CS522 - Exotic and Path-Dependent Options

Transcription

1 CS522 - Exotic and Path-Deendent Otions Tibor Jánosi May 5, Other Otion Tyes We have studied extensively Euroean and American uts and calls. The class of otions is much larger, however. A digital (binary) otion is an otion that has a discontinuous ayo. An examle of such an otion is the Euroean cash-or-nothing otion, whose ayo at the exiration time T is given by the following formula: ; if S > K V B (T; S) = 0; otherwise : As usual, we have denoted the strike rice by K. An Asian otion is an otion whose ayo deends on a suitably de ned average S of the underlying stock s rice between the incetion of the otion and its exiration. The ayo of an Asian call with strike K is then given by C A (T; S) = max( S K; 0): Note that the second argument of C in the formula above is not the nal stock rice, but its average. This average can be comuted continuously, or at discrete oints only: S = T S = n Z T 0 S(t)dt S(t i ), where 0 6 t < t 2 < : : : < t n 6 T A xed lookback call or ut is an otion whose value deends on the maximum or the minimum of the underlying s rice during the lifetime of the otion, resectively: C fixed L (T ) = max( max S(t) K; 0) t2[0;t ] P fixed L (T ) = max(k min S(t); 0) t2[0;t ] Like in the case of Asian otions, the maximum or the minimum stock rice can be comuted by samling the stock rice at discrete moments in time. In the case of a oating lookback otion the strike rice is not xed, but it is chosen based on the best available rice of the underlying over the lifetime of the otion: CL oat (T ) = max(s min S(t); 0) t2[0;t ] PL oat (T ) = max( max S(t) S; 0) t2[0;t ]

2 A large class of otions is that of barrier otions. Barrier otions have knock-in and/or knock-out features. A knock-in feature is a condition that causes the otion to be e ective only if the underlying reaches a certain rice level. A knock-out feature is a condition that causes the otion to terminate immediately if the underlying reaches a certain rice level. There are four main tyes of barrier otions: "u-and-in", "u-and-out", "down-andin", and "down-and-out." The "u-and-in" otion becomes e ective only if the underlying rst reaches a redetermined knock-in barrier from below. Once the barrier is reached, the otion behaves like a lain-vanilla otion. The "down-and-out" otion behaves like a lain vanilla otion as long as the knockout-barrier seci ed for the resective otion is not reached from above. When - and if - that haens, the otion immediately terminates. The de nition of the "u-and-out" and "down-and-in" otions is analogous. Given that we have four tyes of barrier otions, and that they can be either uts and calls with Euroean or American features, we have imlicitly de ned 6 barrier otion tyes, each with its own valuation roblem. Even more comlicated otions can be de ned. For examle, one can de ne otions that knock-in only if the rice of the underlying enters and stays inside a band of rices delimited by an uer and lower barrier. It is not necessary for the underlying of an otion to be a stock. One can de ne otions on otions, leading to comound otions, e.g. a call on a ut. A chooser otion is an otion exiring at time T which allows the holder to choose between a lain-vanilla ut or a call exiring at time T 2 > T. Otions can be de ned on underlying commodities (e.g. otions on gold futures). As you can see, there are many tyes of otions, and there is almost a limitless theoretical ossibility of adding new features. In ractice, however, the choice of features is guided by the needs of the market. Otions with features desired by many market articiants are often traded on exchanges. Tyically, these are simle otions that have been widely studied theoretically and are well understood emirically. Comlicated otions, with many unusual features are often written by big nancial comanies at the request of their clients. Often, such values are hard to value, esecially since they are illiquid. 0.2 Valuing Path-Deendent and Exotic Otions It is sometimes ossible to write closed-form formulas for the value of exotic and athdeendent otions Analytic Valuation: Chooser Otions Assuming that the money market account earns interest at the constant rate r, the time-t value of a chooser otion that gives its holder the right to choose at time time T between 2

