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欧洲股票期权的希腊中立投资组合

The option sensitivity measures familiar to most option traders are often referred to as the希腊人：三角洲，，，，伽玛，，，，vega，，，，兰姆达，，，，罗，，，，andTheta。Delta is the price sensitivity of an option with respect to changes in the price of the underlying asset. It represents a first-order sensitivity measure analogous to duration in fixed income markets. Gamma is the sensitivity of an option's delta to changes in the price of the underlying asset, and represents a second-order price sensitivity analogous to convexity in fixed income markets. Vega is the price sensitivity of an option with respect to changes in the volatility of the underlying asset. For more information, see定价和分析平等衍生品。

特定选项的希腊人是用于定价该选项的模型的函数。但是，如果有足够的选择可以使用，则交易者可以为其希腊人建造具有任何理想值的投资组合。例如，为了使期权投资组合的价值与基础资产价格的微小变化隔离，一位交易者可能会构建一个期权投资组合，其delta为零。然后，这种投资组合被称为“达美航空中立”。另一个交易者可能希望保护期权投资组合免受基础资产价格的更大变化，因此可能会构建一个投资组合，其Delta和Gamma都为零。这样的投资组合既是三角洲和伽马中性的。第三交易者可能希望构建与三角洲和伽马中立性外，基础资产波动率的小变化相关的投资组合。然后，这样的投资组合是Delta，Gamma和Vega中性的。

Using the Black-Scholes model for European options, this example creates an equity option portfolio that is simultaneously delta, gamma, and vega neutral. The value of a particular greek of an option portfolio is a weighted average of the corresponding greek of each individual option. The weights are the quantity of each option in the portfolio. Hedging an option portfolio thus involves solving a system of linear equations, an easy process in MATLAB®。

Step 1

Create an input data matrix to summarize the relevant information. Each row of the matrix contains the standard inputs to Financial Toolbox™ Black-Scholes suite of functions: column 1 contains the current price of the underlying stock; column 2 the strike price of each option; column 3 the time to-expiry of each option in years; column 4 the annualized stock price volatility; and column 5 the annualized dividend rate of the underlying asset. Rows 1 and 3 are data related to call options, while rows 2 and 4 are data related to put options.

datamatrix = [100.000 100 0.2 0.3 0％ 称呼119.100 125 0.2 0.2 0.025％ 放87.200 85 0.1 0.23 0％ 称呼301.125 315 0.5 0.25 0.0333];％ 放

Also, assume that the annualized risk-free rate is 10% and is constant for all maturities of interest.

风险福特= 0.10;

为了清楚起见，分配每一列DataMatrix到名称反映列中财务数据类型的列向量。

StockPrice = datamatrix（：，1）;StrackPrice = datamatrix（：，2）;exprytime = datamatrix（：，3）;波动率= datamatrix（：，4）;rendendrate = datamatrix（：，5）;

Step 2

基于黑色 - choles模型，计算价格，以及四个选项中每个选项的三角洲，伽玛和Vega敏感性。功能blspriceandBlsdelta有两个输出，而BlsgammaandBlsvega只有一个。与伽玛和Vega敏感性相比，呼叫选项的价格和三角洲与原本等效的PUT选项的价格和三角洲不同，这对两个呼叫和看台都是有效的。

[callprices，putprices] = blsprice（Stockprice，Strackprice，...风险舍弃，过期时间，波动率，股息）；[Calldeltas，putdeltas] = blsdelta（Stockprice，...Strikeprice，风险延迟，过期，波动性，...股息）；Gammas = Blsgamma（股票，Strikeprice，风险弗雷特，...到期时间，波动性，派发）'vegas = blsvega（股票，Strikeprice，Riskfreate，...ExpiryTime, Volatility , DividendRate)'
Gammas = 0.0290 0.0353 0.0548 0.0074 Vegas = 17.4293 20.0347 9.5837 83.5225

Extract the prices and deltas of interest to account for the distinction between call and puts.

Prices = [CallPrices(1) PutPrices(2) CallPrices(3)...putprices（4）] deltas = [Calldeltas（1）Putdeltas（2）Calldeltas（3）...PutDeltas(4)]
价格= 6.3441 6.6035 4.2993 22.7694 deltas = 0.5856 -0.6255 0.7003 -0.4830

Step 3

Now, assuming an arbitrary portfolio value of $17,000, set up and solve the linear system of equations such that the overall option portfolio is simultaneously delta, gamma, and vega-neutral. The solution computes the value of a particular greek of a portfolio of options as a weighted average of the corresponding greek of each individual option in the portfolio. The system of equations is solved using the back slash (\ \）讨论的操作员求解同时线性方程。

A = [Deltas; Gammas; Vegas; Prices]; b = [0; 0; 0; 17000]; OptionQuantities = A\b每个选项的数量（数量）％。
选件= 1.0E+04 * 2.2333 0.6864 -1.5655 -0.4511

第4步

最后，计算整个投资组合的市场价值，三角洲，伽玛和Vega作为组件选项相应参数的加权平均值。加权平均值被计算为两个向量的内部产物。

PortfolioValue = Prices * OptionQuantities PortfolioDelta = Deltas * OptionQuantities PortfolioGamma = Gammas * OptionQuantities PortfolioVega = Vegas * OptionQuantities
PortfolioValue = 17000 PortfolioDelta = 1.8190e-12 PortfolioGamma = 0 PortfolioVega = 0

The output for these computations is:

Option Price Delta Gamma Vega Quantity 1 6.3441 0.5856 0.0290 17.4293 22332.6131 2 6.6035 -0.6255 0.0353 20.0347 6864.0731 3 4.2993 0.7003 0.0548 9.5837 -15654.8657 4 22.7694 -0.4830 0.0074 83.5225 -4510.5153

您可以根据需要验证投资组合价值为$ 17,000，并且该选项投资组合确实是Delta，Gamma和Vega Neutral。基于这些措施的树篱仅对基础变量的微小变化有效。

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