# This payoff amount is also the value of the option at

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This payoff amount is also the value of the option at expiration. It represents value because it iswhat the option is worth at that point. If the holder of the option sells it to someone else aninstant before expiration, it should sell for that amount because the new owner would exercise itand capture that amount. To the seller, the value of the option at that point is –Max(0,STX),which is negative to the seller if the option is in the money and zero otherwise.Using the payoff value and the price paid for the option, we can determine the profit from thestrategy, which is denoted with the Greek symbol Π. Let us say the buyer paidc0for the optionat time 0. Then the profit isΠ =Max(0,STX) –c0(profit to the call buyer),To the seller, who received the premium at the start, the payoff iscT= –Max(0,STX)(payoff to the call seller),The profit is2018/8/21...40
Π = –Max(0,STX) +c0(profit to the call seller),Exhibit 3illustrates the payoffs and profits to the call buyer and seller as graphicalrepresentations of these equations, with the payoff or value at expiration indicated by the darkline and the profit indicated by the light line. Note in Panel A that the buyer has no upper limiton the profit and has a fixed downside loss limit equal to the premium paid for the option. Sucha condition, with limited loss and unlimited gain, is a temptation to many unsuspectinginvestors, but keep in mind that the graph does not indicate the frequency with which gains andlosses will occur. Panel B is the mirror image of Panel A and shows that the seller has unlimitedlosses and limited gains. One might suspect that selling a call is, therefore, the worst investmentstrategy possible. Indeed, it is a risky strategy, but at this point these are only simple strategies.Other strategies can be added to mitigate the seller’s risk to a substantial degree.Exhibit 3.Payoff and Profit from a Call Option2018/8/21...41
2018/8/21...42
Now let us consider put options. Recall that a put option allows its holder to sell the underlyingasset at the exercise price. Thus, the holder should exercise the put at expiration if theunderlying asset is worth less than the exercise price (ST<X). In that case, the put is said to bein the money. If the underlying asset is worth the same as the exercise price (ST=X), meaningthe put is at the money, or more than the exercise price (ST>X), meaning the put is out of themoney, the option holder would not exercise it and it would expire with zero value. Thus, thepayoff to the put holder ispT=Max(0,XST)(payoff to the put buyer),If the put buyer paidp0for the put at time 0, the profit isΠ =Max(0,XST) –p0(profit to the put buyer),And for the seller, the payoff ispT= –Max(0,XST)(payoff to the put seller),And the profit isΠ = –Max(0,XST) +p0(profit to the put seller),Exhibit 4illustrates the payoffs and profits to the buyer and seller of a put.

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Term
Fall
Professor
INA
Tags
Derivatives Market, Chartered Financial Analyst