Negative in Financial Modeling

Mark Burgin Department of Mathematics, UCLA Gunter Meissner Shidler College of Business, University of Hawaii, e-mail: [email protected]

Abstract After this, negative probabilities slowly but steadily have become more We first define and derive general properties of negative probabilities. We popular techniques in . Maurice Bartlett (1945) worked out the then show how negative probabilities can be applied to modeling financial mathematical and logical consistency of negative probabilities. However, he options such as Caps and Floors. In trading practice, these options are typi- did not establish a rigorous foundation for negative utilization. cally valued in a Black–Scholes–Merton framework, assuming a lognormal Andrei Khrennikov published numerous papers going back to 1982 related distribution for the underlying interest rate. However, in some cases, such to negative probabilities, some endorsed by Andrey Kolmogorov. In a 2007 as the 2008/2009 financial crisis, interest rates can get negative. Then the interview Khrennikov discusses the p-adic model and finds that negative lognormal distribution is inapplicable. We show how negative probabilities probabilities ‘appear quite naturally’ in this framework. In 2009 Khrennikov associated with negative interest rates can be applied to value interest rate provides the first rigorous mathematical theory of negative probabilities in options. A model in VBA, which prices Caps and Floors with negative prob- the setting of p-adic analysis. abilities, is available upon request. A follow up paper will address bigger Negative probabilities are also used in mathematical finance. The con- than unity probabilities in financial modeling. cept of risk-neutral or pseudo probabilities is a popular concept and has been numerously applied, for example, in credit modeling by Jarrow and Keywords Turnbull (1995), and Duffie and Singleton (1999). Espen Haug (2004) is the negative probabilities, negative interest rates, lognormal distribution, Caps, first author to explicitly support negative probabilities in financial mod- Floors eling. He shows that negative probabilities can naturally occur, for exam- ple in the binomial CRR model, which is a discrete representation of the JEL Classification: C10 Black–Scholes–Merton model discussed below. Haug argues that rather than disregarding the whole model or transforming the negative probabilities to positive ones, negative probabilities can be a valuable mathematical tool in 1. Introduction financial models to add flexibility. The classical probability theory is applied in most sciences and in many of Negative probabilities are able to model a random process if that proc- the humanities. In particular, it is successfully used in physics and finance. ess might become negative in rare events. In this paper we demonstrate this However, physicists found that they need a more general approach than the using negative interest rates. More generally, negative probabilities might classical probability theory. The first was (1932), who intro- also be applied to any random process where limited liability doesn’t elimi- duced a function, which looked like a conventional probability distribution nate losses in all states. For example, suppose an enterpreneur owns compa- and has later been better known as the Wigner quasi-probability distribu- nies x and y. If company x defaults and has net liabilities, assets of company tion because in contrast to conventional probability distributions, it took y might have to be liquidated to cover the liabilities of x. Hence we can argue negative values, which could not be eliminated or made nonnegative. The that the price of x can de facto become negative. In this paper we will con- importance of Wigner’s discovery for foundational problems was not recog- centrate on negative interest rates and show how they can be modeled with nized until much later. Another outstanding physicist, Nobel Laureate Dirac negative probabilities. (1942) not only supported Wigner’s approach but also introduced the physi- The remaining paper is organized as follows. In Section 2, we resolve the cal concept of negative energy. He wrote: “Negative energies and probabilities mathematical issue of the negative probability problem. We build a math- should not be considered as nonsense. They are well-defined concepts mathematically, ematical theory of extended probability as a probability function, which

like a negative of money” (referring to debt). is defined for real numbers and can take both positive and negative values.

