A note on Black–Scholes implied volatility

A note on Black–Scholes implied volatility

ARTICLE IN PRESS Physica A 370 (2006) 681–688 www.elsevier.com/locate/physa A note on Black–Scholes implied volatility Jesu´s Chargoy-Corona, Carlos...

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ARTICLE IN PRESS

Physica A 370 (2006) 681–688 www.elsevier.com/locate/physa

A note on Black–Scholes implied volatility Jesu´s Chargoy-Corona, Carlos Ibarra-Valdez Divisio´n de Ciencias Ba´sicas e Ingenierı´a. Universidad Auto´noma Metropolitana-Iztapalapa Apartado Postal 55-534, Me´xico D.F. 09340, Me´xico Received 15 July 2005; received in revised form 20 February 2006 Available online 27 April 2006

Abstract An approximate formula for the Black–Scholes implied volatility is given by means of an asymptotic representation of the Black–Scholes formula. This representation is based on a variable change that reduces the number of meaningful variables from five to three. It is stated clearly which is the family of functions we are going to work, specially the inverse of the normal accumulative function. Estimates for the error in the resulting approximate formulas for both the option value and the volatility are obtained as well. r 2006 Elsevier B.V. All rights reserved. Keywords: Black–Scholes formula; Implied volatility; Asymptotic and approximate formulae

1. Introduction The celebrated Black–Scholes (BS) formula gives the fair (no arbitrage) price for an European call option, which is quite an important financial instrument. The approximate figure suggests that option trading neighbors half of the world gross product. The BS formula is not too much complicated and is enough important to have obtained general attention, to the extent that it has been used extensively along more than thirty years, in currently millions of financial transactions worldwide. The interest in this formula has been widespread in publications of several kinds, including recently the econophysics section of Physica A, where several papers devoted to the BS formula connections with Finance and Mathematical Physics have been published (see for example [1–5]). The BS formula (as displayed below in Section 2) gives explicitly the option value as a function of five variables, c ¼ f ðr; s; K; t; sÞ, four of them observable (the risk-free interest rate r; the current price of the underlying asset s, the option exercise price K, and the time to maturity t), and one not directly observable and conceptually more involved variable: the current volatility of the asset, s. For several practical reasons, frequently, we got option prices c from the market, and get interested in the corresponding underlying asset volatility s that follows from the BS formula as far as the other three variables remain constant. This is called The Implied or Implicit Volatility. (See [6–8] and specially [9], the discussion in 6.7.2. p. 94). Since BS formula is somewhat complicated and involves transcendental functions, this is usually Corresponding author. Tel.: +52 55 58044657; fax: +52 55 58044660.

E-mail address: [email protected] (C. Ibarra-Valdez). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.03.019

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performed by numerical methods (most frequently by Newton–Raphson with quadratic convergence, or Halley’s method with cubic convergence). The mathematical problem that arises here is that of inverting the BS formula for obtaining the volatility in closed form, as a function of r; s; K; t and the option value c. However, in most finance textbooks and in several research papers dealing with this subject, it is asserted the impossibility of obtaining volatility as a closed form function of the option value and the remaining variables (see [6,8,9]; a similar assertion can also be found in the introduction of the forthcoming paper by Teichmann and Schachermayer [10]). Most works on this problem assume such impossibility and proceed in two broad directions: one theoretical, that attempts to obtain abstract mathematical properties of the implicit volatility, such as partial differential equations governing it or similar approaches (see [11–13]), and the other research direction quite more practical, centering on obtaining approximate formulas and testing them against market data [14–16]. It is worth mentioning that the latter works do not estimate the accuracy of their approximations. Information from Galois Theory and related mathematical areas points towards the fact that obtaining closed form solutions from given equations depend on what functions can be used [17,18]. Thus before attempting to clear a variable from an equation, one must say in which terms (with which family of functions) one is going to work. In this note we shall use the elementary arithmetic operations and functions, together with the normal accumulative function NðÞ, and its inverse, to get asymptotic and approximate formulas for the option value, and an approximate formula for the implicit volatility. Which is similar to others in the literature. The asymptotic BS formula, and the approximate volatility formula can be considered as an extension of an explicit formula for the volatility to a neighborhood of the zero log-moneyness manifold. This is done in Section 2. In Section 3 we establish bounds for the error between the exact and approximate formulas, both for the option value and the implicit volatility. Section 4 concludes. 2. BS formula, change of variables and asymptotic formula The BS formula is c ¼ f ðr; s; K; t; sÞ ¼ sNðd 1 Þ  Kert Nðd 2 Þ with d1 ¼

logðs=KÞ þ ðr þ 1=2s2 Þt pffiffiffi ; s t

d2 ¼

pffiffiffi logðs=KÞ þ ðr  1=2s2 Þt pffiffiffi ¼ d 1  s t, s t

and 1 NðuÞ ¼ pffiffiffiffiffiffi 2p

Z

 t2 exp  dt. 2 1 u



Note that the function of five variables defined above is real-analytic in the positive orthant r; s; K; t; s40. Consider the implicit function problem for s: F ðr; s; K; t; s; cÞ ¼ f ðr; s; K; t; sÞ  c ¼ 0. Since pffiffiffi @F ¼ s tN 0 ðd 1 Þ40, @s then for each initial condition ðr0 ; s0 ; K 0 ; t0 ; s0 ; c0 Þ such that s0 ; t0 40, there exists by the implicit function theorem, a local (implicit) solution s ¼ sðr; s; K; t; cÞ. The problem is to give a reasonably complex formula, involving all the above functions and N 1 , the inverse of the normal accumulative function N. We did not succeed in obtaining such an explicit expression, but we

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obtain an exact asymptotic representation for the BS formula, that gives an approximate formula for the implicit volatility and where the order of magnitude of the error can be estimated. 1 With respect to the function Np , ffisince N : R ! ð0; 1Þ is analytic, strictly increasing and with a nowhere ffiffiffiffiffi 0 vanishing derivative ðN ðuÞ ¼ ð1= 2pÞ expðu2 =2ÞÞ, then by the inverse function theorem, it has an analytic inverse, which happens to be global, and we denote it by j ¼ N 1 : ð0; 1Þ ! R. Theoretical properties, numerical tables, etc. have been widely developed for this function, and thus we can consider it as an elementary function to work with (see for example [19,20]). In the sequel, we shall write Rnþ ¼ fðx1 ; . . . ; xn Þ : xi X0; 8ig;

Rnþþ ¼ fðx1 ; . . . ; xn Þ : xi 40; 8ig.

2.1. Motivation for the change of variables Define the log-moneyness as a ¼ logðs=Kert Þ ¼ logðs=KÞ þ rt, and the zero-log-moneyness manifold M ¼ fðr; s; K; t; sÞ 2 R5þþ : s ¼ Kert g, i.e., Mp ¼ffiffiffi fa ¼ 0g. In this pffiffiffi manifold we can obtain a closed form solution for the volatility, since in this case d 1 ¼ s t=2, d 2 ¼ s t=2, and hence the BS formula becomes pffiffiffi pffiffiffi c ¼ s½Nðs t=2Þ  Nðs t=2Þ. pffiffiffi For any d 2 R it happens that NðdÞ  NðdÞ ¼ 2NðdÞ  1, then c ¼ sð2Nðs t=2Þ  1Þ, from which we get pffiffiffi sM ¼ ð2= tÞ jððc þ sÞ=2sÞ. Our aim is to extend, in certain sense, this explicit formula to a neighborhood of M. Note that given an initial condition c0 ¼ f ðr0 ; K 0 ; t0 ; s0 ; s0 Þ, the intersection M \ fðr; s; K; t; sÞ 2 R5þþ : f ðr0 ; K 0 ; t0 ; s0 ; s0 Þ ¼ c0 g is a manifold of dimension three. Then we are led to look for three relevant variables on which the BS formula depend, and this can be accomplished by finding invariants under the action of an appropriate Lie group. In particular, such action must involve an explicit symmetric function aðr; s; K; t; sÞ whose level hypersurfaces are ‘parallel’ to M. The group action is similar to the action of the additive abelian group R5 on R5þþ given by ðt1 ; . . . ; t5 Þ  ðr; K; t; s; sÞ ¼ ðexpðt1 Þr; expðt2 ÞK; expðt3 Þt; expðt4 Þs; expðt5 ÞsÞ. The relevant variables, which can be obtained by the technique sketched above are pffiffiffi y ¼ s t; a ¼ logðs=KÞ þ rt; s ¼ s. Write c ¼ f ðr; K; t; s; sÞ ¼ uðy; a; sÞ. In the manifold M we have a ¼ 0, therefore c=s ¼ uðy; 0; sÞ=s ¼ 2Nðy=2Þ  1 ¼ gðyÞ. Then an extension to a neighborhood of M must have the form uðy; a; sÞ=s ¼ bðy; aÞgðyÞ þ Rðy; aÞ, which is a decomposition of the type obtained with division theorems like Malgrange’s or Mather’s on the commutative ring of functions C 1 ðRþ  R  Rþ Þ (see for example, [21]). In what follows we give explicitly the variable change, the functions bðy; aÞ; Rðy; aÞ, and estimates for the latter, which can be seen as a remainder.

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2.2. Changes of variables Now let us define c : R5þþ ! R5þþ as cðs; t; r; s; KÞ ¼ ðst1=2 ; t; logðs=KÞ þ rt; s; kÞ. It can be seen that c : R5þþ ! cðR5þþ Þ is a homeomorphism, in fact a biholomorphism (an analytic invertible mapping with analytic inverse) and with f : cðR5þþ Þ ! R5þþ , fðy; t; a; s; kÞ ¼ ðyt1=2 ; t; t1 ða  logðs=KÞÞ; s; KÞ being its inverse: f ¼ c1 . Note that cðR5þþ Þ can be decomposed as R2þþ  G, with G open and connected. Now, by making the variable change given by f, and relabeling the variables by a permutation the BS formula becomes u ¼ uðy; a; sÞ ¼ cðfðy; t; a; s; kÞÞ ¼ s½Nða=y þ y=2Þ  ea Nða=y  y=2Þ.

(1)

Then we have reduced the number of significative variables from five to three. Here y and a stand, respectively, for st1=2 and logðs=KÞ þ rt. Now we take advantage of this reduction obtaining an asymptotic representation of the option value (formula (3) below). Proposition 1. For any ðy; a; sÞ 2 Rþþ  R  Rþþ we have Z 1 uðy; a; sÞ ¼ s½ð1 þ ea ÞNðy=2Þ  ea  þ sða=yÞ N 0 ðy=2 þ ax=yÞð1  eað1xÞ Þ dx. 0

To establish this formula, we use the mean value equality Z 1 f ðt þ hÞ ¼ f ðtÞ þ h f 0 ðt þ xhÞ dx 0

and the property NðtÞ ¼ 1  NðtÞ of the normal accumulative function, to get Z 1 Nða=y þ y=2Þ ¼ Nðy=2Þ þ ða=yÞ N 0 ðy=2 þ xa=yÞ dx, 0

Z

1

N 0 ðy=2 þ xa=yÞ dx

Nða=y  y=2Þ ¼ Nðy=2Þ þ ða=yÞ 0

Z

1

N 0 ðy=2 þ xa=yÞ dx.

¼ 1  Nðy=2Þ þ ða=yÞ 0

Inserting these relations in (1), we arrive to Z 1 ð1=sÞuðy; a; sÞ ¼ Nðy=2Þ þ ða=yÞ N 0 ðy=2 þ xa=yÞ dx  ea þ ea Nðy=2Þ Z  ða=yÞ 0

0 1

ea N 0 ðy=2 þ xa=yÞ dx

Z 1 ¼ ð1 þ ea ÞNðy=2Þ  ea þ ða=yÞ N 0 ðy=2 þ xa=yÞ 0   N 0 ðy=2 þ xa=yÞ  1  ea 0 dx N ðy=2 þ xa=yÞ and since N 0 ðy=2 þ xa=yÞ ¼ eax , N 0 ðy=2 þ xa=yÞ

