Negative Probabilities in Financial Modeling. Abstract

Transcription

1 Negatve Probabltes n Fnancal Modelng Mark Burgn Department of Mathematcs Unversty of Calforna, Los Angeles 405 Hlgard Ave. Los Angeles, CA Gunter Messner Shdler College of Busness Unversty of Hawa 404 Male Way, D0 Honolulu, HI 968 Emal: messner@hawa.edu Abstract We frst defne and derve general propertes of negatve probabltes. We then show how negatve probabltes can be appled to modelng fnancal optons such as Caps, Floors and Swapoptons. In tradng practce, these optons are typcally valued n a Black-Scholes-Merton framework assumng a log-normal dstrbuton for the underlyng nterest rate. However, n some cases, such as the 008/009 fnancal crss, nterest rates can get negatve. Then the log-normal dstrbuton s napplcable. We show how negatve probabltes can be appled to value nterest rate optons n a log-normal framework mplyng a postve probablty for negatve nterest rates. A model n VBA, whch prces Caps, Floors and Swapoptons wth negatve probabltes, s avalable upon request. Key words: Negatve probabltes, negatve nterest rates, Caps, Floors, Swaptons JEL Classfcaton: C10

2 1. Introducton The classcal probablty theory s appled n most scences and n many of the humantes. In partcular, t s successfully used n physcs and fnance. However, physcsts found that they need a more general approach than the classcal probablty theory. The frst was Eugene Wgner (193), who ntroduced a functon, whch looked lke a conventonal probablty dstrbuton and has later been better known as the Wgner quas-probablty dstrbuton because n contrast to conventonal probablty dstrbutons, t took negatve values, whch could not be elmnated or made nonnegatve. The mportance of Wgner's dscovery for foundatonal problems was not recognzed untl much later. Another outstandng physcst, Drac (194) not only supported Wgner s approach but also ntroduced the physcal concept of negatve energy. He wrote: Negatve energes and probabltes should not be consdered as nonsense. They are well-defned concepts mathematcally, lke a negatve of money." After ths, negatve probabltes lttle by lttle have become popular although questonable technques n physcs. Bartlett (1945) worked out the mathematcal and logcal consstency of negatve probabltes. However, he dd not establsh rgorous foundaton for negatve probablty utlzaton. Khrennkov (009) provdes the frst mathematcal theory of negatve probabltes n hs textbook. However, he s dong ths not n the conventonal settng of real numbers but n the framework of p-adc analyss. Negatve probabltes are also used n mathematcal fnance. The concept of rsk-neutral or pseudo probabltes s a popular concept and has been numerously appled, for example, n credt modelng by Jarrow and Turnbull (1995), and Duffe and Sngleton (1999). Haug (007) extends the rsk-neutral framework to allow negatve probabltes and shows how negatve probabltes can help add flexblty to fnancal modelng. The remanng paper s organzed as follows. In secton, we resolve the mathematcal ssue of the negatve probablty problem. We buld a mathematcal theory of extended probablty as a probablty functon, whch s defned for real numbers and can take both postve and negatve values. Thus, extended probabltes nclude negatve probabltes. Dfferent propertes of extended probabltes are found. In secton 3, we gve examples of negatve nomnal nterest rates n fnancal practce and show problems of current fnancal modelng of negatve nterest rates. In Secton 4, we buld mathematcal models of nterest rate optons as Caps and Floors and Swapoptons, ntegratng extended probabltes nto the prcng model to allow for negatve

3 nterest rates. Conclusons are gven n Secton 5. A follow up paper wll specfy >1 probabltes and apply them to fnancal optons.. Mathematcal theory of extended probablty Extended probabltes generalze the standard defnton of a probablty functon. At frst, we defne extended probabltes n an axomatc way and then develop applcaton of extended probabltes to fnance. To defne extended probablty, EP, we need some concepts and constructons, whch are descrbed below. Some of them are well-known, such as, for example, set algebra, whle others, such as, for example, random antevents, are new. We remnd that f X s a set, then X s the number of elements n (cardnalty of) X (Kuratowsk and Mostowsk, 1967). If A X, then the complement of A n X s defned as C X A = X \ A. A system B of sets s called a set rng (Kolmogorov and Fomn, 1989) f t satsfes condtons (R1) and (R): (R1) A, B B mples A B B. (R) A, B B mples A B B where A B = (A \ B) (B \ A). For any set rng B, we have B and A, B B mples A B, A \ B B. Indeed, f A B, then by R1, A \ A = B. If A, B B, then A \ B = ((A \ B) (B \ A)) A B. If A, B B and A B = Ø, then A B = A B B. It mples that A B = (A \ B) (B \ A) (A B) B. Thus, a system B of sets s a set rng f and only f t s closed wth respect unon, ntersecton and set dfference. Example.1. The set CI of all closed ntervals [a, b] n the real lne R s a set rng. Example.. The set OI of all open ntervals (a, b) n the real lne R s a set rng. A set rng B wth a unt element,.e., an element E from B such that for any A from B, we have A E = A, s called a set algebra (Kolmogorov and Fomn, 1989). Example.3. The set BCI of all closed subntervals of the nterval [a, b] s a set algebra. Example.4. The set BOI of all open subntervals of the nterval [a, b] s a set algebra.