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Thus, extended probabilities include negative probabilities. Different prop- Elements from Ω+ that belong to F+ are called elementary positive random erties of extended probabilities are found. In Section 3, we give examples of events or simply, elementary positive random events. negative nominal interest rates in financial practice and show problems of If w ∈ Ω+, then –w is called the antievent of w. current financial modeling of negative interest rates. In Section 4, we build a Elements from Ω– that belong to F – are called elementary negative random mathematical model for Caps and Floors integrating extended probabilities events or elementary random antievents. into the pricing model to allow for negative interest rates. Conclusions are For any set X ⊆ Ω+, we define given in Section 5. A follow up paper will specify >1 probabilities and apply X + = X ∩ Ω+, them to financial options. X – = X ∩ Ω –, –X = {–w; w ∈ X} 2. Mathematical theory of extended probability and In 2009, Andrei Khrennikov derived a rigorous mathematical theory of F – = {–A; A ∈ F +} negative probabilities in his textbook in the framework of p-adic analysis. Our derivation of negative probabilities is in the conventional setting of real If A ∈ F + , then –A is called the antievent of A. – numbers. Elements from F are called negative random events or random antievents. Extended probabilities generalize the standard definition of a probabil- Definition 1. The function P from F to the set R of real numbers is called a ity function. At first, we define extended probabilities in an axiomatic way probability function, if it satisfies the following axioms: and then develop application of extended probabilities to finance. α Ω → Ω To define extended probability, EP, we need some concepts and construc- EP 1 (Order structure). There is a graded involution : , i.e., a α2 Ω tions, which are described below. Some of them are well-known, such as, for mapping such that is an identity mapping on with the following prop- α Ω α Ω+ ⊇ Ω– ∈ Ω+ example, set algebra, while others, such as, for example, random antievents, erties: (w) = –w for any element w from , ( ) , and if w , then α ∉ Ω+ are new. (w) . We remind that if X is a set, then |X| is the number of elements in (cardi- EP 2 (Algebraic structure). F + ≡ {X ∈ F; X ⊆ Ω+} is a set algebra that has Ω+ as nality of) X (Kuratowski and Mostowski, 1967). If A ⊆ X, then the complement a member. of A in X is defined as C A = X\A. + X EP 3 (Normalization). P(Ω ) = 1. A system B of sets is called a set ring (Kolmogorov and Fomin, 1989) if it EP 4 (Composition) F ≡{X; X+⊆ F+ & X –⊆ F – & X + ∩ –X – ≡ ∅ & X – ∩ –X + ≡ ∅}. satisfies conditions (R1) and (R2): (R1) A, B ∈ B implies A ∩ B ∈ B. EP 5 (Finite additivity) ∪ (R2) A, B ∈ B implies A Δ B ∈ B where A Δ B = (A\B) ∪ (B\A). P(A B) = P(A) + P(B) ∈ For any set ring B, we have ∅ ∈ B and A, B ∈ B implies A ∪ B, A\B ∈ B. for all sets A, B Fsuch that ∩ ≡ Indeed, if A ∈ B, then by R1, A\A = ∅ ∈ B. If A, B ∈ B, then A\B = ((A\B) ∪ A B Ø (B\A)) ∩ A ∈ B. If A, B ∈ B and A ∩ B = Ø, then A Δ B = A ∪ B ∈ B. It implies that ∈ Ω ∈ ∈ Ω ∈ EP 6 (Annihilation). {vi , w, –w ; vi , w & i I} = {vi; vi & i I} for any A ∪ B = (A\B) ∪ (B\A) ∪ (A ∩ B) ∈ B. Thus, a system B of sets is a set ring if and element w from Ω. only if it is closed with respect union, intersection and set difference. Axiom EP6 shows that if w and –w are taken (come) into one set, they annihilate one another. Having this in mind, we use two equality Example 2.1. The set CI of all closed intervals [a, b] in the real line R is a symbols = and ≡. The second symbol means equality of elements of sets. set ring. The second symbol also means equality of sets, when two sets are equal Example 2.2. The set OI of all open intervals (a, b) in the real line R is a set when they have exactly the same elements (Kuratowski and Mostowski, ring. 1967). The equality symbol = is used to denote equality of two sets with A set ring B with a unit element, i.e., an element E from B such that annihilation, for example, {w, –w} = Ø. Note that for sets, equality ≡ for any A from B , we have A ∩ E = A, is called a set algebra (Kolmogorov and implies equality =. Fomin, 1989). For equality of numbers, we, as it is customary, use symbol =. Example 2.3. The set BCI of all closed subintervals of the interval [a, b] is a EP 7. (Adequacy) A = B implies P(A) = P(B) for all sets A, B ∈ F. set algebra. For instance, P({w, –w}) = P(Ø) = 0. ∈ + Example 2.4. The set BOI of all open subintervals of the interval [a, b] is a EP 8. (Non-negativity) P(A) ≥ 0, for all A F . set algebra. It is known that for any set algebra A, the empty set Ø belongs to A and Ω Ω A set algebra B closed with respect to complement is called a set field. for any set field B in , the set belongs to A (Kolmogorov and Fomin, 1989). Let us consider a set Ω, which consists of two irreducible parts (subsets) Definition 2. The triad (Ω, F, P) is called an extended probability space. Ω+ and Ω–, i.e., neither of these parts is equal to its proper subset, a set F of Definition 3. If A ∈ F, then the number P(A) is called the extended subsets of Ω, and a function P from F to the set R of real numbers. probability of the event A. Elements from F, i.e., subsets of Ω that belong to F, are called random Let us obtain some properties of the introduced constructions. events. ^ Elements from F+ = {X ∈ F; X ⊆ Ω+} are called positive random events. Lemma 1. α(Ω+) ≡ –Ω+ ≡ Ω– and α(Ω–) ≡ –Ω– ≡ Ω+.

