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数学代写|金融数学代写Intro to Mathematics of Finance代考|Option Spread Strategies

Option spreads are simultaneous long and short positions in either different call options or different put options (but not both calls and puts). Three general types of option spread strategies are vertical spreads, horizontal spreads, and diagonal spreads. The terms describing the three spreads relate to the visualization of a matrix of option prices with strike price forming the vertical axis and expiration date forming the horizontal axis. Thus, in a vertical option spread, the options differ by strike price; in a horizontal option spread, the options differ by expiration date; and in a diagonal spread, they differ by both strike price and expiration date. This section focuses on the strategies with the same expiration date – vertical spreads.

Figure $1.10$ illustrates a vertical call spread known as a bull spread. The payout of a bull spread at expiration is positively related to the underlying asset, hence it is “bullish” with respect to the price of the underlying asset. The bullish nature of a bull spread occurs when the long call option position is in the option with the lower strike price and the short option position is in the asset with the higher strike price. Interestingly, the same bullish diagram can be generated using put options with the same structure: the long put option position is in the option with the lower strike price and the short option position is in the asset with the higher strike price.

Figure $1.11$ illustrates a vertical call spread known as a bear spread. Bear spreads reverse the direction of the options by establishing the long call option position in the option with the higher strike price and the short option position is in the asset with the lower strike price. Like in the case of bull spreads, the same bearish diagram can be generated using put options with the same structure: the long put option position is in the option with the higher strike price and the short option position is in the asset with the lower strike price.

数学代写|金融数学代写Intro to Mathematics of Finance代考|Option Combination Strategies

Option combinations involve simultaneous positions in at least one call option and at least one put option. This section focuses on combinations of calls and puts with the same expiration date and underlying asset. The synthetic long positions in the underlying asset using options (long a call and short a put) and short positions (long a put and short a call) discussed in Section 1.8.1 are option combinations. This section discusses others.

Option straddles have equally sized simultaneous long positions or simultaneous short positions in call and put options with the same strike price (and same expiration date). Figure $1.12$ illustrates the profitability of a long straddle at expiration with a solid line and shows the underlying components of long a call and long a put with dashed lines. The value of a long straddle is positively exposed to the volatility of the underlying asset. A short straddle is the mirror image (i.e., the maximum profit is above $K$ with losses on the far right and left).Option strangles are similar to option straddles, except the call and put options have different strike prices. Figure $1.13$ illustrates a long strangle. Other options strategies can involve option portfolios such as ratio spreads with more calls than puts or vice versa such that the payoffs differ in different directions.

There are many types of stand-alone options on financial assets that differ from simple calls and puts. Some options allow exercise at several specific points in time (a Bermuda option), some are based on functions of prices such as averages or extremes (e.g., Asian options have payoffs based on averaged prices of the underlying asset). Some options cease to exist if the underlying asset reaches a particular level or become exercisable if the underlying asset reaches a particular level such as barrier options.
Financial markets and economic activity in general are full of options. There are options on real assets such as options to: buy land, rent space, return products, extend warrantees, take early retirement, terminate contracts, and on and on.

There are non-traded financial options such as options to pre-pay mortgages and other loans, options to break some bank certificates of deposits (i.e., receive early termination), options to rollover some bank deposits, options to cash out insurance policies, options to increase insurance coverage, and so forth.

数学代写|金融数学代写Intro to Mathematics of Finance代考|MATH424

金融数学代考

数学代写|金融数学代写Intro to Mathematics of Finance代考|Option Spread Strategies

期权价差是不同看涨期权或不同看跌期权(但不是看涨期权和看跌期权)中的同时多头和空头头寸。三种一般类型的期权价差策略是垂直价差、水平价差和对角线价差。描述三种价差的术语与期权价格矩阵的可视化相关,其中行使价构成纵轴,到期日构成横轴。因此,在垂直期权价差中,期权因执行价格而异;在水平期权价差中,期权的到期日不同;在对角线价差中,它们的执行价格和到期日都不同。本节重点介绍具有相同到期日的策略——垂直点差。

数字1.10说明了称为牛市价差的垂直看涨期权价差。到期时牛市价差的支付与标的资产正相关,因此相对于标的资产的价格而言是“看涨”的。当多头看涨期权头寸位于执行价格较低的期权中,而空头期权头寸位于执行价格较高的资产中时,牛市价差的看涨性质就会出现。有趣的是,使用相同结构的看跌期权可以生成相同的看涨图:看跌期权多头头寸位于执行价格较低的期权中,空头头寸位于执行价格较高的资产中。

数字1.11说明了称为熊市价差的垂直看涨期权价差。熊利差通过在具有较高执行价格的期权中建立多头看涨期权头寸而在具有较低执行价格的资产中建立空头期权头寸来反转期权的方向。与牛市价差的情况一样,使用具有相同结构的看跌期权可以生成相同的看跌图:看跌期权多头头寸位于执行价格较高的期权中,空头头寸位于执行价格较低的资产中价格。

数学代写|金融数学代写Intro to Mathematics of Finance代考|Option Combination Strategies

期权组合涉及同时持有至少一种看涨期权和至少一种看跌期权。本节重点介绍具有相同到期日和标的资产的看涨期权和看跌期权的组合。在第 1.8.1 节中讨论的使用期权(多头看涨和空头看跌期权)和空头头寸(多头看跌和空头看涨期权)的基础资产合成多头头寸是期权组合。本节讨论其他内容。

期权跨式在具有相同执行价格(和相同到期日)的看涨期权和看跌期权中具有同等规模的同时多头头寸或同时空头头寸。数字1.12用实线说明到期时多头跨式的盈利能力,并用虚线显示多头看涨期权和多头看跌期权的基本组成部分。多头跨式的价值正向暴露于标的资产的波动性。空头跨式是镜像(即最大利润高于ķ最右边和最左边的损失)。期权绞杀类似于期权跨式,除了看涨期权和看跌期权有不同的执行价格。数字1.13说明了一个漫长的扼杀。其他期权策略可能涉及期权组合,例如看涨期权多于看跌期权的比率价差,反之亦然,这样收益在不同方向上不同。

金融资产的独立期权有多种类型,不同于简单的看涨期权和看跌期权。有些期权允许在几个特定时间点行使(百慕大期权),有些则基于价格函数,例如平均值或极端值(例如,亚洲期权的收益基于标的资产的平均价格)。如果标的资产达到特定水平,则某些期权将不复存在,或者如果标的资产达到特定水平(例如障碍期权),则某些期权将变得可行使。
总体而言,金融市场和经济活动充满了选择。实物资产有多种选择,例如:购买土地、出租空间、退回产品、延长保修期、提前退休、终止合同等等。

有非交易金融期权,例如预付抵押贷款和其他贷款的期权、打破某些银行存款证的期权(即提前终止)、展期某些银行存款的期权、兑现保险单的期权、期权增加保险范围等等。

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