3 a Euroean ut or a call with strike K and exiration date T 2 following formula: > T is given by the V (t; T ; T 2 ; K) = (t; T 2 ; K) + c(t; T ; Ke r(t 2 T ) ); i.e. the value of chooser otion is equal to the value of a Euroean ut with exiration date T 2 and strike rice K, lus the value of a Euroean call with exiration date T and strike rice Ke r(t 2 T ). We rovide the roof of this claim below. When we are at time T, and we hold a chooser otion, the rational decision is to choose the underlying lain-vanilla ut or call with the higher value. For any underlying stock rice S(T ), we will thus choose the call if c(t ; T 2 ; K) > (T ; T 2 ; K); if not, we will choose the ut. Let us denote by A the set of all states of the world at time T for which the underlying call is more valuable than underlying the ut. Let function I A be the indicator function of set A. Since all the relevant arameters (r,, T, T 2, K) are xed, in our model the states of the world at time T are distinguished only by the rice of the stock rice at that very same moment of time T. To avoid comlicating our notation further, we identify these states with their associated stock rice. The time-t 2 ayo of the chooser otion is given by the following formula: max(s(t2 ) K; 0), if I V (T 2 ; T ; T 2 ; K) = A (S(T )) = max(k S(T 2 ); 0), if I A (S(T )) = 0 : The notation for the value of the chooser otion is somewhat unwieldy, but informative. V (t a ; t b ; t c ; K) reresents the time-t a value of a chooser otion exiring at time t b ; the ut or call that the chooser otion make available will exire at time t c and their common strike rice is K. The formula exresses the very simle fact that if at time T we went through a state in set A, then we chose the call, and we will receive its ayo. If we went through a state that is not in A (i.e. we went though a state in the comlement of A, A C ), then we will receive the ayo of the ut. By introducing the notation () + = max(; 0) and the indicator function I A C of the set A C, we get V (T 2 ; T ; T 2 ; K) = (S(T 2 ) K) + I A (S(T )) + (K S(T 2 )) + I A C(S(T 2 )): Since A and A C have no common elements, and their union is the set of all ossible states at time T, we can conclude that I A (S(T )) + I A C(S(T )) =, irresective of the value of S(T ). The time-t 2 of the chooser otion can now be rewritten as follows: V (T 2 ; T ; T 2 ; K) = (S(T 2 ) K) + I A (S(T )) + (K S(T 2 )) + ( I A (S(T 2 ))) = (S(T 2 ) K) + (K S(T 2 )) + I A (S(T )) + (K S(T 2 )) + = (S(T 2 ) K) I A (S(T )) + (K S(T 2 )) + : Strictly seaking, we do not have a chooser otion at T 2 anymore, since it exired at time T. We are really talking about the ayo of the instrument we got when we exercised the chooser otion. For simlicity, however, we will kee talking about the chooser otion s ayo at T 2. 3