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660-65_Meissner_TP_March_2012_Fin610-65_Meissner_TP_March_2012_Fin61 6161 33/23/12/23/12 2:28:032:28:03 PMPM Proof. By Axiom EP1, α(Ω+) ≡ –Ω+ and α(Ω+) ⊇ Ω–. As Ω ≡ Ω+ ∪ Ω– , Axiom (A+ ∩ B+) ∪ (A+ ∩ B–) ∪ (A– ∩ B+) ∪ (A– ∩ B–) ≡ EP1 also implies α(Ω+) ⊆ Ω–. Thus, we have α(Ω+) ≡ Ω–. The first part is proved. (A+ ∩ B+) + (A– ∩ B–) The second part is proved in a similar way. because (A+ ∩ B–) ≡ Ø and (A– ∩ B+) ≡ Ø. Ω+ ∈ Ω– ∈ Thus, if = {wi ; i I}, then = {–wi ; i I }. In a similar way, we prove the following results. As α is an involution of the whole space, we have the following result. Lemma 7. A\B ≡ (A+\B+) + (A–\B–) for any subsets A and B of Ω. Proposition 1. α is a one-to-one mapping and |Ω+| = |Ω–|. Lemma 8. X ≡ X+ + X– = X + ∪ X– for any set X from F. ∈ Ω+ ∈ Ω– Corollary 1. (Domain symmetry) w if and only if –w . Lemma 9. A + B ≡ (A+ + B+) + (A–+ B–) for any sets X and Y from F. Corollary 2. (Element symmetry) – (–w ) = w for any element w from Ω. Axioms EP6 and EP7 imply the following result. Corollary 3. (Event symmetry) – (–X ) ≡ X for any event X from Ω. Proposition 6. P(X + Y) = P(X ∪Y) for any two events X and Y from Ω. Lemma 2. α(w) ≠ w for any element w from Ω. Lemma 10. P(∅) = 0. Indeed, this is true because if w ∈ Ω+, then by Axiom EP1, α(w) ∉ Ω+ and Properties of the structure F + are inherited by the structure F. α ∈ Ω– α thus, (w) ≠ w. If w , then we may assume that (w) = w. However, in this Theorem 1. (Algebra symmetry) If F + is a set algebra (or set field), then F is a α Ω+ α case, (v) = w for some element v from because by Axiom EP1, is a projec- set field (or set algebra) with respect to operations + and ∩. tion of Ω+ onto Ω–. Consequently, we have – α(α(v)) = α(w) = w Proof. At first, we prove that F is a set algebra (or set field). + However, α is an involution, and we have α(α(v)) = v. This results in the Let us assume that F is a set algebra and take two negative random – – equality events H and K from F . By the definition of F , H = –A and K = –B for some + v = w positive random events A and B from F . Then we have ∩ ∩ ∩ Consequently, we have α(v) = v. This contradicts Axiom EP1 because H K = (–A) (–B) = –( A B) + ∩ ∈ + ∩ ∈ – v ∈ Ω+. Thus, lemma is proved by contradiction. As F is a set algebra, A B F . Thus, H K F . In a similar way, we have Ω+ ∩ Ω– ≡ Proposition 2. Ø. H ∪ K = (– A) ∪ (–B) = –( A ∪ B) Proposition 3. F+ ⊆ F , F – ⊆ F and F ⊆ F+ ∪ F –. As F + is a set algebra, A ∪ B ∈ F+. Thus, H ∪ K ∈ F–. Corollary 1 implies the following result. By the same token, we have H\K ∈ F–. Besides, if F+ has a unit element E, then –E is a unit element in F–. Proposition 4. X ⊆ Ω+ if and only if –X ⊆ Ω–. Thus, F – is a set algebra. – ≡ ∈ ⊆ Ω– ∩ Ω– Proposition 5. F {X F; X } = F . Now let us assume that F + is a set field and H ∈ F –. Then by the definition + ∩ – ≡ – + Ω+ Corollary 4. F F Ø. of F , H = –A for a positive random event A from F . It means that CΩ+A = \A Axioms EP6 implies the following result. ∈ F+. At the same time, C H = Ω–\H = (–Ω+ )\(–A) = –(Ω+\A) = –C A Lemma 3. X ∪ –X = Ø for any subset X of Ω. Ω– Ω+ As C A belongs to F+, the complement C H of H belongs to F–. Indeed, for any w from the set X, there is –w in the set X, which annihi- Ω+ Ω– Consequently, F – is a set field. lates w. Let us once more assume that F+ is a set algebra and take two random Let us define the union with annihilation of two subsets X and Y of Ω by events A and B from F. Then by Theorem 1, F – is a set algebra. By Lemma 8, the following formula: A ≡ A+ + A– and B ≡ B+ + B–. By Axiom EP4, A+, B+ ∈ F +, A–, B– ∈ F –, while by X + Y ≡ (X ∪ Y)\[(X ∩ –Y) ∪ (–X ∩ Y)] Proposition 2, A + ∩ A – ≡ ∅, B + ∩ B – ≡ ∅, A ≡ A+ ∪ A–, and B ≡ B+ ∪ B–. Here the set-theoretical operation\represents annihilation, while sets By Lemma 6, A ∩ B ≡ (A+ ∩ B+) + (A– ∩ B–). Thus, (A ∩ B)+ ≡ A+ ∩ B+ and X ∩ –Y and X ∩ –Y depict annihilating entities. (A ∩ B)– ≡ A– ∩ B–. As F + is a set algebra, (A ∩ B)+ ≡ A+ ∩ B+∈ F +. As it is proved Some properties of the new set operation + are the same as properties of that F – is a set algebra, (A ∩ B)– ≡ A– ∩ B–∈ F –. Consequently, A ∩ B ∈ F. the union ∪, while other properties are different. For instance, there is no By Lemma 7, A\B ≡ (A+\B+) + (A–\B–). Thus, (A\B)+ ≡ A+\B+ and (A\B)– ≡ A–\B–. distributivity between operations + and ∩. As F + is a set algebra, (A\B)+ ≡ A+\B+∈ F+. As it is proved that F – is a set algebra, Lemma 4. a) X + X ≡ X for any subset X of Ω; (A\B)– ≡ A–\B–∈ F –. Consequently, A\B ∈ F. b) X + Y ≡ X + Y for any subsets X and Y of Ω; By Lemma 9, A + B ≡ (A+ + B+) + (A– + B–). Thus, (A + B)+ ≡ A+ + B+ and (A + B)– ≡ c) X + Ø ≡ X for any subset X of Ω; A– + B–. As F + is a set algebra, (A + B)+ ≡ A+ + B+ ≡ A+ ∪ B+∈ F +. As it is proved that d) X + (Y + Z) ≡ (X + Y) + Z for any subsets X , Y and Z of Ω; F – is a set algebra, (A + B)– ≡ A– + B– ≡ A– ∪ B– ∈ F –. Consequently, A + B ∈ F. e) X + Y ≡ X ∪ Y for any subsets X and Y of Ω+ (of Ω–); Besides, if F + has a unit element E, then –E is a unit element in F – and E ∪ –E is a unit element in F. Lemma 5. a) Z ∩ (X + Y ) ≠ Z ∩ X + Z ∩ Y; Thus, F is a set algebra. b) X + (Y ∩ Z) ≠ (X ∩ Y) + (X ∩ Z). Now let us assume that F + is a set field and A ∈ F. Then as it is Lemma 6. A ∩ B ≡ (A+ ∩ B+) + (A– ∩ B–) for any subsets A and B of Ω. demonstrated above, F – is a set field. By Lemma 8, A ≡ A+ + A–. By Indeed, as A ≡ A+ ∪ A– and B ≡ B+ ∪ B–, we have Proposition 2, Ω+ ∩ Ω– = Ø, we have ∩ ≡ + ∪ – ∩ + ∩ – ≡ A B (A A ) (B B ) CΩA = CΩ+A + CΩ–A

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+ + Then CΩ+A belongs to F as F is a set field and as it is proved in Theorem 1, future if desired. The reason for this unusual behavior is that investors wanted – – – CΩ–A belongs to F . Consequently, CΩA belongs to F and F is a set field. to invest in the safe haven currency US dollar even if they had to pay for it. Theorem is proved. 3.2. Modeling interest rates In finance, interest rates are typically modeled with a geometric Brownian 3. Negative interest rates and the problem of their motion, modeling dr √ = μ dt + σ ε dt (2) Negative probabilities can help to model interest rates and interest rate r r r derivatives. To show this, let us start with the equation dr: change in the interest rate r μ : drift rate, which is the expected growth rate of r, assumed non-sto- Real interest rate = Nominal interest rate – Inflation rate (1) r chastic and constant where the nominal interest rate is the de facto rate, which is received by dt: infinitely short time period the lender and paid by the borrower in a financial contract. For example, σ r: expected volatility of rate r, assumed non-stochastic and constant the nominal interest rate is the rate, which the lender receives on a saving ε: random drawing from a standardized normal distribution. All draw- account or the coupon of a bond. From equation (1) we see that a real inter- ings at times t are iid. est rate can easily be negative and in reality often is. For example, if the In equation (2), the first term on the right hand side gives the average nominal interest rate on a savings account is 1% and the inflation rate is 3%, expected growth rate of r. The second term on the ride side adds stochastic- naturally, the real interest rate, i.e. the inflation adjusted rate of return for ity to the process via ε, i.e. provides the distribution around the average the lender is –2%. growth rate. Importantly, from equation (1) we can observe that the relative change dr/r is normally distributed, since ε is normally distributed. If the 3. 