LOS e: Explain how the binomial model value converges as time periods are added.

这个LOS可理解成，当binomial model的区间划分的越来越小的时候，binomial model得到的期权价格趋向于同一个价格，换句话说，划分500个区间与划分1000个区间得到的期权价格很接近。见下面这个图有个直观的了解 [pic]

LOS f: Explain the assumptions underlying the Black-Scholes-Merton model. BSM模型有六个假设，分别是，

1. 基础资产的价格遵循自然对数正态分布，注意，是价格遵循自然对数正态分别，收益遵循的是正态分布。

2. 无风险利率已知且不变。

3. 基础资产的波动率已知且不变

4. 无税收和交易成本

5. 在期权寿命期间，基础资产不会产生现金流

6. BSM模型只适用与定价European options The following key assumptions underlie the Black-Scholes-Merton Model: ν1. The underlying price follows a lognormal probability distribution as it advances through time. Log returns are also referred to as continuously compounded returns. If this return follows a normal distribution, it is log normally distributed. ν2. The risk-free rate is known and invariable, or constant 。the model does not allow interest rates to be random. ν3. The volatility of the underlying asset is known and invariable 。volatility is both known, and assumed not to changeover time. This is the most critical assumption of the model. Of course, volatility in reality is not constant. ν4. There are no taxes or transaction costs. ν5.There are no cash flows on the underlying asset. 6.The model is generally only used to value European options. A binomial model with a great number of time periods is better to value American options.

LOS h: Explain how an option price, as represented by the Black-Scholes-Merton model, is affected by each of the input values (the option Greeks). 这个LOS讲了，如何利用BSM公式，来知道期权价格与输入量之间的关系。对于call来说，可以用这个公式帮助记忆，[pic]，变形为[pic]，从这个公式可知，价格与现价成正相关，与X成负相关，与R成正相关，与时间t成正相关。另外，波动率增加，会增加卖权与买权的价值，这个要另外记忆了。 [pic] [pic]

*There is an exception to this, the general rule that the theta of a European put is positive. An increase in time to expiration can decrease the value of a European put, when the price decrease from discounting the exercise price at a higher rate outweighs the positive effect of longer time

LOS i: Explain the delta of an option, and demonstrate how it is used in dynamic hedging. Call的 Delta在binomial model是通过Delta＝(C+-C-)/(S+-S-)计算的，在BSM模型中，Delta＝N(d1),可见，Call的 Delta是在0到1之间的。Call的 Delta的含义是，在基础资产价格变动1单位的时候，call的价格变动delta个单位。这样，可以利用多头基础资产，空头call，或者空头基础资产，多头call构建无风险资产，达到套期保值的目的。如多头100单位的call,hedge ratio 是0.6的话，可以空头60单位的基础资产，这样就构建了一个无风险的资产组合。 但问题是，随着基础资产价格的变化，delta也会发生变化，这样，原来构建的无风险资产又不是完全对冲了。这样，就需要不断的根据delta的变化调整投资组合，称为dynamic hedging，动态套期保值。

Delta defines sensitivity of option price to any change in the underlying asset's price. Delta is constantly changing, thus delta hedging is a dynamic process, referred to as dynamic hedging, meaning that the hedge must be constantly adjusted. If changes in the underlying asset's price are large, or the delta hedge is not adjusted over a fairly long time period, the hedge may not be effective. This is because the delta may be unstable 。this is referred to as the gamma effect. If this effect is great, option price changes will not be very close to changes which had been estimated by the delta times the underlying asset price change.

LOS j: Explain the gamma effect on an option's price and delta. Gamma是期权价格对股票价格的二阶偏导数，_ Gamma在基础资产的现价等于执行价格时候最大，Gamma衡量的是当基础资产的价格发生变动的时候，delta的变化情况，Gamma大，delta变化大，Gamma小，delta变化小。所以，利用delta套期保值，Gamma大的时候，套期保值的效果就差、 Gamma在基础资产的现价等于执行价格时候最大这句话要记住。基础资产的现价大于或小于执行价格，Gamma都会变小。见下图，在S=X的时候，Gamma最大。此时套期保值的效果也最差。 [pic] Gamma is a measure of the sensitivity of the delta to the underlying asset, or how much delta changes. When gamma is large, delta changes quickly and is not a good approximation of how much the option moves for each movement in the underlying stock price change. Gamma is greater when there is greater uncertainty concerning whether the option will expire in-