4 The last equality has been obtained by noticing that the sum (S(T 2 ) K) + (K S(T 2 )) + reresents the ayo of a ortfolio consisting of a long call and a short ut. An alication of the ut-call arity for Euroean otions exiring at time T 2 immediately roduces the equivalent exression. Now that we have an analytic exression for the value of the chooser otion (more recisely, whatever became of it) at time T 2, we can comute its value at time T by determining the discounted exect value of the ayo under the equivalent martingale robability measure: V (T ; T ; T 2 ; K) = e r(t 2 T ) E q (S(T2 ) K) I A (S(T )) + (K S(T 2 )) + = e r(t 2 T ) I A (S(T ))E q [(S(T 2 ) K)] + e r(t 2 T ) E q (K S(T2 )) + = e r(t 2 T ) I A (S(T ))E q [(S(T 2 ) K)] + (T ; T 2 ; K) = I A (S(T )) e r(t 2 T ) E q [S(T 2 )] Ke r(t 2 T ) + (T ; T 2 ; K) = I A (S(T )) S(T ) Ke r(t 2 T ) + (T ; T 2 ; K): The rst term in the exression above has been rewritten by rst removing the indicator function from under the exectation oerator. This ste is justi ed by noting that at time T the value of the underlying stock S(T ), and thus the value of the indicator function has been totally determined; this quantity is not random from time T on. Noting that the exectation of a constant (K) is the constant itself, and that the discounted exected stock S(T 2 ) is equal to the stock rice at time T (why?), we can comletely eliminate the exectation oerator. The second term corresonds to the de nition of the time-t value of a Euroean call with exiration T 2 and strike rice K. Let us now revisit the condition that de nes states in set A: c(t ; T 2 ; K) > (T ; T 2 ; K) c(t ; T 2 ; K) (T ; T 2 ; K) > 0 S(T ) Ke r(t 2 T ) > 0 The last relation has been obtained by using the ut-call arity for Euroean uts. Note that the exression that de nes the set of states A is the same as the exression multilying the indicator function in the exression for V (T ; T ; T 2 ; K). By observing that I A (S(T )) S(T ) Ke r(t 2 T ) S(T ) Ke = r(t 2 T ), if S(T ) Ke r(t 2 T ) > 0 ; 0, otherwise we can rewrite the leftmost term in V (T ; T ; T 2 ; K) as S(T ) Ke r(t 2 T ) +. We now get: V (T ; T ; T 2 ; K) = S(T ) Ke r(t 2 T ) + + (T ; T 2 ; K) = c(t ; T ; Ke r(t 2 T ) ) + (T ; T 2 ; K): 4

5 Thus, at time T the value of the chooser otion is given by the value of a Euroean call with exiration T and strike Ke r(t 2 T ), lus the value of a Euroean ut exiring at T 2 with strike rice K. A simle arbitrage argument immediately leads to the conclusion that the equality must hold for all times t between 0 and T : V (t; T ; T 2 ; K) = c(t; T ; Ke r(t 2 T ) ) + (t; T 2 ; K): A chooser otion aears to be, and in some sense is, a comlicated instrument. Careful reasoning and the knowledge we gained from studying lain-vanilla otions allowed us to decomose the chooser otion into a ortfolio of one Euroean ut and one Euroean call. These instruments, in turn, can be valued using the Black-Scholes formulas. It is often ossible to decomose comlex instruments (not only otions) into simler ones. Sometimes, like here, the valuation of the simler instruments is straightforward, but this is not always the case. Can you derive an alternative valuation formula for chooser otions? Hint: Exress the value of I A as a function of I A C Binomial Method: General, but Slow If an instrument can not be valued using closed-form formulas, then one must look for numerical alternatives. As we have seen earlier, aroximate values for simle otions are easy to comute using the binomial method. We have also seen, however, that certain features of the underlying otion make the binomial method to become resource intensive, when analyzed from an algorithmic ersective. In articular, we have shown that if a stock ays discrete dividends, then each node that corresonds to a given dividend ayment "srouts" a new subtree whose nodes will not recombine with the nodes of subtrees associated with its neighboring nodes. This yields to an exonential increase in the total number of nodes in a binomial tree, when viewed as a function of the number of dividend ayments. The resence of discrete dividends can greatly increase the running time and memory consumtion needed for evaluating even simle otions. While we did not address this toic extensively, it should be clear to an informed reader that discrete dividends will comlicate valuation methods based on di erential equations or inequalities as well (they induce discontinuities in the solution). Path deendent otions imose another tye of comlexity, which also yields to an exonential increase in the number of states. Let us consider an Euroean instrument whose ayo is an arbitrary function of all the stock rices encountered on the seci c ath followed from time 0 to exiration. This means that each individual node in which we could end u at exiration must be associated with the entire history of stock rice evolution. There are several ways in which one could achieve this association. The easiest one, and the one tyically given in textbooks, is to not to recombine the nodes. If this aroach is followed, then each node is associated with a unique ath from the root of the tree (time- 0) to the resective node. Hence, knowing what node we are in allows us to determine 5