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we get that a

ð1=sÞuðy; a; sÞ ¼ ð1 þ e ÞNðy=2Þ  e

Z

a

1

N 0 ðy=2 þ xa=yÞ½1  eað1xÞ  dx

þ ða=yÞ 0

which is equivalent to the formula of Proposition 1. Now let Z

1

Qðy; aÞ ¼ ða=yÞ

N 0 ðy=2 þ ax=yÞð1  eað1xÞ Þ dx,

0

then, by Proposition 1, uðy; a; 1Þ ¼ ð1=sÞuðy; a; sÞ ¼ ð1 þ ea ÞNðy=2Þ  ea þ Qðy; aÞ.

(2)

In the integral defining Q, we make l ¼ ax=y and integrate by parts getting Z a=y aþyl l¼a=y Qðy; aÞ ¼ Nðy=2 þ lÞð1  e Þjl¼0  Nðy=2 þ lÞðyeaþyl Þ dl ¼  ð1  ea ÞNðy=2Þ þ ea y

Z

0

a=y

Nðy=2 þ lÞeyl dl.

0

From the latter equality and (2), we obtain the following Theorem 1 (Asymptotic formula for Black Scholes option value). For any ðy; a; sÞ 2 Rþþ  R  Rþþ , it follows " # Z a=y uðy; a; sÞ ¼ sea 2Nðy=2Þ  1 þ y Nðy=2 þ lÞeyl dl . (3) 0

Remark. Note that this formula is global, and hence the original BS formula can be completely recovered from here. From Theorem 1 we define the approximate option value ua ðy; a; sÞ ¼ sea ½2Nðy=2Þ  1.

(4)

Now let us obtain an approximate formula for the volatility. We use j ¼ N 1 . From (4) we get pffiffiffi ya ¼ sa t ¼ 2jððua ert þ KÞ=2KÞ or pffiffiffi sa ¼ ð2= tÞjððua ert þ KÞ=2KÞ.

(5)

Remark. Note that from equality (3) that if a ¼ 0, then u ¼ ua , and sa ¼ sM . 3. Estimating the error The following estimate has as special feature to be global. Theorem 2. For any ðy; a; sÞ 2 Rþþ  R  Rþþ , it holds juðy; a; sÞ  ua ðy; a; sÞjpsjaj expðða þ jajÞ=2Þ. This estimation is easily established by bounding the integral term in (3). Then it is a consequence of the following estimation, which has a direct proof: for any y 2 Rþþ ; a 2 R it holds   Z   a=y   yl (6) Nðy=2 þ lÞe dlpjaj expðða þ jajÞ=2Þ. y   0

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Now we shall enunciate our most delicate result: to give tight bounds for the error between the implied volatility and its approximation. We have Theorem 3. There are a; b40, with b small enough, such that for all y 2 Rþþ ; jajoa, and s4ð1 þ bÞu0 it holds pffiffiffiffiffiffi jy  ya jp2 2pjajejaj exp½ð1=2Þjðð1 þ ejaj ð1 þ aÞ1 þ jajejaj Þ2 =2Þ, or pffiffiffi pffiffiffiffiffiffi js  sa jpð2= tÞ 2pjajejaj exp½ð1=2Þjðð1 þ ejaj ð1 þ aÞ1 þ jajejaj Þ2 =2Þ.

(7)

A sketch of the proof goes as follows. Give an initial condition (r0 ; s0 ; K 0 ; t0 ; s0 , c0 Þ where we can assume that s0 4c0 . This is reasonable from general financial considerations (see [6,7]). Therefore, there is an a40 such that s0 4ð1 þ aÞc0 . Also, by properties of exponentials and logarithms there is a b40 such that for any jtjob it holds ðaÞ

ð1 þ aÞ1 et o1

and

ðbÞ ð1 þ aÞ1 et þ tet o1.

