# 微积分网课代修|导数代写Derivatives theory代考|LT013086 Long Put plus Stock: Insurance

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修导数代写Derivatives theory代考|Long Put plus Stock: Insurance

Options can also be used to provide insurance. For example, suppose you run a pension fund and already own stocks whose current price on 15 July (on NYSE) is $S_{0}=\$ 72$. But you are worried about a fall in price of the stocks between now and 25 October when your stocks will be sold to provide lump sum payments to pensioners. Well, you can ‘insure’ your stocks by buying an October-put option with a strike price of, say,$K=\$70$ (with maturity date 25 October). Note that in this example you hold two assets: the stock-XYZ and a put option (on stock-XYZ).

If stock prices in New York fall to $S_{T}=\$ 30$on 25 October, then instead of selling your stocks in New York at$S_{T}=\$30$, you can exercise your October-put option in Chicago, which means delivering your stock-XYZ in Chicago and you will receive $K=\$ 70$for each stock (from the options clearing house). By buying the put option on 15 July, you have guaranteed a minimum price of$K=\$70$ on 25 October at which you can sell the stocks-XYZ, held by the pension fund. The cost of this ‘insurance’ is the put premium $P=\$ 2.2$paid on 15 July. True, the pension fund has lost$\$2$ per stock as the initial price of the stock was $S_{0}=\$ 72$in July since the pension fund can only obtain$K=\$70$ when they deliver the stock and exercise the put option in Chicago – the $\$ 2$is the ‘deductible’ in the put insurance contract.${ }^{5}$Losing$\$2$ per stock because you had the foresight to take out insurance by buying a put option (with $K=\$ 70$), is a lot better than if you had not purchased the put, since then your stocks-XYZ would have fallen in value by$\$42(=72-30)$ on the NYSE.

## 微积分网课代修导数代写Derivatives theory代考|Hidden Options

Aristotle in Book I of Politics, mentions the Greek philosopher Thales who developed a ‘financial device’ which was in fact an option. One winter he ‘read the stars’ and decided that next autumn would result in an exceptionally good olive harvest. He therefore quietly went around the owners of olive presses and paid them a small retainer (i.e. the call option premium) to secure the right to be first to use their olive presses in the autumn, for a fixed price (the strike price), if he so wished. Come autumn, the harvest was good and therefore the demand for the olive presses was high and Thales could charge a high price to the olive growers to let them use the olive presses, but Thales only paid the lower strike price to the owners of the olive presses. Even if Thales had been wrong about the harvest, the most he could have lost was the small option premium he initially paid to the owners of the olive presses.

Although some people may not be aware of it, they probably hold options. For example, consider rural bus services whose fares are often subsidised via local government taxes (e.g. sales taxes and community charges). If you live out of town, you have the option to take the bus into town by paying the known fixed fare (= strike price). You will do this if the value of your journey on that day by bus exceeds the fixed fare (strike price). Hence, if you live out of town you are holding an implicit call option and the call premium is that part of your local taxes that goes to subsidise the bus company. You may never use the bus but the option to use the bus (e.g. if your car breaks down) has a positive value to you and hence you may be willing to see the rural bus service subsidised by local taxes.

Next, suppose in January you have been offered a place at one of several universities, if you achieve a grade $B$ (or above) in your examinations in June. You will make your final decision about attending a specific university or not, in September. The (implicit) option premium you pay is the time and effort you put into studying between January and June. You have nine months to decide on your choice of ‘the best’ university for you (i.e. the time to maturity of the option), which is conditional on getting appropriate grade B or above in the June exams.

In September, if you decide to go to a university, you will have to ‘pay’ the strike price (i.e. tuition and living expenses and income foregone while attending the course). In September you will choose that university with the largest net payoff $S_{T}-K>0$ where $S_{T}$ is the (expected) present value of your additional earnings after graduating from a particular university. If $S_{T}-K>0$ then you will ‘take delivery’ of one of the university courses, so the option you have is a ‘call option’. Of course if $S_{T}<K$ then you will choose not to go to university (and instead look for a job) – that is, you will not exercise your ‘call option’, as your extra post-university earnings do not cover the costs of attending university.