6 ath ath ath 2 or 3 ath 2 ath 3 ath 4 ath 4 Figure : Tracking aths in binomial trees with recombining and non-recombining nodes. fully the rice history of the underlying instrument, as illustrated in gure. This idea is easy to imlement, and very general, but it results in the doubling of the number of nodes at each level of the binomial tree. Such a binomial tree that contains n intervals (thus n + levels) will have 2 n nodes on the last level, and a total of 2 n+ nodes. The desire to use high values for n, which is necessary to increase recision, thus con icts with the resource-consumtion limits imosed by the size of the tree. From an algorithmic ersective, it is not strictly necessary to use a non-recombining tree in order to maintain the rice history of the underlying stock. In rincile, it is ossible to design data structures that allow the user to maintain in each node a list of all ossible incoming aths, and comute and store searately all the relevant values that are associated with those aths. Such solutions, while logically equivalent to a binomial tree, are comlex to imlement and non-uniform. In the non-recombining tree each node can be reached only through one ath from the root, and only only one value (set of values) characterizing that ath must be accounted for. In our alternative solution, nodes far from the root can be reached through many aths, which rogressively comlicates accounting for the aths, and for the values associated with them. A more careful reasoning will show that the overall comlexity and resource requirements of such alternative methods are not less than those of the corresonding simle non-recombining binomial tree. There are no shortcuts if the entire rice history must be accounted for. From a rogrammer s ersective, one big advantage of the binomial methods is that the valuation of many di erent instruments can be imlemented in a very uniform manner. It is ossible to write very general valuation algorithms based on the binomial method such that all the seci c functionality is factored out into user-de ned functions. You have solved a roblem of this avor when you built a general valuation function on to of a recombining binomial tree. Of course, e ciency matters, and one is thus motivated to look for more e cient 6

7 alternatives. Consider, for examle, the case of Asian otions with discrete rice averaging. At exiration, the only rices that matter are the values samled at times t i, i = ; n; rices at intermediate times are not relevant. As long as we associate each node at exiration with its own sequence of samled rices, our binomial model will be useful for valuation. Indeed, we can solve this roblem in a manner analogous to the treatment of discrete dividend ayments, by "srouting" a new indeendent subtree at each node that occurs at any of the samling times t i (and making sure that nodes from di erent subtrees never recombine). In-between samling times, as well as between time 0 and the rst samling time, the subtrees will be recombining. Can you say why? While the total number of nodes needed to imlement this variation of the binomial model will still be exonential in the number of samling times, avoiding the doubling of the number of nodes at each time ste is can lead to a huge decrease in the number of nodes needed if there are many more time intervals reresented in the tree than samling times Numerical Solutions to Di erential Equations and Inequalities The solution of many valuation roblems can be stated in terms of the solution of a roblem involving (a system of) di erential equations and inequalities with suitable initial, nal, and boundary conditions. Some of these roblems can be solved analytically, but many can only be comuted numerically. Many mathematical roblems that arise in the context of valuation have been studied extensively in other elds, such as hysics and engineering. We do not exand on this toic further Monte Carlo Methods There are no closed-form formulas for valuing discretely samled Asian otions. We will now study these otions to gain insights into a new valuation technique, the Monte-Carlo method. As usual, we assume that the stock rice follows a log-normal distribution. For concreteness, we will focus on Asian calls. The treatment of Asian uts is analogous. The time-t value of the otion is, as usual, the discounted exect value of the ayo at exiration, where the exectation is comuted under the equivalent martingale robabilities q: V (t; T ; K) = e r(t t) E q max( S K; 0) : Let us assume for the moment that we have a general - but aroximate - method of numerically estimating exectations E q []. One is often not interested only in determining the value of a articular instrument, but also on simultaneously determining the sensitivity of the resective value to changes in various underlying arameters. Knowing the value of an instrument is imortant for trading and accounting uroses, knowing its sensitivities is imortant for hedging. Assume that we can comute the aroximate value of an Asian otion, and that the resective value is ~ V (t; T ; K; ). Note that we have modi ed the notation to make 7

