衍生品经典文献：波动率曲面构建

近年来，大量研究提供了不同的波动率曲面构建方法，本期推荐的三篇经典文献分别使用参数法和非参数法对波动率曲面进行建模。Heston 模型和 SABR 模型是比较流行的参数法，它们都具有近似解析解；Avellaneda 相对熵方法是一种非参数方法，拟合的波动率曲面需要进行无套利修正。
1
A closed-form solution for options with stochastic volatility with applications to bond and currency options
——Steven L. Heston

Abstract
I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility.
The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options.
Simulations show that correlation between volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model.
The solution technique is based on characteristic functions and can be applied to other problems.

作者使用一种新技术推导出了标的资产服从随机波动模型的欧式看涨期权的近似解析解。
随机波动模型允许波动率和现货资产收益之间存在任意相关性。此外，作者介绍了随机利率模型，并说明了如何将模型应用于债券期权和外汇期权。
模拟表明波动率和现货资产价格之间的相关性对于解释 BS 模型中的收益偏斜和行权价格偏差很重要。
近似解析解的推导方法基于特征函数，该方法可以应用于其他问题。

2
Managing smile risk
——Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, and Diana E. Woodward

Abstract
Market smiles and skews are usually managed by using local volatility models a la Dupire. We discover that the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices.
Due to this contradiction between model and market, delta and vega hedges derived from the model can be unstable and may perform worse than naive Black-Scholes’ hedges.
To eliminate this problem, we derive the SABR model, a stochastic volatility model in which the forward value satisfiesand

and the forward
and volatility
are correlated:
We use singular perturbation techniques to obtain the prices of European options under the SABR model, and from these prices we obtain explicit, closed-form algebraic formulas for the implied volatility as func tions of today’s forward price
and the strike K.
These formulas immediately yield the market price, the marke t risks, including vanna and volga risks, and show that the SABR model captures the correct dynamics of the smile. We apply the SABR model to USD interest rate options, and find good agreement between the theoretical and observed smiles.

波动率微笑和偏斜通常用Dupire的局部波动率模型来管理。作者发现由局部波动模型预测的波动率微笑的动态变化与观察到的市场行为相反：当标的价格下降时，局部波动模型预测该微笑向更高的价格移动；当价格上涨时，这些模型预测波动率微笑会向更低的价格移动。
由于模型与市场之间存在矛盾，因此从该模型衍生的delta和vega套期保值可能会不稳定，并且其性能可能会比单纯的Black-Scholes套期保值更差。为了解决这个问题，作者提出了SABR模型，这是一个随机波动率模型，其中远期价格满足

远期价格
和波动率
是相关的：
作者使用奇异摄动技术获得了SABR模型下的欧式期权价格，并从这些价格中推导出了隐含波动率的显示近似表达，这个表达式是关于今天的远期价格
和行权价格K的函数。
这些公式可以立即得到市场价格，市场风险（包括Vanna和Volga风险），并证明SABR模型捕获了正确的波动率微笑动态变化。最后，作者将SABR模型应用于美元利率期权，并证明了观察到的波动率微笑和理论之间保持了良好的一致性。

3
Calibrating volatility surfaces via relative-entropy minimization
——Marco Avellaneda,Craig Friedman,Richard Holmes and Dominick Samperi

Abstract
A framework for calibrating a pricing model to a prescribed set of options prices quoted in the market is presented. Our algorithm yields an arbitrage-free diffusion process that minimizes the Kullback-Leibler relative entropy distance to a prior diffusion. It consists in solving a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns to be solved for in the optimization step is equal to the number of option prices that need to be calibrated, and is independent of the mesh-size used for the scheme. This results in an efficient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy.
The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics.
The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschmark over-the-counter options and computing interpolated implied-volatility curves.

作者提出了一种用期权市场报价校准定价模型的方法。该算法产生了无套利扩散过程，并且使扩散的Kullback-Leibler相对熵距离最小。算法使用Bellman抛物线方程的有限差分法结合基于梯度的优化例程来解决约束（最小极大）最优控制问题。在优化步骤中要解决的未知数等于需要校准的期权价格数，并且与差分法所使用的网格大小无关。
这推出了一种有效的非参数校准方法，该方法可以将任意数量的期权价格匹配到任何所需的精确度上。这种算法可用于在行权价或到期日方向上对隐含波动率进行插值，从而有助于奇异期权定价。
此外，作者讨论了使用该算法计算波动率曲面稳定性和可靠性，包括贝叶斯先验对表面形状和隐含的波动微笑/偏斜的影响。最后，作者通过校准美元-德国马克场外期权的市场价格，获得了内插的隐含波动率曲线。

参考文献：
[1] Heston S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options[J]. The review of financial studies, 1993, 6(2): 327-343.
[2] Hagan P S, Kumar D, Lesniewski A S, et al. Managing smile risk[J]. The Best of Wilmott, 2002, 1: 249-296.
[3] Avellaneda M, Friedman C, Holmes R, et al. Calibrating volatility surfaces via relative-entropy minimization[J]. Applied Mathematical Finance, 1997, 4(1): 37-64.

关于镒链
上海镒链金融科技公司专注于风险管理、衍生品估值、金融数据、交易等前台核心技术，是全球FinTech的建设者，中国金融技术的领先者之一。

镒链场外衍生品交易管理系统镒链场内衍生品交易管理系统扫码申请试用