1. Examples of negative nominal interest rates relative change of a variable is normally distributed, it follows that the vari- However, in rare cases, also the nominal interest rate can be negative. able itself is lognormally distributed with a pdf   An example of this would be that the lender gives money to a bank, and 2 1 − 1 ln(x)−μ additionally pays the bank an interest rate. This happened in the 1970s in √ e 2 σ (3) xσ 2π Switzerland. The lender had several motives In the equation (3), μ and σ are the mean and standard deviation of ln(x) a) Switzerland is considered an extremely safe country to place capital respectively. Figure 1 shows a lognormal distribution. b) Investors were speculating on an increase of the Swiss franc The logarithm of a negative number is not defined, hence with the pdf c) Some investors avoided paying taxes in their home country equation (3), negative values of interest rates cannot be modeled. However, Another example of negative nominal interest rates occurred in Japan as discussed above, negative interest rates do exist in the real financial in 2003. Banks lent Japanese Yen and were willing to receive a lower Yen world. Here negative probabilities come into play. We will explain this with amount back several days later. This means de facto a negative nominal options on interest rates. interest rate. The reason for this unusual practice was that banks were eager to reduce their exposure to Japanese Yen, since confidence in the Japanese economy was low and the Yen was assumed to devalue. 4. How negative probabilities allow more adequate Similarly, in the US, from August to November 2003, ‘repos’, i.e. repurchase interest rate modeling agreements traded at negative interest rates. A repo is just a collateralized loan, i.e. the borrower of money gives collateral, for example a Treasury bond, to the 4.1. Modeling interest rate options lender for the time of the loan. When the loan is paid back, the lender returns The two main types of options are call options and put options. A call the collateral. However, in 2003 in the US, settlement problems when return- option is the right but not the obligation to pay a strike price and receive an ing the collateral occurred. Hence the borrower was only willing to take the risk of not having the collateral returned if he could pay back a lower amount Figure 1: Lognormal distribution with μ = 0 and r =1. than originally borrowed. This constituted a negative nominal interest rate. A further example of the market expecting the possibility of negative nominal interest rates occurred in the worldwide 2008/2009 financial crisis, when strikes on options on Eurodollars Futures contracts were quoted above 100. A Eurodollar is a dollar invested at commercial banks outside the US. A Eurodollar futures price reflects the anticipated future interest rate. The rate is calculated by subtracting the Futures price from 100. For example, if the 3 month March Eurodollar future price is 98.5, the expected interest rate from March to June is 100 – 98.5 = 1.5, which is quoted in per cent, so 1.5%. In March 2009, option strikes on Eurodollar future contracts were quoted above 100 on

