# 数学代写|金融数学代写Intro to Mathematics of Finance代考|TFIN101

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

The previous section mostly discussed speculators in option contracts. Options can also be valuable opportunities for hedgers whose primary purpose is to reduce risk. The following examples illustrate the use of options for hedging purposes while illustrating the variety of options available to hedgers.
A producer of rings for college graduations may by concerned that the price of gold is going to rise prior to the spring when most graduation ceremonies are held. However, the jeweler needs to announce prices for the rings earlier in the year before the orders for rings are placed. If the manufacturer knew how many rings would be ordered in the months prior to graduation, the manufacturer could buy the gold now or could enter a forward contract to purchase the gold. But the manufacturer does not want to be heavily exposed to the risk that their forecasts may be quite erroneous and the firm would end up with a large surplus or shortage of gold. Options can come to the rescue. The firm might enter forward contracts for a low estimate of their gold usage and purchase a call option on the additional amount of gold to reach the high end of their estimates. The purchase price of the call option on gold may be reasonable when compared to the potential outlay involved with ordering too much or too little gold in the spot, forward, or futures markets.

Next, consider the uncertainty faced by farmers, food processors, and food exporters with regard to the prices of grains near an important harvest season. The farmer fears that the price of her crop will fall before it can be harvested and sold. But the farmer is concerned about entering a short position in a forward contract since if her crop fails she will still be responsible for settling the promise made to deliver the crop to satisfy the terms of the forward contract. A potentially attractive solution to the farmer’s dilemma is to purchase a put option on some or all of the anticipated harvest. The put option will lock in a minimum price while limiting the farmer’s downside risk to the purchase price of the option.

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Put-Call Parity and Arbitrage

An important relationship exists between call option prices, $C$, and put option prices, $P$. The price of a European put option and European call option with the same underlying asset, same strike price $K$, and same expiration date $T$ must satisfy the following relationship known as put-call parity:
$$C+K e^{-r T}=P+S_0$$
From left to right, Equation $1.14$ can be expressed as “owning a call plus a bond is equivalent to owning a put plus the stock.” This well-known equation is often arranged differently or expressed with different notation. This particular arrangement is convenient for demonstrating why the relationship must be true in a well-functioning market with no arbitrage. The left and right sides of Equation $1.14$ can be viewed as payoff to two portfolios, A and B.
Portfolio A (left side): Buy a call on a stock + buy a zero-coupon bond that pays $K$ at time $T$.
Portfolio B (right side): Buy a put on the same stock + buy the stock.
Note that the two payoff columns on the right side of Table $1.2$ demonstrate that the total payoff to Portfolio A ( $S_T$ or $K$ ) equals the total payoff to Portfolio B $\left(S_T\right.$ or $\left.K\right)$ whether the call option is exercised at expiration (i.e., $S_T>K$ ), or whether the put option is exercised $\left(K>S_T\right)$. Obviously, when $S_T=$ $K$ neither option has a value at expiration and portfolios A and B have equal values $(K)$. Since under all scenarios both portfolios have the same total payoffs, they must therefore be worth the same today. This means that Equation $1.14$ holds:
$$C+K e^{-r T}=P+S_0 .$$
Another way to explain put-call parity is that the payout at time $T$ for call (C) minus put $(P)$ is the same as $S_T-K$. (This can be seen by combining the payout diagrams). The cost today to replicate $S_T-K$ by buying the stock today and borrowing $K e^{-r T}$ dollars from the bank is $S_0-K e^{-r T}$.

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Put-Call Parity and Arbitrage

$$C+K e^{-r T}=P+S_0$$

$$C+K e^{-r T}=P+S_0 .$$

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