al Formula For Valuing AmericanPuts. 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. Itis simply because all the investors could lose if the option is not exercisedbefore the expiration rate is just the option price (that is the premium) thathe or she has paid earlier. Call options give the investor the right to buy theunderlying stock at the exercise price, X; while the put options give theinvestor the right to sell the underlying security at X. However only Americaoptions can be exercised at any time during the life of the option if the holdersees fit while European options can only be exercised at the expiration rate,and this is the reason why American put options are normally valued higher thanEuropean options.
Nonetheless it has been proved by academics that it isimpossible to derive an analytical formula for valuing American put options andthe reason why will be discussed in this paper as well as some main suggestedtechniques that are used to value them. According to Hull, exercising an American put option on a non-dividend-payingstock early 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 thestock price is virtually zero. By exercising early at this point of time, aninvestor makes an immediate gain of $20. On the contrary, if the investor waits,he might not be able to get as much as $20 gain since negative stock prices areimpossible.
Therefore it implies that if the share price was zero, the putwould have reached its highest possible value so the investor should exercisethe option early at this point of time. Additionally, in general, the early exerices of a put option becomes moreattractive as S, the stock price, decreases; as r, the risk-free interest rate,increases; and as , the volatility, decreases. Since the value of a put isalways positive as the worst can happen to it is that it expires worthless sothis can be expressed as where X is the strike price Therefore for an American putwith price P, , must always hold since the investor can execute immediateexercise any time prior to the expiry date. As shown in Figure 1,Here provided that r > 0, exercising an American put immediately always seems tobe optimal when the stock price is sufficiently low which means that the valueof the option is X – S. The graph representing the value of the put thereforemerges into the put’s intrinsic value, X – S, for a sufficiently small value ofS which is shown as point A in the graph.
When volatility and time toexpiration increase, the value of the put moves in the direction indicated bythe arrows. In other words, according to Cox and Rubinstein, there must always be somecritical value, S`(z), for every time instant z between time t and time T, atwhich the investor will exercise the put option if that critical value, S(z),falls to or below this value (this is when the investor thinks it is the optimaldecision to follow). More importantly, this critical value, S`(z) will dependon the time left to expiry which therefore also implies that S`(z) is actually afunction of the time to expiry. This function is referred to, according toWalker, as the Optimum Exercise Boundary (OEB). However in order to be able to value an American put option, we need to solvefor the put valuation foundation and then optimum exercise boundary at the sametime.
Yet up to now, no one has managed to produce an analytical solution tothis problem so we have to depend on numerical solutions and some techniqueswhich are considered to be good enough for all practical purposes. (Walker,1996)There are basically three main techniques in use for American put optionvaluations, which are known as the Binomial Trees, Finite Difference Methods,and the Analytical Approximations in Option Pricing. These three techniqueswill be discussed in turns as follows. Cox et al claim that a more realistic model for option valuation is one thatassumes stock price movements are composed of a large number of small binomialmovements, which is the so-called Binomial Trees (Hull, p343, 3rd Ed). Binomial trees assume that in each short interval of time, , over the life ofthe option a stock price either moves up from its initial value of S to , ormoves down to .
In general, ; 1 and ; 1. The probability of an up movementwill be denoted by thus, the probability for a down movement is . The basicmodel of this simple binomial tree is shown in Figure 2. Furthermore, the risk-neutral valuation principle is also in use when using a binomial tree, whichstates that any security dependent on a stock price can be valued on theassumption that the world is risk neutral.
Therefore the risk-free interest rateis the expected return from all traded securities and future cash flows can bevalued by discounting their expected values at the risk-free interest rate. Theparameters p, u, and d must give correct values for the mean and variance ofstock price changes during a time interval of length . By using the binomial tree, options are evaluated by starting at the end of thetree (that is time T) and working backward. The value of the option is known attime T. As a risk-neutral world is being assumed, the value at each node at timeT – can be calculated as the expected value at time T discounted at rate r for atime period . Similarly the value at each node at time T – can be calculated asthe expected value at time T – discounted for a time period at rate r, and so on.
When we are dealing with American options, it is necessary to check at eachnode to see if early exercise is optimal rather than holding the option for alonger while. Therefore by working the binomial backward through all the nodes,the value of the option at time zero is obtained. For example, consider a five-month American put option on a non-dividend-payingstock when the stock price is $50, the strike price is $50, the risk-freeinterest rate is 10% per annum, and the volatility is 40% per annum. With ourusual notation, this means that S = 50, X = 50, r = 0.
10, = 0. 40, and T = 0. 4167. Suppose that we break the life of the option into five intervals of length onemonth (= 0. 0833 year) for the purposes of constructing a binomial tree.
Then =0. 0833 and using the formulas,The top value in the tree diagram above shows the stock price at the node whilethe lower one shows the value of the option at the node. The probability of anup movement is always 0. 5076; the probability of a down movement is always0.
4924. Here the stock price at the jth node (j = 0, 1, . . .
, i) at time is calculated as. Also the option prices at the penultimate nodes are calculated from theoption prices at the first final nodes. First we assume no exercise of theoption at the nodes. This means that the option price is calculated as thepresent value of expected option price in time . For example at node E theoption price is calculated as while at node A it is calculated asThen it is possible to check if early exercise of the option isworthwhile. At node E, the option has a value of zero as both the stock priceand strike price are $50.
Thus it is best to wait and the correct value at nodeE is $2. 66. Yet the option should be exercised at node A if it is reachedbecause the option would be worth $50. 00 – $39. 69 or $10. 31, which is obviouslyhigher than $9.
90. Options in earlier nodes are calculated in a similar way. As we keep on calculating backward, we find the value of the option at theinitial node to be $4. 48, which is the numerical estimate for the option’scurrent value. However in practice, a smaller value of would be used by whichthe true value of the option would be $4.
29. (Hull, p347, 3rd Ed)The second technique that is commonly used is the so-called Finite DifferenceMethods. These methods value a derivative by solving the differential equationthat the derivative satisfies. The differential equation is converted into aset of difference equations and the difference equations are solved repeatedly. For instance, in order to value an American put option on a non-dividend-payingstock by using this method, the differential equation that the option mustsatisfy isThe Finite Difference Methods are similar to tree approaches inthat the computations work back from the end of the life of the derivative tothe beginning.
There are two different methods involved; one is called theExplicit Finite Difference Method and the other is the Implicit FiniteDifference Method. The former is functionally the same as using a trinomialtree. The latter is more complicated but has the advantage that the user doesnot have to take any special precautions to ensure convergence. The maindrawback of these methods is they cannot easily be used in situations where thepay-off from a derivative depends on the past history of the underlying variable. Finally there is also an alternative to the numerical procedures which is knownas a number of analytic approximations to the valuation of American options.
The best known of these is a quadratic approximation approach proposed byMacMillan and then extended by Barone-Adesi and Whalley. This method involvesestimating the difference, v, between the European option price and the Americanoption price since v must satisfy the differential equation for both. They thenshow that when an approximation is made, the differential equation can be solvedusing standard methods. The techniques mentioned in this paper are those commonly used in practise.
Although they are not perfect, they are still considered good enough forpractical purposes. So far no one has managed to create a direct analyticalvaluation method for valuing American put options.Category: Business