the CME, Chicago Mercantile Exchange. This means that market participants ^ could buy the right to pay a negative nominal interest on US dollars in the

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660-65_Meissner_TP_March_2012_Fin630-65_Meissner_TP_March_2012_Fin63 6363 33/23/12/23/12 2:28:062:28:06 PMPM underlying asset. A put option is the right but not the obligation to receive a 2) We can add a location paramet  er to the lognormal distribution. Hence 2 2 strike price and deliver an underlying asset. − 1 ln(x)−μ − 1 ln(x−α)−μ equation (3) √1 e 2 σ becomes 1 √ e 2 σ , In the interest rate market, the main types of options are Caps and Floors, xσ 2π (x−α)σ 2π Bond options, and Swap options. A Cap is a series of Caplets and a Floor is a where α is the location parameter. For α > 0, the lognormal distribu- series of Floorlets. We will discuss Caplets and Floorlets in this paper. tion is shifted to the left. As a result, the probability distribution Caplets and Floorlets are typically valued in the Black–Scholes–Merton allows negative interest rates with a positive probability while main- framework: taining the skew and kurtosis of the lognormal distribution. The problem is that shifting the entire probability distribution means Caplet = m PA e–rl tl {r N(d ) – r N(d )} (4) ts,tl f 1 k 2 also shifting the initial current value of the interest rate r. Hence the Floorlet = m PA e–rl tl {r N(–d ) – r N(–d )} (5) modeled starting value of r will be different from the current mar- ts,tl k 2 f 1 ket interest rate r. This is inconsistent, especially for American style rf 1 2 ln( )+ σ tx √ options, which determine early exercise opportunities. rk √ 2 where d1 = and d = d – σ tx σ tx 2 1 3) A consistent way to model options on negative interest rates is to Caplet : optioni on an interest rate from time t to time t , t > t , i.e. the ts,tl s l l s apply negative probabilities to equations (4) and (5). We add a param- right but not the obligation to pay the rate rK at time tl. eter β to equations (4) and (5) and for simplicity we set m = 1 and PA =1. Floorlet : option on an interest rate from time t to time t , t > t , i.e. the ts,tl s l l s Hence we derive right but not the obligation to receive the rate rK at time tl. m: time between t and t , expressed in years Caplet = e–rl tl {r [N (d )–β] – r [N(d )–β]} β ∈ ℜ (7) s l ts,tl f 1 k 2 PA: principal amount Floorlet = e–rl tl {r [N(–d )–β] – r [N(–d )–β]} (8) t : option maturity, t ≤ t < t    ts,tl k 2 f 1 x x s l ,   dft 1 r : forward interest rate, derived as = s − where df is a rf + 1 σ 2 √ f rfts,t df 1 t −t w ln r 2 tx l tl l s = k √ σ where d1 σ and d2 = d1 – tl discount factor, i.e. dft = 1/(1+ry). tx y This brings us to negative probabilities which imply negative interest rK: strike rate i.e. the interest rate that the Caplet buyer may pay and the rates. Let’s show this. From basic option theory we know that the value of a Floorlet buyer may receive from time ts to time tl. Equations (4) and (5) give the arbitrage-free tradable price of a Caplet and Caplet is divided into intrinsic value IV and time value TV: a Floorlet in the Black–Scholes–Merton framework. Partially differentiating Caplet = IV + TV ≥ 0 (9) equations (4) and (5) results in the risk parameters, which underlie hedging ts,tl Caplet Caplet the risks of Caplets and Floorlets. See www.dersoft.com/greeks.doc for details. The intrinsic value is defined The functions (4) and (5) satisfy the famous Black–Scholes–Merton PDE IV = max (r – r , 0) (10) 2 Caplet K ___∂D __1 ___∂D __1 ___∂ D __1 2 2 D = + S + 2 σ S (6) ∂t i ∂S 2 ∂S i where r is the current value of the underlying interest rate. D: financial derivatives as for example a Caplet or a Floorlet Let’s investigate the case of the Caplet being in-the-money and the i: discount rate 2 Floorlet being out-of the money, i.e. r > rK. Hence equation (10) changes to S: modeled variable IV = r – r (11) σ: volatility of S Caplet K 1 showing that the equations (4) and (5) are arbitrage free . Since a Caplet does not pay a return, the time value of a Caplet is bigger than 0, 4.2. Applying negative probabilities to Caplets and Floorlets TVCaplet ≥ 0 (12) Our original problem is that the market applied lognormal distribution, β which is underlying the valuation of interest rate derivatives, cannot model With a positive , negative probabilities may emerge for certain input β β negative interest rates. Several solutions to this problem are possible. parameter constellations of in equations (7) and (8). I.e. N(d1)– , N(d2)– , β β N(–d2)– and N(–d1)– may become negative. In this case, from equations 1) We can model interest rates with an entirely different distribution (7) and (8), the resulting Caplet price can become, especially for low implied as for example the normal distribution, which allows negative inter- volatility, smaller than the intrinsic value, i.e. Caplet t ,t < IVCaplet . From equa- est rates. This is done by Vasicek (1977), Ho and Lee (1986), and Hull s l tions (9), (11), and (12), this implies that r < 0 for small rK. Hence we have an and White (1990). However, empirical data shows that interest rate extension of the lognormal distribution, with negative probabilities associ- distribution is far from normal, see for example Chen and Scott 2002. ated with negative values for r (see Figure 2). Thus, the suggested solutions do not correctly reflect the reality. One The higher the value of β, the more likely it is that negative probabilities could further argue that the use of the empirical interest rate distri- with associated negative interest rates will emerge. This lowers Caplet prices bution should underlie the option valuation process, as for example applied in the Omega function (see Keating and Shadwick 2002). and increases the Floorlet price, which is a desired result, since it adjusts However, there are two drawbacks of this approach. a) The empirical the Caplet and Floorlet prices for the possibility of negative interest rates. distribution might not be a good indication of future interest rate The magnitude of the parameter β, that a trader applies, reflects a trader’s distributions. b) Numerical methods as Monte Carlo are necessary to opinion on the possibility of negative rates. A trader will use more extreme value the option. Our approach outlined in point 3) below results in a β-values if he/she believes strongly in the possibility of negative interest convenient closed form for solution. rates, vice versa.

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Figure 2: Extended distribution with negative probabilities associated dersoft.com), Adjunct Professor of Mathematical Finance at NYU (www.cims.nyu.edu) and Director with negative interest rates. of the Master in Financial Engineering program at the Shidler College of Business at the University of Hawaii (www.mfe.shidler.hawaii.edu). He has published numerous papers on derivatives and is a frequent speaker on conferences and seminars. He is author of 4 books, including his 2008 edition of “The Definitive Guide to CDOs – Application, Pricing, and Risk Management”.

ENDNOTES 1. For a derivation of equation (6) see www.dersoft.com/BSMPDEgeneration.pptx. For a proof that the functions (4) and (5) satisfy the PDE (6), see www.dersoft.com/bspdeproof.doc

