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Explain Why It Has Proved Impossible To Derive An Analytical Formula For Valuing

1647 words - 7 pages

It's a very useful essay for those who are specialising in finance -Explain why it has proved impossible to derive an analytical formula for valuingAmerican Puts, and outline the main techniques that are used to produceapproximate valuations for such securitiesInvesting in stock options is a way used by investors to hedge against risk. It issimply because all the investors could lose if the option is not exercised before theexpiration rate is just the option price (that is the premium) that he or she has paidearlier. Call options give the investor the right to buy the underlying stock at theexercise price, X; while the put options give the investor the right to sell theunderlying security at X. However only America options can be exercised at any timeduring the life of the option if the holder sees fit while European options can only beexercised at the expiration rate, and this is the reason why American put options arenormally valued higher than European options. Nonetheless it has been proved byacademics that it is impossible to derive an analytical formula for valuing Americanput options and the reason why will be discussed in this paper as well as some mainsuggested techniques that are used to value them.According to Hull, exercising an American put option on a non-dividend-paying stockearly if it is sufficiently deeply in the money can be an optimal practice. For example,suppose that the strike price of an American option is $20 and the stock price isvirtually zero. By exercising early at this point of time, an investor makes animmediate gain of $20. On the contrary, if the investor waits, he might not be able toget as much as $20 gain since negative stock prices are impossible. Therefore itimplies that if the share price was zero, the put would have reached its highestpossible value so the investor should exercise the option early at this point of time.Additionally, in general, the early exerices of a put option becomes more attractive asS, the stock price, decreases; as r, the risk-free interest rate, increases; and as , thevolatility, decreases. Since the value of a put is always positive as the worst canhappen to it is that it expires worthless so this can be expressed aswhere X is the strike priceTherefore for an American put with price P, , must always hold since theinvestor can execute immediate exercise any time prior to the expiry date. As shownin Figure 1,Here provided that r > 0, exercising an American put immediately always seems to beoptimal when the stock price is sufficiently low which means that the value of theoption is X - S. The graph representing the value of the put therefore merges into theput's intrinsic value, X - S, for a sufficiently small value of S which is shown as pointA in the graph. When volatility and time to expiration increase, the value of the putmoves in the direction indicated by the arrows.In other words, according to Cox and Rubinstein, there must always be some criticalvalue, S`(z), for every time...

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