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## Abstract

An option’s market price reflects the risk-neutral probability that it will end up in the money. Research has been increasing in recent years that shows how, given a set of market prices for options covering a range of strikes, an estimate of the entire risk-neutral probability distribution can be obtained. The technique is based on the fact that the second partial derivative of the option pricing function with respect to the strike price is the risk-neutral density (discounted from option expiration). This idea is generally attributed to Breeden and Litzenberger’s 1978 paper. In this article, Zimmermann shows that the connection between the second partial derivative of the option price with respect to the exercise price and risk-neutral probabilities has a much longer history, including a little-known 1974 note by Fischer Black, and going all the way back to Bachelier in 1900.

The second partial derivative of option price with respect to the strike price plays an important role in the valuation of contingent claims, in theoretical, empirical, and numerical work alike. Breeden and Litzenbeger showed that the derivative is directly related to Arrow–Debreu state prices. This note directs the attention towards two unrecognized old origins of this derivative (Black, Bronzin) and their relation to the modern interpretation, as well as to Bachelier’s statement of the relation.

In an important paper, Breeden and Litzenberger [1978] show that state claims (or Arrow–Debreu securities) can be synthetically created (approximately) through a specific combination of options across three consecutive strike prices, or so-called butterfly spreads. As a consequence, in the absence of arbitrage, state prices can be recovered from the cost of purchasing butterfly spreads. The positivity of state prices, implied by the absence of (weak) arbitrage in turn implies that option prices must be a convex function of the underlying strike prices. In the limit of a continuous state space, the second derivative of the option price with respect to the strike, evaluated in any state of the distribution, can be used directly as the corresponding Arrow–Debreu state price. This is an important result because it renders much practical relevance to the state preference model, which has often been criticized as having little empirical content.^{1} This claim was falsified by Banz and Miller [1978], who show how option prices can be used to infer state prices for determining market risk premiums in capital budgeting decisions. Moreover, the insight obtained from the second derivative plays an important role in estimating empirical state price densities from option data^{2} and in the numerical implementation of derivative pricing models.

While all of this is common knowledge and widely applied in the option pricing literature, it is not well known that the pricing kernel property of the second partial derivative was used (or at least recognized) before Breeden and Litzenberger [1978]. In fact, it appeared in a short unpublished paper by Black [1974] and in a largely forgotten book by Bronzin [1908], both of which are discussed in this article. Bachelier’s thesis [1900] also contains the derivative, however, he did not offer an explanation of it; this too is briefly discussed in this article.

## OPTIONS AND STATE PRICES: A REVIEW

This section provides a short review of the Breeden and Litzenberger [1978] model: it is assumed that the state space is characterized by a discrete distribution of (stock) prices *S*_{T} at a future date *T*, with adjacent realizations separated by Δ*S*. Call options^{3} are traded with strike prices *K* available for any *S*_{T}. A simple or symmetric butterfly spread can then be constructed, which is a combination of two short positions on strike *K*, a long position on *K* − Δ*K* and one on *K* + Δ*K*, where Δ*K* ≡ Δ*S*_{T} denotes the increment between adjacent strike prices. This strategy delivers a payoff of Δ*K* if the stock price is *S*_{T} = *K*, and zero otherwise. If an investor buys 1/Δ*K* units of the strategy, she would get an Arrow–Debreu state security, i.e., an asset with a payoff of 1 for a stock price *S* = *K*, and zero otherwise. The principle of no arbitrage implies that the amount invested in the strategy must be equal to the current price of the state security (state price).

Denote the price of a European option at *t* with maturity at *T* and strike *K* by *C*(*S _{t}*,

*K*,

*T*) ≡

*C*(

*K*). The value at

*t*of a symmetric butterfly spread is denoted by ψ(

*S*,

_{t}*K*,

*T*; Δ

*K*), which is a function of the increments of the stock price distribution, and hence of the size of Δ

*K*. Thus, a portfolio with 1/Δ

*K*units of this strategy costs

where *p**(*S _{t}*,

*K*,

*T*; Δ

*K*) denotes the state price at

*t*of a claim paying 1 unit on date

*T*upon

*S*=

_{T}*K*, and zero otherwise. In the limit of a continuous probability space and a continuum of strike prices, its payoff on date

*T*can be characterized by

The limiting state price for a continuous state space can be derived as follows: First, notice that from (1), the state price in relation to Δ*K* can be stated as

which converges to

4Absence of (weak) arbitrage requires positive state prices *p**, which implies a convex relationship between *K* and *C*(*K*). The limiting state price is therefore

Hence, the capital invested in the 1/(*dK*) spreads delivers a unit payoff upon *S _{T}* =

*K*with vanishing probability and costs an infinitesimal amount of money, namely,

*p** =

*C*(

_{KK}*K*) ·

*dK*. This is the continuous state-space analogue to the “classical” Arrow–Debreu securities.

In a complete market, the arbitrage-free price of a given payoff at *T*, which depends on *S _{T}*, denoted by

*g*(

*S*

_{T}), can be characterized by

where *f**(*S _{T}*) is the unique, positive state price density for a continuum of states. Notice that the price density is related to the risk-neutral probability density (RND)

*q*(

*S*

_{T}), which is more commonly used in the modern option pricing theory by

where *r* is the riskless rate. However, for reviewing the models in this article, it is more convenient to work with the price density *f**(*S*_{T}).

Obviously, *f**(*S _{T}*)

*dS*=

_{T}*p**(

*S*,

_{t}*K*,

*T*;

*dS*) is the state price and can be substituted using (5) and recognizing

_{T}*dS*

_{T}=

*dK*. Therefore,

Hence, the second strike-derivative *C _{KK}*(

*K*) is a state price density:

An immediate consequence is that the price of any *S*-contingent claim can be priced using *C _{KK}* and integrating over

*K*instead of

*S*

_{T}. This fact is not explicitly mentioned in the Breeden–Litzenberger paper, but it is essential.

## BLACK AND THE DIRAC DELTA FUNCTION

In a largely unrecognized short unpublished manuscript,^{4} ,Black [1974] uses the second strike-derivative as pricing kernel for general derivatives. However, his derivation is, at first glance, not based on butterfly spreads or state prices at all. Indeed, one of the few references to this paper can be found in footnote 7 of Breeden and Litzenberger [1978], where the authors state: “The result (…) was noted by Black [1974] for when *c*(*X*, *T*) is given by the Black–Scholes option-pricing equation. However, the result was noted as a mathematical curiosity rather than being derived as a general proposition, as in this section.”

Although the derivation is rather obscure and not very straightforward, the result is by no means stated as a mathematical “curiosity,” but rather it is related to Dirac’s delta function, which is an extremely general and elegant characterization of the valuation problem.^{5} But unfortunately, and not unusual in Black’s papers, the relationship between the Dirac function and the option-based (or state pricing–based) pricing kernel is far from obvious—at least for the mathematically unsophisticated reader—and it is not explained at all. Moreover, unlike the Breeden–Litzenberger paper, it does not offer an economic explanation of the second derivative in terms of butterfly spreads and state prices. However, that relationship can be established easily.

Instead of investing 1/Δ*K* units in the butterfly spread centered at strike *K* as in (1), we can do this with 1/(Δ*K*)^{2} units. The limiting payoff for Δ*K* → 0 is

which seems to be only slightly different from (3), but the mathematical implication is substantially different: the function (10) has the key property of the Dirac delta function δ(*S _{T}*-

*K*), i.e., has all its (infinite) mass at

*S*

_{T}=

*K*, and is zero otherwise. Therefore,

Discounting the Dirac payoff by using the state pricing function implies the current value of the strategy

12 13where (13) represents the *sifting property* of the Dirac delta function. Hence, the capital invested in the 1/(*dK*)^{2} spreads delivers an infinite payoff upon *S _{T}* =

*K*with vanishing probability and costs

*f**(

*K*). However, notice that the current value of the 1/(Δ

*K*)

^{2}units invested in butterfly spreads can be derived from (1)

with limiting value *C _{KK}*(

*K*) as derived in (4). Therefore, (13) implies

which is the earlier result that the second derivative is a state price density. This demonstrates the consistency of the state pricing and Dirac delta approach.

Of course, the Dirac delta approach can be directly applied to the general valuation function of a call:

16where (*S _{T}* –

*K*)

^{+}restricts the payoff to non-zero values. The second strike-derivative is

The option payoff can be expressed by the Heavyside function *H*(·), which takes the constant 1 if the expression inside the brackets is positive, and zero otherwise:

The first derivative with respect to *K* is

The first derivative of the Heavyside function (and thus the second derivative of the call payoff) is the Dirac delta function^{6}

which is infinity (or, if interpreted as distribution, has infinite mass) at *S _{T}* =

*K*, and zero otherwise. Therefore (17) can be stated as

where the last expression, as mentioned earlier, is the sifting property of the Dirac delta function.

While we have used the Dirac delta function as an analytical alternative to a continuum of Arrow–Debreu prices in characterizing the valuation function *f**(·), Black uses the Dirac function in the following way:

**Black’s pricing equation for general contingent claims.** Black denotes the Black–Scholes value of a call option at *t* with maturity *t** by *w*(*x*, *t*, *c*, *t**), where *x* is the underlying asset price, and *c* is the strike. The value of a *general* claim on the same underlying is denoted by *y*(*x*, *t*) and has the terminal payoff at maturity

The key insight of Black is that the value of this general claim can be expressed as

23where *w*_{33} denotes the second partial derivative of a call option with respect to *c*, the strike price. The key result is that the value of a general derivative claim on *x* can be determined by extracting price information from a cross-section of call options on *x* and integrating over the strike *c* (respectively *u* in the preceding formula).

He shows (or rather provides an informal reasoning) that this solution satisfies a) the standard dynamic hedging differential equation for derivatives (assuming constant volatility of the underlying and constant interest rate) and b) its boundary condition (22). About a), he notes that “any derivative of a solution to the differential equation with respect to one of its parameters is also a solution to the differential equation.” About b), he notices that the second derivative evaluated in *t**, *w*_{33}(*x*, *t**, *u*, *t**), “is the Dirac delta function δ(*x* - *u*)” which implies

by the sifting property. The current value of the claim in (23) results directly from replacing δ(*x* – *u*) = *w*_{33}(*x*, *t**, *u*, *t**) by the second derivative evaluated at *t*, *w*_{33}(*x*, *t*, *u*, *t**). He also mentions that an alternative way to show that (23) satisfies the boundary condition is by integrating the r.h.s. of (23) by parts.

Unfortunately, Black remains silent about the Arrow–Debreu interpretation of his Dirac delta approach. This might also explain why the approach seems like a mathematical curiosity. Perhaps the interpretation was obvious to him.

In the context of static option replication, the Dirac delta function (DDF) has been used successfully by Carr and Madan [1998] for expanding and decomposing option payoffs. More generally, the DDF is an essential ingredient in the implementation of Fourier transforms in the pricing of complex options. Carr and Madan [1999], and more recently, Kwok, Leung, and Wong [2012] provide overviews. None of these papers, however, credit Black for having presented the first formula using the DDF for pricing complex options.

## BRONZIN: DIFFERENTIATION OF INTEGRALS

In his textbook treatment, Ingersoll [1987, p. 145] shows that the interpretation of the second strike-derivative (9) can be derived directly, without reference to butterfly spreads, by applying Leibniz’s rule for differentiation of integrals. Denote the continuous state price density by *f**(*S*_{T}); the price of a call option is

Leibniz’s rule implies

26 27which is the same result as in (9). Interestingly, this derivation, paired with a very accurate interpretation of the expression, can be found in Bronzin [1908], a largely ignored option pricing monograph published in German just a few years after Bachelier’s pathbreaking work. Bronzin’s contribution is only briefly addressed here because it has been analyzed in detail elsewhere.^{7}

This characterization emphasizes that Bronzin’s “probability” density is in fact a “pricing” density that is directly related to priced state securities. Moreover, after deriving the put-call parity, Bronzin shows that the equivalent relations can be derived using put option prices.

**Bronzin’s “remarkable relation.”** In section 8 of the first chapter in Part II [pp. 50ff in the original German version], Bronzin analyzes the relationship between option prices and a valuation function. The premium of a call option with strike *M* is denoted by *P*_{1} (the put option premium by *P*_{2}). The time to maturity plays no role in his analysis since, in these days, the premiums were paid at maturity. The value of the underlying security at maturity is denoted by *x*. Notice that *M* as well as *x* are defined as deviations from the forward price. The valuation problem for a call option is stated as^{8}

where *f*(*x*) is probability function of *x*. The equation is derived from the principle of “fairness” (in Bronzin’s wording: *Rechtmässigkeit*) saying that the mathematical expectation (*Hoffnungswert*) of the net cash flows, including the option price, must be zero for the buyer and seller of the contract [p. 42]. The probability function is only specified in the second chapter, and remains unspecified here. He applies the rules of differentiation of integrals to get (Equation 16)

which^{9} he calls a remarkable relation, from which he derives two insights (Equations 17 and 19):

(*C* is a constant that can be determined easily from the payoff function). The first expression is our well-known second derivative, while the second expression shows that “the determination of *P*_{1} as a function of *M* can be accomplished in a fashion quite different compared to the direct evaluation of its integral, which in turn may be of great advantage, depending upon which form function *f*(*x*) takes.” [Translated from p. 51].

The interesting insight from Bronzin’s analysis is that integration over *x* (the underlying price) can be replaced by integration over *M* (the strike prices) for pricing calls. This is an analogous result to Black’s key Equation (23). In his subsequent derivation of option-pricing formulas for alternative distributional assumptions of the underlying price,^{10} he makes extensive use of this analytical simplification.