4 A set algebra B closed wth respect to complement s called a set feld. Let us consder a set Ω, whch conssts of two rreducble parts (subsets) Ω + and Ω -,.e., nether of these parts s equal to ts proper subset, a set F of subsets of Ω, and a functon P from F to the set R of real numbers. Elements from F,.e., subsets of Ω that belong to F, are called random events. Elements from F + = {X F; X Ω + } are called postve random events. Elements from Ω + that belong to F + are called elementary postve random events or smply, elementary postve random events. If w Ω +, then w s called the antevent of w. Elements from Ω - that belong to F - are called elementary negatve random events or elementary random antevents. For any set X Ω +, we defne X + = X Ω +, X - = X Ω -, X = { -w; w X} and F - = { A ; A F + } If A F +, then A s called the antevent of A. Elements from F - are called negatve random events or random antevents. Defnton 1. The functon P from F to the set R of real numbers s called a probablty functon, f t satsfes the followng axoms: EP 1 (Order structure). There s a graded nvoluton α: Ω Ω,.e., a mappng such that α s an dentty mappng on Ω wth the followng propertes: α(w) = -w for any element w from Ω, α(ω + ) Ω -, and f w Ω +, then α(w) Ω +. EP (Algebrac structure). F + {X F; X Ω + } s a set algebra that has Ω + as a member. EP 3 (Normalzaton). P(Ω + ) = 1. EP 4 (Composton) F {X; X + F + & X - F - & X + -X & X - -X + }. EP 5 (Fnte addtvty) P(A B) = P(A) + P(B) for all sets A, B F such that

5 A B Ø EP 6 (Annhlaton). { v, w, - w ; v, w Ω & I } = { v ; v Ω & I } for any element w from Ω. Axom EP6 shows that f w and - w are taken (come) nto one set, they annhlate one another. Havng ths n mnd, we use two equalty symbols = and. The second symbol means equalty of elements of sets. The second symbol also means equalty of sets, when two sets are equal when they have exactly the same elements (Kuratowsk and Mostowsk, 1967). The equalty symbol = s used to denote equalty of two sets wth annhlaton, for example, { w, - w } = Ø. Note that for sets, equalty mples equalty =. For equalty of numbers, we, as t s customary, use symbol =. EP 7. (Adequacy) A = B mples P(A) = P(B) for all sets A, B F. For nstance, P({ w, - w }) = P(Ø) = 0. EP 8. (Non-negatvty) P(A) 0, for all A F +. It s known that for any set algebra A, the empty set Ø belongs to A and for any set feld B n Ω, the set Ω belongs to A (Kolmogorov and Fomn, 1989). Defnton. The trad (Ω, F, P) s called an extended probablty space. Defnton 3. If A F, then the number P(A) s called the extended probablty of the event A. Let us obtan some propertes of the ntroduced constructons. Lemma 1. α(ω + ) -Ω + Ω - and α(ω - ) -Ω - Ω +. Proof. By Axom EP1, α(ω + ) -Ω + and α(ω + ) Ω -. As Ω Ω + Ω -, Axom EP1 also mples α(ω + ) Ω -. Thus, we have α(ω + ) Ω -. The frst part s proved. The second part s proved n a smlar way. Thus, f Ω + = { w ; I }, then Ω - = { -w ; I }. As α s an nvoluton of the whole space, we have the followng result. Proposton 1. α s a one-to-one mappng and Ω + = Ω -. Corollary 1. (Doman symmetry) w Ω + f and only f w Ω -. Corollary. (Element symmetry) - (- w ) = w for any element w from Ω. Corollary 3. (Event symmetry) - (- X ) X for any event X from Ω. Lemma. α(w) w for any element w from Ω.