8 exlicit the deendence of the value on some underlying arameter. We can estimate the sensitivity of V ~ to small changes in the arameter by using any of the numerical methods we have introduced for comuting derivatives. Here is a method using symmetric di erences: V ~ (t; T ; K; ) V ~ (t; T ; K; + ) V ~ (t; T ; K; ) : Such an aroach might have the downside that the valuation method for the resective otion must be used twice in order to comute the desired sensitivity. We oint out, however, that this is not necessarily the case, as some methods might generate values for an entire range of arameters. Think, for examle, of the valuation roblem for Euroean or American otions using nite di erences, where we simultaneously comute the values of the otion for an many oints between S min and S. Deending on the ste size of the grid, one might be able to use these values to estimate the delta of the resective otion Sensitivities of Asian Otions as Exectations Since we have an exact exression giving us the value of the Asian otion as an exectation, we can, in rincile, comute the sensitivity with resect to in nitesimal changes in the value of arameter : V = e r(t t) E q max( S K; 0) = e r(t t) E q max( S K; 0) + e r(t t) E q max( S K; 0) : We assumed that we have a method of comuting numerical values for comuting exectations under q, but that does not imly that we know how to comute exressions like E q []. Given our assumtions, it would be ractical if we could interchange the order of the di erential and exectation oerator. Do the Exectation and Di erential Oerator Commute? Is this ossible? If yes, the following equality should hold in general, irresective of the exression relaced by : E q [] = E q : To make things more seci c, let us examine the validity of this urorted equality by using a digital "all-or-nothing" otion. Let us consider one of the in nitely many ossible aths that the stock rice might follow between time 0 and the exiration time T. At exiration, the ayo of the digital otion is di erentiable for all underlying stock rices, excet at S(T ) = K. It is easy to see that irresective of the arameter at hand, we have that V digital V (T; T ; K) = 0 for all S 6= K, i.e. digital (T; T ; K) = 0 a.s. h i Vdigital ("almost surely"). From this, we immediately conclude that E q (T; T ; K) = 0. If 8

9 the di erential and exectation oerator were commutable, the last relation would imly that E q [V digital (T; T ; K)] = 0, which, in turn, means that V digital (t; T ; K) = e r(t t) E q [V digital (T; T ; K)] : It is easy to see why this equality can not be true for all arameters. The last relation imlies that the value of a digital otion is not sensitive to changes in the volatility of the underlying stock (), or the stock rice at time t. This can not be the case. Thus, in general, the di erential and the exectation oerator do not commute. With this caveat, we note that it is ossible for the two oerators to commute, assuming that the function reresented by above is uniformly integrable. In the following, we will commute the di erential and exectation oerator without roving the uniform integrability of the functions at hand. The reader should feel assured, however, that the condition is satis ed. Modeling the Stock Price at Samling Times The assumed log-normal distribution of the underlying stock rices imlies that for any two moments of time t a < t b, the following relation holds: S(t b ) = S(t a ) ex r 2 2 (t b t a ) + Z a;b tb t a : Note that Z a;b is a value drawn from a standard normal distribution. Consider now an Asian otion with discrete rice averaging, with n samling times such that at least two samling times occur after time 0. Now take two successive samling times t i and t i+, such that 0 = t 0 < t i < t i+. In simulating the rice of the otion at time t i+, we have the choice 2 of roducing a random rice starting from time 0, or for roducing a random rice starting from time t i, as illustrated in the two exressions below: S(t i+ ) = ex r t i+ + Z 0;i+ tt+ S(t i+ ) = S(t i ) ex r (t i+ t i ) + Z i;i+ tt+ t i : Are these two formulations equivalent? If not, which version should we use for modeling the evolution of the stock rice? Both questions can be answered easily. The two methods are not equivalent, in that they do not roduce the same robability distribution for the rices at time t i+. A direct calculation will convince you of this. Intuitively, it is clear that the equivalence can not hold. Assume, for examle, that t i and 2 We have other choices as well. For examle, we could simulate the stock rice at time t i+ by starting at time t i and "skiing" over time t i. As the subsequent discussion makes it clear, these alternative methods would not correct. As they do not rovide any additional insights, we do not consider them exlicitly. 9

10 t i+ are very close to each other, but far away from t 0 = 0. Under these conditions, it is highly likely that S(t i+ ) will be close to the rice at S(t i ). This condition is satis ed when we simulate S(t i+ ) starting from S(t i ), but it will not necessarily hold when we start the simulation at t 0 = 0: The roblem is that when we restart the simulation from t 0 = 0, we disregard the fact that the stock rice has reached S(t i ) and time t i, and that this fact constrains the future evolution of the stock rice (changes the distribution of the future stock rices comared to the unconstrained distribution generated by starting from t 0 ). As long as we make sure that the generates rices are not indeendent of each other, we can still write the rice for each intermediate samling time t i, i = ; n by using the rice of the underlying stock at time t 0 : " # Xi S(t i ) = ex r 2 2 t i + Z l;l+ tl+ t l : Note that Z l;l+ are indeendent standard normal random variables. l= The Delta of An Asian Call Using the insights gained above, we get: = V = e r(t t) E q max( S K; 0) + e r(t t) E q max( S {z } =0 = :e r(t t) E q max( S K; 0) : S(t) K; 0) Focusing on the di erential oerator only, we can now write the following: max( S K; 0) = S ( = max( S K; 0) S S(t) S, if S > K S(t) 0, if S < K = I S>K S S(t) a.s. Note that max( S K; 0) is not di erentiable with resect to S at the oint S = K. This is not a roblem, however, as the set of oint of non-di erentiability is of 0-measure. The notation I S>K denotes the indicator function of the event S > K. Comuting the 0

11 value of S S(t) S is straightforward: = = n = n = n = n = " n # S(t i ) S(t i ) " # Xi ex r 2 2 t t+ + Z l;l+ tl+ t l l= " # Xi ex r 2 2 t t+ + Z l;l+ tl+ t l l= S(t i ) S : Putting everything together, we get that = V = e r(t t) E q I S>K S : Ignoring non-random factors, the delta of the Asian call has been written as an exectation. By noting that a regular Euroean call is just an Asian call with a single samling eriod at the exiration date, we can also write the formula for the delta of Euroean calls: EU = V EU = e r(t t) E q IS(T )>K S(T ) : This exression can easily be evaluated directly, through a comutation similar in avor to that undertaken when we derived the Black-Scholes formulas. The Vega of an Asian Call derivation: We roceed in a manner analogous to the revious = V = e r(t t) E q max( S K; 0) + e r(t t) {z } E q max( S =0 = e r(t t) E q max( S K; 0) : K; 0)

12 Focusing on the di erential oerator, we can write the following: max( S K; 0) = S max( S S = I S>K a.s. K; 0) S The derivation of the S term is also easy: S " # = S(t i ) n = S(t i ) n " = # Xi ex r n 2 2 t i + Z l;l+ tl+ t l = n = n = n l=! Xi t i + Z l;l+ tl+ t l S(t i ) l= t i + log S(t i) log S(t i) r r t i S(t i ) 2 2 t i S(t i ) Putting everything together, we write the vega of the Asian call as an exectation: " = V = e r(t t) E q I n S>K log S(t i) r + 2 # 2 t i S(t i ) : Again, we can immediate derive the formula for a Euroean call s vega: EU = V EU = e r(t t) E q I S(T )>K S(T ) log S(T ) r T : The formulas for an Asian call s delta and vega do not aear articularly simle, nor useful. Their advantages will be revealed as we examine Monte-Carlo methods below A Primer on Monte-Carlo Methods Theorem (The Weak Law of Large Numbers) If random variables X, X 2, X 3,... are indeendent, E [X i ] = m, V ar [X i ] 6 K <, and S n = P n X i, then Sn! m in L2 n and robability. 2

