On a Closed Formula for the Derivatives of e f(x) and Related Financial Applications

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1 International Mathematical Forum, 4, 9, no. 9, On a Closed Formula or the Derivatives o e x) and Related Financial Applications Konstantinos Draais 1 UCD CASL, University College Dublin, Ireland Abstract We give a closed ormula or the derivative o arbitrary order o the unction gx) expx)) and count the number o the summands therein, as well as the maximal number o distinct derivatives o appearing as actors in each summand. We also suggest some exotic derivatives which beneit rom this ormula. Mathematics Subject Classiication: 5A1, 5A17, 91B, 91B8, 91B7 Keywords: Faà di Bruno ormula, composite exponential unction, number o summands, number o actors 1 Introduction The computation o derivatives o unctions o the orm expx)) is very useul and ubiquitous in analysis, when one is interested in boundedness properties o such unctions. The Blac-Scholes model or underlying asset prices in inance, in particular, is o this unctional orm, and its derivatives are needed or various estimates, in particular the computation o the sensitivity o derivative prices on various parameters, such as the starting price or the volatility o the underlying assets these sensitivities are also nown as grees ). In this paper we give a closed orm or [expx))] n), n N. Our result is a special case o the amous Faà di Bruno ormula [5], but, in the case o the special unctional orm we are studying, the ormula taes a simple orm, allowing us to count explicitly the number o summands involved and the maximal number o distinct derivatives o present as actors in such a summand. We do not 1 The author is also ailiated with the School o Mathematics, University College Dublin, Ireland, as well as with the Claude Shannon Institute Address: UCD CASL, University College Dublin, Belield, Dublin 4, Ireland Konstantinos.Draais@ucd.ie

2 4 K. Draais need and we mae no use o the general Faà di Bruno ormula, but rather we derive everything rom irst principles. Motivation In order to motivate the study o derivatives o expx)) in a inancial context, let us turn our attention to the ollowing exotic path derivative: assume the buyer o the derivative wishes to target a particular underlying asset, and to request rom the seller money at a pre-speciied time-dependent rate per unit o price o the underlying asset and unit o time over a pre-speciied time interval. The buyer s motive or requesting such a derivative might, or example, be a speculation that the asset s price will be aected at some particular point or short interval in time by some event, which will allow the buyer to mae a proit. The seller, on the other hand, will charge or such a derivative its air price, as usual: assuming that St) is the price o the underlying asset over the time interval [,T] o interest and that gt) is the desired rate, the expected total proit or the buyer, and hence the price o the derivative, will be: T ) P E St)gt)dt. Assume now urther that the price o the underlying asset ollows the Blac- Scholes model or some r and σ: ) ) St) S) exp r σ t + σwt) where W is a Wiener process W t) N, t)); imposing also some mild and reasonable conditions on g, or example that it be integrable and bounded over [,T], it ollows that: T E St) ) gt) dt < and, thereore, by Fubini s theorem, the order o the mean and the integral can be interchanged. Further, it is also simple to see that, or any n N: T ) d n St) E dσ n gt) dt < whence, by Lebesgue s dominated convergence theorem and Fubini s theorem, it ollows that the dierentiation, the integration, and the mean can be interchanged.

3 Closed ormula or the derivatives o e x) 43 A closed ormula or the nth derivative o S, then, which is essentially a unction o the orm expx)), will acilitate the analysis o the sensitivity o P over σ. And, although most o the time only the ormulas or n 1orn are needed or this purpose, the case can be made and indeed has been made [4]) that higher order derivatives must be considered or a more comprehensive and detailed analysis o sensitivity. A more extreme perhaps orm o a path derivative where the derivatives o expx)) play a crucial role would result rom speculation on the sensitivity o a higher order) o the asset price rather than the price itsel. In the notation used above, we are now interested in: T P n E ) d n St) dσ gt)dt n and the previous analysis applies verbatim here. Having so motivated the study o the derivatives o expx)), we now proceed with the derivation o the ormula. 3 Notation Let us denote by N the collection o sequences o natural numbers, with the additional restriction that only a inite number o the elements o the sequence can be positive: { } N { i } i N, i N i < In other words, n N : i > n, i ; in such cases we will also be writing 1,,..., n,,...) 1,,..., n ), i.e. we will use vector notation. We also deine the constant s 1,, 3,...), i.e. i N,s i i. n We will write! i! i!, and also s)) ) i) i. 4 The general ormula We claim that the derivative o order n o the unction given is o the orm: ) e x) n) ) e x) s) 1) We will proceed now to prove that this sum extends over the we claim it does, and also ind the exact value o the coeicients R. Notice that the n in this i

