# 数学代写|金融数学代写Intro to Mathematics of Finance代考|ACF1003

Consider a wiring company projecting use of 100,000 pounds of copper in the next quarter. The company wishes to lock in its copper acquisition costs. The firm can buy a three-month call option on 100,000 pounds of copper with an exercise or strike rate of $\$ 2.31$per pound. Let’s look at the payoffs of the firm’s call option position while ignoring the firm’s other cash flows. The payoff of the call option to the call buyer at expiration (ignoring the option’s initial cost or premium) is Call payoff to buyer$=\operatorname{Max}\left(S_T-K, 0\right)$where$K$is the strike price and$S_T$is the value of the underlying commodity at the option’s expiration date and where$\operatorname{Max}\left(S_T-K, 0\right)$takes on the greater of the value$S_T-K$or 0 . For a copper producer, the decision may be to buy a put option to sell its copper production next quarter. The payoff of the put to the put buyer at expiration (ignoring the option’s initial cost or premium) is Put payoff to buyer$=\operatorname{Max}\left(K-S_T, 0\right)$Note that the payoff of the put option is not the mirror image of the payoff of the call. The payoff of the put is 0 when$S_T>K$, while the payoff of the call is 0 when$S_T<K$. Let’s look at the payoffs of the call and put to the party that writes or sells the options. Call payoff to seller$=-\operatorname{Max}\left(S_T-K, 0\right)$Put payoff to seller$=-\operatorname{Max}\left(K-S_T, 0\right)$The call buyer hopes that the price of the asset underlying the option will rise above$K$, so that she can gain$S_T-K$. The put buyer hopes that the underlying asset will fall below$K$, so that she can gain$K-S_T$at the expiration of the option. Note that the option seller’s payoff is the opposite of the buyer’s payoff because the option is a zero-sum game. An option’s payoff is not a deterministic number; It is a function of$S_T$and therefore depends on the future asset price$S_T$. It is this uncertainty that makes options interesting to study and useful to manage risk. ## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Profit and Loss Diagrams for Analyzing Option Exposures Option profit and loss diagrams express the potential profits and losses to an option or a portfolio that contains one or more options measured at the expiration date of the option. The profits and losses in these diagrams typically ignore the time value of money and frictions such as transactions, costs, or taxes. Option premiums are not included in option payoff diagrams but they are included in profit-loss diagrams. Figure$1.3$illustrates the potential profits and losses to a simple long position in a call option as of the date that the option expires. Note that the call buyer suffers a loss when$S_T$is less than or equal to$K$. That loss will be equal to the price or premium paid to purchase the option. For each dollar by which the price of the underlying asset exceeds the strike price, the call option buyer receives$\$1$. The breakeven point (ignoring the time value of money) is when the underlying asset’s price exceeds the strike price by the cost of the option to the buyer (i.e., the premium paid).

The price of the option just prior to the moment that it expires will be driven by arbitragers in a frictionless market to be equal to the payoff described in the previous section (e.g., $\operatorname{Max}\left(S_T-K, 0\right)$ ) for an option buyer. If the market price of the option differs from $\operatorname{Max}\left(S_T-K, 0\right)$ at expiration, then an arbitrager can buy an underpriced in-the-money option, exercise that option and collect a profit. If the option is overpriced, the arbitrager can write the option, buy the underlying asset for $S_T$, and deliver the asset and receive $K$. In both cases, the arbitrager would profit without risk. Therefore, options tend to be priced at expiration such that market participants can receive appropriate value by closing their positions rather than taking or making delivery, and can often lower their transactions costs at the same time. As is often the case with derivatives, many or most market participants close their derivative positions without taking or making delivery of the derivative’s underlying asset because their goal in establishing a derivative position is usually to obtain or hedge exposure, not to obtain or deliver an asset.

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|The Payoffs of Calls and Puts

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