6 Indeed, ths s true because f w Ω +, then by Axom EP1, α(w) Ω + and thus, α(w) w. If w Ω -, then we may assume that α(w) = w. However, n ths case, α(v) = w for some element v from Ω + because by Axom EP1, α s a projecton of Ω + onto Ω -. Consequently, we have α(α(v)) = α(w) = w However, α s an nvoluton, and we have α(α(v)) = v. Ths results n the equalty v = w Consequently, we have α(v) = v. Ths contradcts Axom EP1 because v Ω +. Thus, lemma s proved by contradcton. Proposton. Ω + Ω - Ø. Proposton 3. F + F, F - F and F F + F -. Corollary 1 mples the followng result. Proposton 4. X Ω + f and only f - X Ω -. Proposton 5. F - {X F; X Ω - } = F Ω -. Corollary 4. F + F - Ø. Axoms EP6 mples the followng result. Lemma 3. X -X = Ø for any subset X of Ω. Indeed, for any w from the set X, there s -w n the set X, whch annhlates w. Let us defne the unon wth annhlaton of two subsets X and Y of Ω by the followng formula: X + Y (X Y) \ [(X -Y) (-X Y)] Here the set-theoretcal operaton \ represents annhlaton, whle sets X -Y and X -Y depct annhlatng enttes. Some propertes of the new set operaton + are the same as propertes of the unon, whle other propertes are dfferent. For nstance, there s no dstrbutvty between operatons + and. Lemma 4. a) X + X X for any subset X of Ω; b) X + Y X + Y for any subsets X and Y of Ω; c) X + Ø X for any subset X of Ω; d) X + (Y + Z) (X + Y) + Z for any subsets X, Y and Z of Ω; e) X + Y X Y for any subsets X and Y of Ω + (of Ω - ); Lemma 5. a) Z (X + Y ) Z X + Z Y ;

7 b) X + (Y Z) (X Y) + (X Z). Lemma 6. A B (A + B + ) + (A - B - ) for any subsets A and B of Ω. Indeed, as A A + A - and B B + B -, we have A B (A + A - ) (B + B - ) (A + B + ) (A + B - ) (A - B + ) (A - B - ) (A + B + ) + (A - B - ) because (A + B - ) Ø and (A - B + ) Ø. In a smlar way, we prove the followng results. Lemma 7. A \ B (A + \ B + ) + (A - \ B - ) for any subsets A and B of Ω. Lemma 8. X X + + X - = X + X - for any set X from F. Lemma 9. A + B (A + + B + ) + (A - + B - ) for any sets X and Y from F. Axoms EP6 and EP7 mply the followng result. Proposton 6. P(X + Y ) = P(X Y) for any two events X and Y from Ω. Lemma 10. P( ) = 0. Propertes of the structure F + are nherted by the structure F. Theorem 1. (Algebra symmetry) If F + s a set algebra (or set feld), then F s a set feld (or set algebra) wth respect to operatons + and. Proof. At frst, we prove that F - s a set algebra (or set feld). Let us assume that F + s a set algebra and take two negatve random events H and K from F -. By the defnton of F -, H = -A and K = -B for some postve random events A and B from F +. Then we have H K = (- A) (-B) = -( A B) As F + s a set algebra, A B F +. Thus, H K F -. In a smlar way, we have H K = (- A) (-B) = -( A B) As F + s a set algebra, A B F +. Thus, H K F -. By the same token, we have H \ K F -. Besdes, f F + has a unt element E, then E s a unt element n F -. Thus, F - s a set algebra.

8 Now let us assume that F + s a set feld and H F -. Then by the defnton of F -, H = -A for a postve random event A from F +. It means that C Ω+ A = Ω + \ A F +. At the same tme, C Ω- H = Ω - \ H = (-Ω + ) \ (-A) = -(Ω + \ A) = -C Ω+ A As C Ω+ A belongs to F +, the complement C Ω- H of H belongs to F -. Consequently, F - s a set feld. Let us once more assume that F + s a set algebra and take two random events A and B from F. Then by Theorem 1, F - s a set algebra. By Lemma 8, A A + + A - and B B + + B -. By Axom EP4, A +, B + F +, A -, B - F -, whle by Proposton, A + A, B + B, A A + A -, and B B + B -. By Lemma 6, A B (A + B + ) + (A - B - ). Thus, (A B) + A + B + and (A B) - A - B -. As F + s a set algebra, (A B) + A + B + F +. As t s proved that F - s a set algebra, (A B) - A - B - F -. Consequently, A B F. By Lemma 7, A \ B (A + \ B + ) + (A - \ B - ). Thus, (A \ B) + A + \ B + and (A \ B) - A - \ B -. As F + s a set algebra, (A \ B) + A + \ B + F +. As t s proved that F - s a set algebra, (A \ B) - A - \ B - F -. Consequently, A \ B F. By Lemma 9, A + B (A + + B + ) + (A - + B - ). Thus, (A + B) + A + + B + and (A + B) - A - + B -. As F + s a set algebra, (A + B) + A + + B + A + B + F +. As t s proved that F - s a set algebra, (A + B) - A - + B - A - B - F -. Consequently, A + B F. Besdes, f F + has a unt element E, then E s a unt element n F - and E -E s a unt element n F. Thus, F s a set algebra. Now let us assume that F + s a set feld and A F. Then as t s demonstrated above, F - s a set feld. By Lemma 8, A A + + A -. By Proposton, Ω + Ω - = Ø, we have C Ω A = C Ω+ A + C Ω- A Then C Ω+ A belongs to F + as F + s a set feld and as t s proved n Theorem 1, C Ω- A belongs to F -. Consequently, C Ω A belongs to F - and F - s a set feld. Theorem s proved.