2. In a follow up paper we will discuss the case r < rK and bigger than unity probabilities.

REFERENCES Bartlett, S. Negative Probability. Math Proc Camb Phil Soc 41: 71–73, 1945. Black, F. and Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political Economy May-June 1973, p. 637–659. Black, F. The Pricing of Commodity Contracts. Journal of Financial Economics 3, March 1976, 5. Concluding Summary 167–79. Chen, R. and Scott, L. Stochastic Volatility and Jumps in Interest Rates: An Empirical Analysis We have defined extended probabilities, which include negative probabili- http://papers.ssrn.com/sol3/papers.cfm?abstract_id=686985 ties, and derived their general properties. Then we have applied extended Dirac, P.A.M. The Physical Interpretation of . Proc. Roy. Soc. London (A probabilities to financial modeling. We have shown that negative nominal 180), pp. 1–39, 1942. interest rates have occurred several times in the past in financial practice, Duffie, D. and Singleton, K. Modeling Term Structures of Defaultable Bonds. Review of Financial as in the 2008/2009 global financial crisis. This is inconsistent with the Studies 12, 1999, p. 687–720. conventional theoretical models of interest rates, which typically apply a Haug, E.G. Why so Negative to Negative Probabilities Wilmott Magazine, Sep/Oct 2004, lognormal distribution. In particular, when Caps and Floors are valued in a pp 34– 38. lognormal Black–Scholes–Merton framework, then the probability of nega- Ho, T. and Lee, S. Term Structure Movements and Pricing Interest Rate Contingent Claims. tive interest rates is zero. Here negative probabilities come into play. We have Journal of Finance, 41, 1986, p.1011–1029. shown that integrating negative probabilities in the Black–Scholes–Merton Hull, J. and White, A. Pricing Interest Rate Derivatives Securities. Review of Financial Studies, 3, framework allows to consistently model negative nominal interest rates, no.4 (1990), p. 573–592. Jarrow, R. and Turnbull, S. Pricing Derivatives on Financial Securities Subject to Credit Risk. which exist in financial practice. Journal of Finance Vol L, No. 1, March 1995, p.53–85. Keating, C. and Shadwick, W. Omega: A Universal Performance Measure. Journal of Mark Burgin is currently a Visiting Scholar at UCLA. Before, he was a Visiting Professor at UCLA, Performance Measurement, Spring 2002. Professor at Institute of Education, Kiev; at the International Solomon University, Kiev; at the Kiev Khrennikov, A. Equations with infinite-dimensional pseudo-differential operators. Dokl. State University, Ukraine; and the Head of the Assessment Laboratory in the Research Center of Academii Nauk USSR, v.267, 6, p.1313–1318, 1982. Science at the National Academy of Sciences of Ukraine. He is a member of New York Academy of Khrennikov, A. p-adic probability and statistics. Dokl. Akad. Nauk, v.322, p.1075–1079, 1992. Sciences and an Honorary Professor of the Aerospace Academy of Ukraine. He is a Chief Editor of Khrennikov, A. Andrei Khrennikov on Negative Probabilities, in Derivatives, Models on Models, the international journals Integration and Information. He is doing research, has publications, and Editor Espen Haug, John Wiley 2007. taught courses in mathematics, computer science, information sciences, system theory, artificial Khrennikov, A. Interpretations of Probability. Walter de Gruyter, Berlin/New York, 2009. intelligence, software engineering, logic, psychology, education, social sciences, and methodology Kolmogorov, A. Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse der Mathematik of science, authorizing and co-authorizing more than 500 papers and 17 books. He originated such (English translation: (1950) Foundations of the Theory of Probability, Chelsea P.C.), 1933. theories as the mathematical theory of technology, system theory of time, general information Kolmogorov, A. and Fomin, S. Elements of Function Theory and Functional Analysis, Nauka, theory, theory of named sets, and neoclassical analysis (in mathematics), making essential contri- Moscow, 1989 (in Russian). butions to such fields as foundations of mathematics, theory of knowledge, theory of intellectual Kuratowski, K. and Mostowski, A. Set Theory, North Holland P.C., Amsterdam, 1967. activity, theory of algorithms, and complexity studies. Meissner, G. Trading Financial Derivatives, Pearson 1998. Meissner, G. Arbitrage Opportunities in Fixed Income Markets Derivatives Quarterly, April 1999. After a lectureship in mathematics and statistics at the Economic Academy Kiel, Gunter Meissner Merton, R. Theory of Rational Option Pricing, Bell Journal of Economics and Management joined Deutsche Bank in 1990, trading interest rate futures, swaps, and options in Frankfurt and New Science, Spring 1973, 4(1), 141–183. York. He became Head of Product Development in 1994, responsible for originating algorithms for Vasicek, O. An Equilibrium Characterization of the Term Structure, Journal of Economics, 5, new derivatives products, which at the time were Lookback Options, Multi-asset Options, Quanto 1977, 177–188. Options, Average Options, Index Amortizing Swaps, and Bermuda Swaptions. In 1995/1996 Gunter Wigner, E. P. On the quantum correction for thermodynamic equilibrium, Phys. Rev. v. 40, pp. Meissner was Head of Options at Deutsche Bank Tokyo. From 1997 to 2007, Gunter was Professor 749–759, 1932. of Finance at Hawaii Pacific University. Currently, he is President of Derivatives Software (www.

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