作为一种补偿形式，期权授予变得更加普遍, 考虑到科技和生命科学领域初创企业的激增. 他们的定价, 然而, 是否被广泛误解，许多员工认为期权是通往未来财富的一张令人困惑的门票.

There 是 consequences to not setting the price of options at or near fair market value (FMV) at the time of grant, 就像 IRC 409 在美国，对低于FMV的期权征收惩罚性税率.

鉴于此, I’ve written this article to cover the basics of option pricing, to make it as widely useful as possible, it’s not bound to any specific tax code or jurisdiction. The principles discussed primarily apply to traded options on listed stock but many of the heuristics can be applied to non-traded options or options on non-traded stock.

Basics of 选项 Valuation

Value of 选项 at Expiry

期权以看涨期权和看跌期权的形式出现，赋予买方一种权利，但不是义务. 结果是, plain vanilla options can be worth something or nothing at expiry; they cannot be worth a negative value to a buyer since there 是 no net cash outflows after purchase. A seller of plain vanilla options is on the opposite side of the trade and can only lose as much as the buyer gains. It is a zero-sum game when this is the only transaction.

建模的要求

A call on a stock grants a right, 但不是以行权价格购买标的的义务. 如果现货价格高于执行价，看涨期权的持有者将在到期时行使期权. The payoff (not profit) at maturity can be modeled using the following call option formula and plotted in a chart.

Excel formula for a Call: = MAX (0, 分享 Price - Strike Price)

建模了

同样，以执行价格卖出的看跌期权可以建模如下.

看跌期权的Excel公式: = MAX(0, Strike Price - 分享 Price)

Moneyness of an Option and Its Relevance

Based on the strike price and stock price at any point of time, the option pricing may be in, at, 或者没有钱:

• 当执行权和股票价格相同时，期权是平价的.
• 当看涨期权的执行价低于股票价格时，它就是现价期权(看跌期权则相反)。.
• 当看涨期权的执行价高于股票价格时(看跌期权的执行价与之相反)，它就处于价外.

Out-of-the-money and at-the-money options don’t have any intrinsic value to them but may have a time value before maturity. The distinction of moneyness is relevant since options trading exchanges have rules on automatic exercise at expiration based on whether an option is in-the-money or not. For instance:, the CBOE’s rules 是:

The 选项 Clearing Corporation has provisions for the automatic exercise of certain in-the-money options at expiration, a procedure also referred to as exercise by exception. 一般, OCC将自动行使任何到期的股票认购权或将其存入客户账户，该账户为0美元.1美元或更多的现金, and an index option that is $0.1美元或更多的现金. 然而, 特定经纪公司对这种自动行使的门槛可能与OCC相同，也可能不同.

因此，期权定价将取决于到期日的现货价格是高于还是低于执行价格. 直观地, the value of an option prior to expiry will be based on some measure of the probability of it being in-the-money with the cash flow discounted at an appropriate interest rate.

Black-Scholes-Merton (BSM) Option Valuation Model

虽然期权从希腊的历史时期就开始使用了, Roman and Phoenician civilizations, Fisher Black originally came up with this option pricing model in 1973，现在广泛使用，将其与物理学中传热公式的推导联系起来. Scholes和Merton对模型进行了修改，使其演变为Black-Scholes-Merton模型. The formula looks as follows:

• 调用:\(C_t = S_t e^{- \ T} N \left (d_1 \右)- K e^{\左(- r T \右)}N \left (d_2 \右)\)
• 说:\ (p_t = K e ^{\离开(- r T \右)}N (d_2)——S_t e ^{-δT} \ N (d_1) \)
• \(d_1 = \ln\left ( \frac{S_0}{K} \right ) + \left ( r + \frac{\sigma ^2}{2} \right )\left ( \frac{T}{\sigma \sqrt{T}} \right )\)
• \(d_2 = d_1 - \sigma \sqrt{T}\)

Let’s not get overwhelmed by these elaborate put and call price formulas and first understand what the model is actually showing. 为调用, 它们在到期前的价值将取决于标的股票的现货价格及其贴现价值, then the strike price and its discounted value and finally, some measure of probability. The components of this break down as follows:

• \(e^{ \left ( - r T \right )}\) and \(e^{- \delta T}\) 是 ways of applying continuous compounding to a cash outflow and cash inflow from exercising the option.
• K and S 是 the strike and spot prices, respectively.

The remainder of the calculation is all about discounting the cash outflow at a continuously compounded discount rate, adjusting for any dividends, or cash flows before maturity and, for probability using a normal distribution.

概率的假设

BSM模型假定为正态分布(钟形曲线分布或 高斯分布) of continuously compounded returns. 该模型还表明，随着当前股票价格与行权价格之比的增加, the probability of exercising the call option increases, taking N(d) factors closer to 1, 这意味着不行使期权的不确定性降低了. As the N(d) factors get closer to 1, 该公式的计算结果更接近于看涨期权的内在价值. The other implication is that when variance (σ) increases, N(d) factors diverge and make the call option more valuable.

