[...]... that the initial value of the asset in question must be equal to the initial value of the replicating portfolio The no-arbitrage pricing principle can be used in other ways to determine prices For example, if the initial values of two portfolios are equal, then it cannot be 8 Introduction to the Mathematics of Finance that one portfolio always yields a higher payoff than the other, regardless of the. .. investor instructs his broker to buy or sell an option, the broker transmits this request to the firm’s floor broker on the appropriate options 16 Introduction to the Mathematics of Finance exchange, who attempts to locate another floor broker (or other official) who has instructions to perform the opposite transaction on behalf of another investor The trade is then made and both brokers record the. .. commissions.) 20 Introduction to the Mathematics of Finance Profit Profit K Stock Price -cost cost K Long Call Stock Price Short Call Profit Profit K Stock Price -cost Long Put cost K Stock Price Short Put Figure 1.3: Profit curves These payoff curves are very informative Here are some of the things we can immediately see from these curves Let O be the strike price, let W be the strock price, let G be the initial... expense by the company 2) The Sarbanes–Oxley Act of 2002 prohibits the backdating of options and strengthens the requirements for reporting stock option grants for public companies 3) The IRS changed the tax laws with regard to the granting of in -the- money stock options Introduction 3 The requirements contained in the FASB statement brought to the forefront the problem that is the subject of this book:... expiration date of the call We will generally reserve the letter O for the strike price of an option and the letter W for the price of the underlying stock The cost of a call will be denoted by G and the cost of a put by T Although it will not be required for our mathematical analysis, we want to give some details about how stock options work S Roman, Introduction to the Mathematics of Finance: Arbitrage... It then inserts itself between the buyer and the seller, playing the role of the buyer for the seller and the role of the seller for the buyer Hence, each investor deals only with the OCC (indirectly) and not the other investor The OCC has sufficient resources to make good on any amounts owed as well as to enforce any collection, should that be required The OCC also plays a role in the exercise of. .. Of course, the downside to the call options is that if the stock does not rise before the expiration date, the investor will receive nothing from the options and will have lost the price of these options, whereas the stockholder still owns the stock Profit and Payoff Curves Generally speaking, when the expiration date arrives, the owner of an option will exercise that option if and only if there is a... E If not, then the seller pays nothing to the buyer This is a financial derivative since its value at time >" depends on the value U of the underlying Moreover, since there is risk involved in selling such an 4 Introduction to the Mathematics of Finance instrument, the seller will not be willing to enter into such a contract without some monetary compensation at the time >! of formation of the contract... estimate the current fair value of the option to buy (or the option to sell) a given asset over some period of time in the future This is done by assuming that the asset in question will have one of several possible values in the future and trying to determine a current fair value of the option based on these possible future values The option to buy (or the option to sell) a stock for a fixed value in the. .. to buy can be exercised at any time on or before the expiration date of the call 2) In a put option, the buyer has the right to sell the underlying stock to the writer at the strike price O per share a) In a European put, the right to sell can only be exercised on the expiration date of the call b) In an American put, the right to sell can be exercised at any time on or before the expiration date of . students of mathematics. Nevertheless, since the subject of this book is the of finance, I havemathematics not watered down the mathematics in any way appropriate to the level of the( book, of course. Pricing, Undergraduate Texts in Mathematics 3582 2 1 2 Introduction to the Mathematics of Finance Indeed, as late as the 1990s, the federal government encouraged the use of stock options as a form of executive. with regard to the granting of in -the- money) stock options. Introduction 3 The requirements contained in the FASB statement brought to the forefront the problem that is the subject of this book: