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Pricing and Analyzing Equity Derivatives

Introduction

这se toolbox functions compute prices, sensitivities, and profits for portfolios of options or other equity derivatives. They use the Black-Scholes model for European options and the binomial model for American options. Such measures are useful for managing portfolios and for executing collars, hedges, and straddles:

  • 一个collar是一个利率option that guarantees that the rate on a floating-rate loan will not exceed a certain upper level nor fall below a lower level. It is designed to protect an investor against wide fluctuations in interest rates.

  • 一个hedge是一项证券交易，可降低或抵消现有投资头寸的风险。

  • 一个straddle是用于交易期权或期货的策略。它涉及以相同的行使价格和到期日期同时购买和致电选项，当基础证券的价格非常波动时，这是最有利可图的。

Sensitivity Measures

这re are six basic sensitivity measures associated with option pricing: delta, gamma, lambda, rho, theta, and vega — the “greeks.” The toolbox provides functions for calculating each sensitivity and for implied volatility.

三角洲

三角洲of a derivative security is the rate of change of its price relative to the price of the underlying asset. It is the first derivative of the curve that relates the price of the derivative to the price of the underlying security. When delta is large, the price of the derivative is sensitive to small changes in the price of the underlying security.

Gamma

Gamma衍生安全性的是相对于基础资产价格的变更速率；也就是说，相对于安全价格，期权价格的第二个导数。当伽玛很小时，三角洲的变化很小。这种敏感性度量对于确定调整对冲位置多少很重要。

Lambda

Lambda，也称为选项的弹性，代表了一种选择价格的百分比变化，相对于基础安全性的价格变化1％。

Rho

Rho是the rate of change in option price relative to the risk-free interest rate.

Theta

Theta是相对于时间的衍生安全性价格的变化率。Theta通常很小或负数，因为选项的价值随着成熟度而倾向于下降。

Vega

Vega是相对于基础安全性波动性的衍生产品价格的变化率。当Vega很大时，安全性对波动率的小变化敏感。例如，期权交易者通常必须决定是否购买对冲Vega或Gamma的期权。所选的树篱通常取决于一个重新平衡对冲位置的频率以及基础资产价格（波动性）的标准偏差。如果标准偏差正在迅速变化，则最好与VEGA保持平衡。

暗示波动

这implied volatility选项的是标准偏差，其期权价格等于市场价格。它有助于确定股票未来波动率的市场估计，并为其他黑色choles功能提供输入波动（在需要时）。

分析模型

Toolbox functions for analyzing equity derivatives use the Black-Scholes model for European options and the binomial model for American options. TheBlack-Scholes model对基础证券及其行为做出了一些假设。黑色 - choles模型是Fischer Black和Myron Scholes开发的第一个用于定价选项的完整数学模型。它检查了市场价格，打击价格，波动性，到期时间和利率。它仅限于某些类型的选项。

这binomial model另一方面，对选项的基础过程的假设更少。二项式模型是一种定价选项或其他权益衍生品的方法，其中每个可能价格的概率遵循二项式分布。基本假设是，在任何短时间内，价格只能移至两个值（一个较高和一个较低）。有关进一步的解释，请参见约翰·赫尔（John Hull）的期权，期货和其他衍生品参考书目。

黑色choles模型

Using the Black-Scholes model entails several assumptions:

  • 这prices of the underlying asset follow an Ito process. (SeeHull, 222页)。

  • 该选项只能在其到期日期（欧洲选项）行使。

  • Short selling is permitted.

  • 这re are no transaction costs.

  • 所有证券都是可分开的。

  • 这re is no riskless arbitrage (where套利是the purchase of securities on one market for immediate resale on another market to profit from a price or currency discrepancy).

  • Trading is a continuous process.

  • 无风险的利率是恒定的，并且对于所有成熟度仍然相同。

If any of these assumptions is untrue, Black-Scholes may not be an appropriate model.

To illustrate toolbox Black-Scholes functions, this example computes the call and put prices of a European option and its delta, gamma, lambda, and implied volatility. The asset price is $100.00, the exercise price is $95.00, the risk-free interest rate is 10%, the time to maturity is 0.25 years, the volatility is 0.50, and the dividend rate is 0. Simply executing the toolbox functions

[OptCall, OptPut] = blsprice(100, 95, 0.10, 0.25, 0.50, 0) [CallVal, PutVal] = blsdelta(100, 95, 0.10, 0.25, 0.50, 0) GammaVal = blsgamma(100, 95, 0.10, 0.25, 0.50, 0) VegaVal = blsvega(100, 95, 0.10, 0.25, 0.50, 0) [LamCall, LamPut] = blslambda(100, 95, 0.10, 0.25, 0.50, 0)
OptCall = 13.6953 OptPut = 6.3497 CallVal = 0.6665 PutVal = -0.3335 GammaVal = 0.0145 VegaVal = 18.1843 LamCall = 4.8664 LamPut = -5.2528

To summarize:

  • 这option call priceOptCall= $ 13.70

  • 这option put price选择= $ 6.35

  • 呼叫的三角洲callval= 0.6665 and delta for a putputval= -0.3335

  • 伽玛GammaVal= 0.0145

  • vegaVegaVal= 18.1843

  • lambda for a callLamcall= 4.8664 and lambda for a putLamPut= –5.2528

Now as a computation check, find the implied volatility of the option using the call option price fromblsprice。

波动率= blsimpv（100、95、0.10、0.25，optcall）
波动率= 0.5000

这function returns an implied volatility of 0.500, the originalblspriceinput.

二项式模型

这binomial model for pricing options or other equity derivatives assumes that the probability over time of each possible price follows a binomial distribution. The basic assumption is that prices can move to only two values, one up and one down, over any short time period. Plotting the two values, and then the subsequent two values each, and then the subsequent two values each, and so on over time, is known as “building a binomial tree.”. This model applies to American options, which can be exercised any time up to and including their expiration date.

This example prices an American call option using a binomial model. Again, the asset price is $100.00, the exercise price is $95.00, the risk-free interest rate is 10%, and the time to maturity is 0.25 years. It computes the tree in increments of 0.05 years, so there are 0.25/0.05 = 5 periods in the example. The volatility is 0.50, this is a call (标志= 1），股息利率为0，在三个期间（ExividendendEdent日期）后支付5.00美元的股息。执行工具箱功能

[StockPrice, OptionPrice] = binprice(100, 95, 0.10, 0.25,...0.05、0.50、1、0、5.0、3）

returns the tree of prices of the underlying asset

股票价格= 100.0000 111.2713 123.8732 137.9629 148.6915 166.2807 0 89.9677 100.0495 111.3211 118.8981 132.9629 0 0 80.9994 90.0175 95.0744 106.3211 0 0 0 72.9825 76.0243 85.0175 0 0 0 0 60.7913 67.9825 0 0 0 0 0 54.3608

and the tree of option values.

选件= 12.1011 19.1708 29.3470 42.9629 54.1653 71.2807 0 5.3068 9.4081 16.3211 24.3719 37.9629 0 0 0 1.3481 2.7481 2.7402

来自二项式功能的输出是二进制树。阅读股票价格矩阵这样：第1列显示了周期0的价格，第2列显示了第1期的上下和下降价格，第3列显示了上升，上下和下跌的价格，等等。忽略零。这选项优惠矩阵给出了价格树中每个节点的相关选项值。忽略与价格树中零相对应的零。

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