4 44 K. Draais notation is superluous, as it can be inerred by, but it will prove convenient later, as a marer o the order o the derivative the coeicient belongs to. Theorem 1. 1) holds i a n+1 e 1 + i +1)+e i e i+1, where the e i,i N denote the sequences in N whose elements are all, except the ith element which is 1. Proo. For n 1 the ormula is true, as e x)) e x) x), so that the summation extends indeed over all s such that s 1, i.e. only on 1,,,...) 1, and a Assume now that the ormula is true or n; we need to show it true or n + 1. But the ormula or n + 1 can be obtained by the ormula or n by dierentiation: e ) x) s) e x) e x) e x) s)) + e x) s) ) ) ) s) +e 1 + a n i ) i+1) s)) e i ) s) +e 1 + a n ) ) i s) e i +e i+1... Observe here that + e 1 and e i + e i+1, i 1,...,n are solutions o the equation s x n + 1. On the other hand, i x is a solution o this equation, it has at least one positive entry, hence at least one o the vectors x e 1, x + e i e i+1, i 1,...,n will belong in N. But as all such vectors that are in N are solutions o the equation s y n, and as, by our inductive assumption, all solutions o this latter equation in N appear in the summation o 1), there will exist a N so that either x + e 1 or x e i + e i+1 or some i 1,...,n. In other words, the summation in the last ormula o the above derivation extends on all { N, s n +1}. So, we can continue with the derivation, by rearranging terms: [ ]... e x) e 1 + a ) n s) +e i e i+1 { N, sn+1} This completes the proo. e x) { N, sn+1} a n+1 s) )

5 Closed ormula or the derivatives o e x) 45 We are still not inished, though: we need to ind an explicit ormula or the coeicients. All we proved so ar about them is that they can be determined recursively, hence uniquely, as this recursive determination maes them unique. The notation we are about to use is compatible with the notation used in [6]. Theorem. Let N and n N so that s n. Then, 1 n! n n i! s n!. For values o,n that do not satisy the equality above, we set i!) i!s!). Proo. All we need to show is that this deinition is consistent with the recursive deinition o the coeicients in the previous theorem. To begin with, observe that, assuming all quantities appearing are positive, and +e j e j+1 e )! n i i! s n! n i s i!) i 1 j + 1)! j+1 1)! n,i j,j+1 i! n! s j!) j+1 s j+1!) j+1 1 n,i j,j+1 s i!) i j+1 j +1 1s 1 n +1 an+1 s j+1! a n+1 j+1 s j! j +1 s j+1a n+1 Now, observe that the equalities we have obtained are true even i not all the intermediate quantities are positive, as then the coeicients on both sides are, and so that the equalities become. Then, ) e 1 + i +1)+e i e i+1 an+1 s s i+1 i+1 n +1 a n+1 s n +1 an+1 Finally, we see immediately that a 1 1 1! 1, as it should be. This completes 1!1! the proo. 5 Number o summands How many are the solutions o the equation s n N in N? They are as many as the dierent ways in which we can write n as the sum o nonnegative integers not greater than n, i.e. the number o partitions o n, as it

6 46 K. Draais is nown in the literature see [1] or details), and is oten denoted by P n) [1, 3]. Partitions have been extensively studied; although there exists an exact ormula or P n) [3], it is not practical at all. Fortunately, simple and accurate bounds also exist [1]. 6 Non-zero elements o the solutions We come now to the main contribution o this wor: how many non-zero elements does have or a given n? In other words, how many distinct derivatives o appear in each summand o 1)? Obviously, at least one. On the other hand, the irst time s non-zero elements will appear in will be when they are all 1 and they occupy the lowest possible positions within the sequence, i.e. ss +1) the irst s ones. Hence, s n becomes n s. So, or a given n, what is the maximum number o non-zero elements o there can be? As any nonzero element is as least 1, we need to solve ss +1) n, ors + s n, which leads to: 7 Conclusion 1+8n 1 s Given the notation we stated in section 3, we proved the identity: e x) ) n) e x) ) n!!s!) s) ). 3) We urther showed that { N, s n} P n), the number o partitions o n, and that s, the number o non-zero elements o in the solutions o s n, is bounded as in ). In section we oered some examples in derivative pricing and sensitivity analysis where this ormula proves to be useul. From this point o view this paper is quite original and unusual, in that it combines mathematical inance and discrete mathematics. Acnowledgements The author is indebted to Dr. Sotirios Sabanis School o Mathematics, University o Edinburgh, Scotland) or introducing him to this problem.

7 Closed ormula or the derivatives o e x) 47 Reerences [1] G. Andrews. The Theory o Partitions, Cambridge 1984) [] F. Blac and M. Scholes. The Pricing o Options and Corporate Liabilities, Journal o Political Economy, Vol. 81, No. 3, 1973, pp [3] J. H. Conway and R. Guy. The Boo o Numbers, Springer-Verlag 1996) [4] L. Ederington and W. Guan. Higher Order Grees, The Journal o Derivatives, Spring 7 [5] S. Roman. The Formula o Faà di Bruno, American Mathematical Monthly, Vol. 87, pp , 198 [6] R. P. Stanley. Enumerative Combinatorics, Vol. 1, Cambridge 1997) Received: August, 8