13 This, and similar theorems, rovide the theoretical basis for evaluating exectations using Monte-Carlo methods. Consider the formulas we derived for Asian otions: V (t; T ; K) = e r(t t) E q max( S K; 0) = e r(t t) E q I S>K ( S K) = e r(t t) E q I S>K S " V = e r(t t) E q I n S>K V log S(t i) r + 2 # 2 t i S(t i ) Any of the quantities under the exectation oerator can be identi ed with the random variables X i. All we have to do is to generate "many" samle values and average them to get estimates of the resective random quantity s exectation. But how many samle values do we need to average? A measure of how far from the true value our average is given by the standard deviation of S n : Sn = V ar [S n ] = q P n n 2 V ar [X i] 6 methods is O n 2 q K. By this estimate of the error, the convergence of Monte-Carlo n. As we know, this is not the whole story, as asymtotic comarisons between the erformance of various methods might occasionally be misleading. Put simly, the "big-o" notation hides constants, and these constants can surrise us. Still, for simle roblems, this rate of convergence is too slow comared to other methods. The advantage of Monte-Carlo methods, however, is that their rate of convergence is not sensitive to many roblem arameters that degrade the erformance of alternative methods. Hence, Monte-Carlo methods are referred, or at least cometitive, for solving many comlex roblems. The valuation of Asian otions is one of the simlest roblems where the advantages of these methods are borne out. It is ossible to increase the rate of convergence by generating samle values of that are not entirely random. Such techniques, known as "seudo Monte-Carlo" methods can increase the rate of convergence to (almost) O(n ). We do not discuss these methods further. Of course, in ractice we will often not know the variance of the quantity that we are trying to comute. We can get an unbiased estimate of the variance using the formula: gv ar [S n ] = n 2 X i n S n : These being said, let us return to the roblem of the value, the delta, and the vega of an Asian call. It is clear that one can generate random samle values for S. Once values for S are available, it takes only a small - and constant, er ath simulated - amount of work to comute the value of exressions I S>K ( S K) and I S>K S, which aear under the exectation oerator in the formulas for the value and delta of the call. Assuming that the cost of generating S dominates the cost of these two simle exressions, it is ossible to comute both the value of the call and its delta will little additional overhead. 3

14 This is in contrast with the requirement of reeating the e ort of comuting the otion value if we had used, say, a symmetric di erence method to comute delta. Things get even better. There are no obvious simle ways to generate samle values for S without generating samle values for the stock rice at all relevant times t i, i = ; n. If so, then all the elements for comuting vega are generated whenever we comute either the call value, or its delta (or both). Again, if the cost of generating the samle values S(t i ) dominates the cost of comuting the value of the exressions under the exectation oerators, then we can comute vega with minimal overhead. To comute delta and vega using nite di erence aroximations would require at least trile the e ort needed for a simle valuation, while using the exectation form of delta and vega we can get away with little more than the e ort needed for a simle valuation Generating Random Numbers Given the distribution of a random variable, how do we samle it? In other words, how do we generate random values drawn from a given distribution? The answer to this question is surrisingly subtle and comlex. Let ' be the.d.f., and be the c.d.f. of a given distribution. From the de nition of, we have that (z) = Prob(z 0 6 z) = Z z '(z)dz: Let u be a random variable drawn from a uniform distribution over the interval (0; ), i.e. u~u(0; ). 3 Let z u the value that satis es the equality (z u ) = u; i.e. z u = (u). It is easy to rove that z u will have the same distribution as that seci ed by ' and. So, if we can generate random values drawn from a uniform distribution on the interval (0; ), then we can generate variables with any distribution. But how to generate such uniformly distributed values? The roblem of generating uniformly distributed random variables is comlicated by the limits of comuter hardware. Let us assume for a moment that we have an algorithm that draws samles from a (mathematically) uniform random distribution, and mas them to the oating-oint number in the interval (0; ) closest to the generated number. 4 Let F (0;) be the set of all oating oint numbers that can be reresented in the interval (0; ). When lined u on the real axis, these values are not equidistant; seci cally, values closer to the left end of the interval (0) will be closer to their neighbors, than values closer to the right end of the interval (). This means that the size of the interval within which a oint is closer to a given value f from F (0;) than any other value in F (0;) deends on the magnitude of f. Hence, certain values from F (0;) will be chosen more frequently than others. So the fact that we can not reresent the entire continuum of values in (0; ) imacts the uniform 3 Note that we did not include the ends of the interval, i.e. values 0 and. Can you see why? 4 Of course, if such an algorithm existed, it could not work this way. This is because the algorithm would have to use the very same oating oint reresentation whose limits we are now discussing. The algorithm could generate its values, however, as if this maing would truly occur after a corresonding value has been drawn from the underlying truly uniform distribution. 4