(8)

Let us define ae ¼ logð1 þ aÞ: Then given the initial condition ðy0 ; 0; s0 Þ (remember that we are working about a ¼ 0) we have ya ¼ 2jðð1 þ ea u0 =sÞ=2Þ, and jNðy=2Þ  Nðya =2Þjpð1=2Þjajejaj ,

(9)

a; bg. A complete derivation of inequality (9) is somewhat long and involved. for each s0 4ð1 þ aÞu0 ; jajo minfe Details can be obtained by request. Now let us recall that j ¼ N 1 : Since this function is strictly increasing it follows that sup jðNðya =2Þ þ x½Nðy=2ÞNðya =2ÞÞpjðð1 þ ejaj ð1 þ aÞ1 þ jajejaj Þ=2ÞÞ.

(10)

0pxp1

Now, for each 0oxo1; we have that pffiffiffiffiffiffi j0 ðxÞ ¼ 1=N 0 ðjðxÞÞ ¼ 2p expðjðxÞ2 =2Þ. Then using this, (10) and the mean value inequality, we obtain jy=2  ya =2j ¼ jjðNðy=2ÞÞ  jðNðya =2ÞÞj p sup j0 ðNðya =2Þ þ x½Nðy=2Þ  Nðya =2ÞÞjNðy=2Þ  Nðya =2Þj 0pxp1

pffiffiffiffiffiffi p 2pjajejaj sup expðjðNðya =2Þ þ x½Nðy=2Þ  Nðya =2ÞÞ2 =2Þ 0pxp1

and (7) follows. Final remarks: 1. We can consider asymptotic formula (3) together with (5) as an extension of sM to a neighborhood of the zero log-moneyness manifold fðr; s; K; tÞ 2 R4þþ : s ¼ Kert g  Rþþ . pffiffiffi 2. In the approximate formula sa ¼ ð2= tÞjððc þ sÞ=2sÞ (this is formula (5), but in the simpler form that we called sM ), make a Taylor development of order one for j about 12, and since jð12Þ ¼ 0; j0 ð12Þ ¼ 1 we get the following ‘‘approximation of the approximate formula’’ pffiffiffiffiffiffiffiffiffiffi sB ¼ 2p=t c=s,

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which, according to Teichmann and Schachermayer [10] was first established by Bachelier, the father of Mathematical Finance, and which is the celebrated Brenner and Subrahmanyam formula, considered by many authors as the best ratio quality/cost approximate formula [14,15]. 3. If BS formula is used in markets—and it is—then it is unavoidable that some of its properties can be present in markets behavior. In particular, since the option value as a function of volatility alone is an injective function, given two option values c; c0 taken from the market (or obtained by some other means), we get two different volatility values s; s0 . Also, as regarding the implicit volatility as a function of the strike price, by using BS formula we arrive to different volatility values, and moreover, as it can be easily verified, this functional dependence has a graph with the form of a half-smile or half-smirk. Similar conclusions are obtained by using our approximates BS and volatility formulas. 4. Conclusions The problem of obtaining the volatility of the underlying asset from the BS formula and given market option values has become a very important issue in Finance, and hence there have been developed two main lines of research, one quite theoretical, and the other very practical. Both assume the impossibility of obtaining closed form solutions for the implied volatility. As we mentioned in the Introduction that latter problem depends on which family of functions we are going to work with. A deep understanding of this question is provided in Galois’ Theory and related fields, and for the case of algebraic equations, we highly recommend Umemura [17], for appropriate considerations on this matter. In this note we used the elementary arithmetic operations and functions, together with the Normal Accumulative Function and its inverse, that we denoted by j. This function has been studied in detail, and as far as there are available many properties, numerical tables, etc. we can consider it as an elementary function. We took advantage of some variable changes that reduce the relevant variables in the BS formula from five to three. This allows one to obtain an asymptotic representation for the BS formula in the new variables, which in turn give approximate formulas for the option value and the implied volatility. Finally, we got bounds for the errors in such approximations. Our contribution in this note is mainly theoretical, and hence we did not test our results against market data. Nevertheless, we hope that some of our formulas and representations will have some practical interest, since the approximate volatility formula is similar to other ones published in the literature. Acknowledgment We thank Dr. Antonio Garcı´ a for helpful remarks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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