N(D2)是股票到期日价格高于行权价格的概率. N(D1) is the term for calculating the expected value of cash/stock inflow at maturity only if the stock price is above the strike price. N(D1) is a conditional probability.

看涨期权买方的收益来自于到期时发生的两个因素:

1. The spot has to be above strike price. (方向).
2. 到期日现货价格与执行价格之差(量子).

Imagine, a call at strike price $100. 如果股票的现货价格为101美元或150美元，则满足第一个条件. The second condition is about whether the gain is $1 or $50. D1项将这两个组合成一个条件概率，如果到期时现货高于行权, what will be its expected value in relation to current spot price.

Setting Up the BSM Model in Excel

The following model is what I use in Excel for BSM calculations (the shaded cells 是 calculations linked to other cells):

The formula for this is as follows:

单元格B2 =估价日期 Cell B3 = Stock/Spot Price 单元格B4 =执行价格 Cell B5 = Implied Volatility Cell B6 = Risk-free rate annualized 单元格B7 =以年为单位的到期时间(计算公式为(B10-B2)/365) Cell B8 = Dividend yield (Calculated as B11/B3) 细胞B9 = No. 选项(将其设置为1，用于计算不基于合约的值) 单元格B10 =到期日期 Cell B11 = Annual Dividend in currency terms 细胞十三区最= D1 = (LN ((B3 \ EXP (b8 \ B7)) / B4) + ((B6 + ((B5) ^ 2) / 2) \ B7)) / ((B5) \ SQRT (B7)) Cell B14 = D2 = B13-B5SQRT(B7) Cell B15 = N(D1) = NORMSDIST(B13) Cell B16 = N(D2) = NORMSDIST(B14) Cell B17 = Call = (B3\EXP(-B8\B7))\B15-B4\EXP(-B6\B7)\B16 Cell B18 = Put = (B17-(B3\EXP(-B8\B7))+B4\EXP(-B6\B7)

Cash Flows of the Underlying

看涨期权可以让买家享受股票的上行空间，而不必真正持有一段时间直到到期. 直观地, if the upside is paid out during the period of holding, then the calls should be less valuable since the right to that upside is not being derived by the option holder. Of course, the reverse applies in the case of puts. 这种直觉可以在下面的图表中看到，一个支付股息的股票为0%, 2%, 5%的股息. 该模型假设股息也以连续复合率支付.

现在，由于美国税法的变化，人们正在讨论特别股息, it is worth mentioning that you will see an adjustment factor to traded options for one-time dividends above a certain percentage of the stock price. One-time special dividends have a big impact on option pricing. In 2004, 当时微软宣布每股额外派发3美元的一次性特别股息，而平时的股息为0美元.08 quarterly, the options were 调整.

Sensitivity to Factors or Option Greeks

期权行业委员会(OIC)有一个免费的计算器，它将显示交易期权的价值和希腊. 我有 分析了 the values for AAPL from 1st October 2018 from the 选项 Industry Council’s website.

Delta and Gamma or Spot Price

The following graph is for AAPL Puts expiring Oct 12,2018年10月1日, 2018 with the vertical line indicating last price.

以下是2018年10月1日至2018年10月12日到期的苹果电话.

The last traded call and put option prices 是 clearly correlated to the strike price and form this hockey stick-esque graph. The reason the dots don’t align to a line is because some of the options were not traded on Oct 1 and the last traded price of these options 是 older, especially for deep in-the-money options.

What happens when the spot price changes for AAPL? AAPL’s price changes by nanoseconds at the exchange. 直觉上，基于BSM模型，期权定价也应该发生变化. This is measured by Delta, 哪一个是期权价值随现货价格变化的近似值. 它是标的价格每变动1美元，期权价值变动的近似值.

Delta is used as a hedging ratio. 如果你想用一个delta为0的期权对冲潜在头寸.5, you will need two options (2 x 0.5) to completely hedge the position (and make it delta-neutral). Delta is an approximation, though. 它适用于价格的小波动和短时间. We see the relationship of the call to changes in stock price below as well as the change in delta over the same range of stock prices. The call prices don’t move smoothly as a line and consequently, the calculated delta moves 就像 a curve. This becomes more noticeable ne是r to the strike price.

对于底层价值1美元的变化，delta的变化叫做Gamma. Gamma总是正值，Delta对于看涨期权为正值，对于看跌期权为负值(对于买方). It also means that for a call, 最高百分比的变化将发生在从钱外变成钱内的时候, 反之亦然. Gamma or the rate of change in delta approaches zero as the strike price moves away from the spot price (for deep out-of-the-money or in-the-money option positions).

即时间值

An option’s price depends on how long it has to run to expiry. 直观地, the longer the time to expiry, the higher the 就像lihood that it will end up in-the-money. 因此，期限较长的期权往往具有较高的价值，无论是看跌期权还是看涨期权. 时间值 subsequently decays to 0 as it nears expiry.

The rate of decay is not a straight line. 用一个球从斜坡上滚下来的比喻来理解它更容易. The speed picks up as the ball rolls further down the slope—slowest being at the top and fastest at the bottom (at expiry). 衰减率由Theta表示，对于看涨和看跌期权是正的.