15 distribution in a subtle way. The inherent discretization introduced by the oating oint reresentation could negatively in uence the quality of the distributions roduced using (u). Could it be that the non-uniform random distributions that we generate are in some sense worse if we start from our "uniform" distribution that if we had generated them directly using some alternate method? These are imortant questions, but we do not elaborate on them. We can, in rincile, generate random values in (0; ) that are uniform in the sense that all values that can be roduced are equally likely to be chosen. This can be done, for examle, by dividing the interval (0; ) into a number of equal-length subintervals, and choosing the midoint of each such interval as the reresentative for the interval. In the case of the oating oint reresentation with a mantissa of N bits, for examle, we can set the exonent to 0, and then create an algorithm that generates all ossible bit con gurations for the mantissa. 5 Excet for scaling by 2 N, the roblem of generating the uniform distribution is reduced to selecting any random integer in the range to 2 N with equal robability. Many random number generators use similar ideas. We have reduced the roblem of generating a uniform random distribution on (0; ) to the roblem of generating integers in a given range. The latter roblem has been studied extensively. Since our comuters are deterministic, true randomness is hard to come by. Assume that we have a source that roduces one random bit b at a time, such that Prob(b = ) =Prob(b = 0) = 0:5. By generating a sequence of N such random bits, we can generate all numbers in the range from to 2 N (but we have to throw away the con gurations for 0). So how do we generate a truly random bit b with the roerty that Prob(b = 0) = 0:5? It turns out that we can reduce this roblem to the roblem of generating a random bit b with the roerty that Prob(b) =, where is xed, but unknown. Indeed, assume that we generate two biased random bits b, and b 2. As long as b = b 2, we kee generating a new air of bits. If b 6= b 2, then we set b to the value of b. Show that this method generates an unbiased bit from a biased bit! Generating a truly random - but erhas biased - bit is still not feasible in a deterministic comuter. This feat is tyically achieved using secial hardware. Hardware that roduces random bits by samling the electronic noise roduced by secial-urose circuit elements is commercially available, but often exensive. It turns out that for many alications true randomness is neither needed, nor articularly useful. Indeed, it is often su cient to generate seudo-random sequences of integers. Such sequences are otherwise erfectly deterministic, but they aear as if they were random in that they ass statistical tests that a truly random sequence would be exected to ass. Pseudo-random sequences are often "seeded" with values, which - while 5 Floating oint reresentations normalize the mantissa, so that the rst digit in base 2 or 6 is nonzero. Normalization might force us to use a non-zero exonent for some values that we generate, but this is a detail that we can easily account for. 5

16 deterministic themselves - are not easily redictable or reroducible (unless the user reuses such a value directly). Good seeds are rovided by the total time, measured in miliseconds or microseconds, elased from a reference date, by the time of continuous comuter utime, or by the number of microrocessor instructions executed since the last reboot. Saved seeds allow for the reetition of the entire seudo-random sequence. This is a great advantage when testing and debugging rograms, for examle. As the brief discussion above shows, generating random or seudo-random number sequences is not trivial. Without exanding further, we conclude by arahrasing two ieces of advice from Donald Knuth s "Art of Comuter Programming:" a) Do not trust your random number generator; test it. Indeed, it is not uncommon for random number generators, esecially for those in general-urose ackages, to be awed. This roblem has been mitigated by the develoment of high-quality mathematical software in the ast few years, but "surrises" in this area are still likely to be abundant. b) Do not choose your random number generator randomly! 6