recognition on financial markets. One of its shortcomings, however, is that it is inconsistent with most observed option prices. Although the model can still be used very efficiently, it has been proposed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a stochastic process. There are two well-known approaches to achieve this goal. In the first class of models, the volatility is assumed to depend on the variables t (time) and S, giving rise to the so-called local volatility models. The second one, conceptually more ambitious, considers that the volatility has a stochastic component of its own. In the latter, the number of factors is increased by the amount of stochastic factors entering the volatility modeling. Both models are of practical interest. In these contexts, it is relevant to express the resulting prices in terms of implied volatilities. Given a price, the Black-Scholes implied volatility is determined, for each given product (that is for each given strike and expiry date defining, say, the call option) as the unique value of the volatility parameter for which the Black-Scholes pricing formula agrees with that given price. Actually, it is common practice on trading floors to quote and to observe prices in this way. A great advantage of having prices expressed in such dimensionless units is to provide easy comparison between products with different characteristics. In principle, the implied volatility can be inferred from computed options prices by inverting the Black-Scholes formula. It is more convenient, however, to directly analyze the implied volatility. Indeed, this approach allows us to shed light on qualitative properties that would otherwise be more difficult to establish. In particular, we derive here several asymptotic formulae that are of practical interest, for example, in the calibration problem. The latter—an inverse problem that consists