利率

利率通过作为折现率对期权价值产生影响. 直观地, 看涨期权意味着在不支付全部价格的情况下获得持有标的股票的好处. 因为看涨期权买家不需要购买股票的全价, 股票全价和看涨期权之间的差额理论上可以投资，因此, 对于较高的贴现率，看涨期权应该具有较高的价值. The sensitivity to interest rates is measured by Rho, 利率越高，看涨期权的价值越高，看跌期权的价值反之亦然.

织女星还是波动性

维加, though not actually in the Greek alphabet, is used to denote the sensitivity of option value to volatility. 波动性是指价格上涨或下跌的可能幅度. The higher the volatility from a spot price, the higher the 就像lihood that the price may reach the strike. Hence, the higher the volatility, the higher the price of options.

Volatility is usually back filled using implied volatility (I”). 隐含波动率是用BSM模型计算的，使用期权的交易价格. 通过VIX期权，IV本身已经成为一种可交易的资产类别.

If you buy an option in a very calm market and there is a sudden uptick and downtick in the price of the underlying, with the price ending back where it was before, you may see that option pricing has increased in value. This is from a revision of its IV estimate.

To summarize the effect of 维加, and indeed the other Greeks, on the prices of options please refer to the following table.

Put-call Parity and Use Cases

想象一下，你有一个文件夹，创造性地命名为“a”，它只有一个 欧洲的电话 2018年12月21日到期的250美元的苹果股票，以及一股苹果股票;

Then you create another portfolio, “B”, which has only a 欧洲的电话 2018年12月21日到期的250美元的苹果股票，以及美国政府 短期国库债券 maturing on the same day for a maturity value of $250.

如你所见，投资组合A和投资组合B在到期时都有相同的收益. This principle is called put-call parity. Another way of stating it is:

This equation can be rearranged to mimic other positions:

1. 持有标的期权和看跌期权，通过以无风险利率借入资金，你创造了一个 合成的电话.
2. 在持有短期国库券和看涨期权的同时做空标的股票，就得到了合成看跌期权.
3. If you want to earn treasury (i.e., 无风险的在持有标的股票的同时，持有看跌期权并做空看涨期权.
4. 你也可以通过持有看涨期权、做空看跌期权和持有国库券来模拟持有标的.

这只适用于欧式的到期、看涨和看跌期权，且执行价格相同.

Employee Non-traded 选项

The employee stock options for non-traded companies 是 different from exchange-traded options in a manner of different ways:

1. There is no automatic exercise when it is in-the-money.
2. Vesting requirements restrict liquidity.
3. 交易对手风险较高，因为你是直接与私人公司打交道, over a collateralized exchange.
4. 投资组合集中度也更为极端，因为可用的多样化措施更少.

In addition to these, as we know, 估值 is also a completely different ball game for private companies. 正如我们讨论过的, Delta(股价), Theta(时间值), Rho(利率)和vega(波动率)是期权估值的重要决定因素. These make 估值 of employee stock options more challenging, 因为δ, Gamma and Volatility 是 especially hard to determine, since the stock itself may not be traded.

对于持有股票期权的雇员，要牢记的关键因素是:

1. Volatility has a key impact on 估值.
2. Option decay due to time value is not linear in nature. Remember the ball rolling down the hill analogy.
3. Option 估值 is both intrinsic value and time value. 仅仅因为没有内在价值并不意味着期权毫无价值, time heals all wounds and may also close the gap. When you receive an option grant, it is usually at-the-money or may be out-of-the-money, 没有内在价值. Tracking intrinsic value as the stock rises is intuitive, 但是时间价值, the opportunity cost of an early exercise, is not always intuitive or accounted for. Due to this opportunity cost, 只有在一些合理的理由下，比如需要现金流，你才应该尽早行使期权, portfolio diversification or stock outlook.

Parting Thoughts and Glossary

当您了解了选项的组成部分后，它们就不那么复杂了. Think of them as more flexible building blocks for allowing you to construct and manage financial portfolios in a less capital intensive way. 理解希腊人的含义是理解他们行为的第一步.

作为一个简短的术语表, below 是 some key terms mentioned throughout the article, summarized in a concise manner:

• 看涨期权和看跌期权 Call is an option with no obligation to buy the underlying asset at an agreed price on or before a specified date. Put is an option with no obligation to sell the underlying asset at an agreed price on or before a specified date.
• 保险费, 买方支付给期权卖方(卖方)的价格称为期权费. It is the 估值 of an option at the time of the trade.
• Exercise/strike and spot prices – 行权价或行权价是使用期权买卖标的资产的指定价格. 现货价格是标的资产在现货市场上的价格.
• 回报, The net cash flow on expiry of an option. 其中一个现金流量是行权价格，另一个是资产的市场价值.
• European and American exercise – 欧式期权只能在到期前的指定期限内执行. 美式期权可以在到期日或到期日之前的任何时间行使.
• Time value and intrinsic value – 时间值 is the premium at a time minus the intrinsic value. 期权的内在价值是任何时候执行价格和现货价格之间的差额.

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