## BACHELIER

It is interesting to note that the derivative can also be found in Bachelier’s pathbreaking study. In Bachelier’s thesis [1900, p. 51], after deriving the general structure of the option formula, he writes the derivative down, but without interpretation or further use in that work.

For the subsequent discussion, remember that in Paris and other European exchanges where options were traded, the price for the option was paid at maturity (as in Bronzin’s model). Therefore, it can be simply subtracted from the option payoff at maturity without discounting.

Unfortunately, Bachelier does not pursue this important insight further; he offers no interpretation or potential applications.

**Bachelier.** On p. 50 of the original French text, Bachelier derives the option price *h* of a call with strike *m* on an underlying price *x* using a zero expected profit condition (in Bachelier’s wording: *principe de l’espérance totale*) applied to the expected cash flow of the option at maturity, including the option price. The expectation is calculated for three price ranges of the probability density *p*, namely (-∞, *m*), (*m*, *m* + *h*), and (*m* + *h*, ∞)

which can be simplified to

32This corresponds (slightly rearranged) to Bachelier’s expressions found on p. 50 of his thesis. One page later, he adds: “On differentiating, the differential equation obtained for the option price [écarts de primes] is

33*p _{m}* being an expression for the probability in which

*x*has been replaced by

*m*” [Translated from p. 51].

## DISCUSSION AND SUMMARY

The second strike-derivative of option prices plays an important role in the pricing of derivative contracts. Breeden and Litzenberger [1978] have shown that the derivative is directly related to Arrow–Debreu state prices. At least three sources prior to their paper can be identified that contain the derivative. Most explicitly, Fischer Black recognizes its use as pricing kernel for general contingent claims, but the derivation is somewhat obscure. He merely shows that the specification satisfies the differential equation and, by relying on the Dirac delta function, the boundary condition of general claims. The Dirac delta function has a close analytical relation to Arrow–Debreu security payoffs in continuous time, as shown in this paper. However, this point is not explicitly addressed in Black’s paper. In his 1900 thesis, Vinzenz Bronzin also addresses the issue; his major observation is that the first and second strike-derivatives make it possible to express option prices as integrals over a continuum of strike prices (instead of the underlying asset price), which is shown to simplify the derivation of option prices for alternative distributional assumptions substantially. Finally, Bachelier’s thesis also contains the derivative and its relationship with the probability (pricing) density; however, it does not offer further usage or explanation. It would be interesting to search for additional historical sources of this analytical feature.

## ENDNOTES

I owe special thanks to the editor, Stephen Figlewksi, for very useful comments on an earlier draft of this paper. William Margrabe originally brought my attention to Black’s paper, sometime in the 1980s. A discussion with Jörg Urban and comments by Yvonne Seiler were also very helpful. The research on Bronzin’s early contribution to option pricing is based on joint work with Wolfgang Hafner.

↵

^{1}See, for example, Michel C. Jensen in his classic review article: “While the state preference approach is perhaps more general than the mean-variance approach and provides an elegant framework for investigating theoretical issues, it is unfortunately difficult to give it empirical content.” Jensen [1972], p. 357.↵

^{2}Early papers include Jackwerth and Rubinstein [1996] and Aït-Sahalia and Lo [1998]; see Figlewski [2018] for an authoritative survey of recent research.↵

^{3}The focus is on call options, but all would apply to put options as well.↵

^{4}Neither the Merton and Scholes [1995] obituary in the*Journal of Finance*, which includes an Appendix (p. 1361) with a “complete list of Fischer Black’s publications” (which in fact also includes his unpublished papers), nor the biography by Mehrling [2011], nor the bibliographies in the collection of papers edited by Lehmann [2004] contain a reference to the paper. Apart from Breeden and Litzenberger [1978], a reference to a 1975 version of Black’s paper can be found in Margrabe [1978].↵

^{5}Kosowski and Neftci [2008] give a short introduction to the Dirac delta function and some applications.↵

^{6}Notice that δ(*x*) = δ(−*x*).↵

^{7}Vinzenz Bronzin was a professor of mathematics at the Accademia di Commercio e Nautica in Trieste in the first half of the 20th century. A reprint of the original text along with an English translation can be found in Hafner and Zimmermann [2009]; an appreciation of Bronzin’s contribution is Zimmermann and Hafner [2007].↵

^{8}In contrast to Bronzin’s original notation, the upper bound of his integral ω, which can be finite or infinite depending on the shape of the probability density, is replaced by ∞.↵

^{9}Unlike common notation, Bronzin’s cumulative density*F*(*M*) runs from*M*to infinity.↵

^{10}This is contained in the second chapter in Part II of his book.

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