9 3. Negatve nterest rates and the problem of ther modelng Negatve probabltes can help to model nterest rates and nterest rate dervatves. To show ths, let us start wth the equaton Real nterest rate = Nomnal nterest rate Inflaton rate (1) where the nomnal nterest rate s the de facto rate, whch s receved by the lender and pad by the borrower n a fnancal contract. For example, the nomnal nterest rate s the rate, whch the lender receves on a savng account or the coupon of a bond. From equaton (1) we see that a real nterest rate can easly be negatve and n realty often s. For example, f the nomnal nterest rate on a savngs account s 1% and the nflaton rate s 3%, naturally, the real nterest rate,.e. the nflaton adjusted rate of return for the lender s -% Examples of negatve nomnal nterest rates However, n rare cases, also the nomnal nterest rate can be negatve. An example of ths would be that the lender gves money to a bank, and addtonally gves pays the bank an nterest rate. Ths happened n the 1970s n Swtzerland. The lender had several motves a) Swtzerland s consdered an extremely safe country to place captal b) Investors were speculatng on an ncrease of the Swss franc c) Some nvestors avoded payng taxes n ther home country Another example of negatve nomnal nterest rates occurred n Japan n 003. Banks lent Japanese Yen and were wllng to receve a lower Yen amount back several days later. Ths means de facto a negatve nomnal nterest rate. The reason for ths unusual practce was that banks were eager to reduce ther exposure to Japanese Yen, snce confdence n the Japanese economy was low and the Yen was assumed to devalue.

10 Smlarly, n the US, from August to November 003, repos,.e. repurchase agreements traded at negatve nterest rates. A repo s just a collateralzed loan,.e. the borrower of money gves collateral, for example a Treasury bond, to the lender for the tme of the loan. When the loan s pad back, the lender returns the collateral. However, n 003 n the US, settlement problems when returnng the collateral occurred. Hence the borrower was only wllng to take the rsk of not havng the collateral returned f he could pay back a lower amount than orgnally borrowed. Ths consttuted a negatve nomnal nterest rate. A further example of the market expectng the possblty of negatve nomnal nterest rates occurred n the worldwde 008/009 fnancal crss, when strkes on optons on Eurodollars Futures contracts were quoted above 100. A Eurodollar s a dollar nvested at commercal banks outsde the US. A Eurodollar futures prce reflects the antcpated future nterest rate. The rate s calculated by subtractng the Futures prce from 100. For example, f the 3 month March Eurodollar future prce s 98.5, the expected nterest rate from March to June s = 1.5, whch s quoted n per cent, so 1.5%. In March 009, opton strkes on Eurodollar future contracts were quoted above 100 on the CME, Chcago Mercantle Exchange. Ths means that market partcpants could buy the rght to pay a negatve nomnal nterest on US dollars n the future f desred. The reason for ths unusual behavor s that nvestors wanted to nvest n the safe haven currency US dollar even f they had to pay for t. 3.. Modelng nterest rates In fnance, nterest rates are typcally modeled wth a geometrc Brownan moton, dr : change n the nterest rate r; dr = μ r dt + σ r ε dt () r µ r : drft rate, whch s the expected growth rate of r, assumed non-stochastc and constant dt : nfntely short tme perod σ r : expected volatlty of rate r, assumed non-stochastc and constant

11 ε : random drawng from a standardzed normal dstrbuton. All drawngs at tmes t are ndependent from each other. In equaton (), the frst term on the rght hand sde gves the average growth rate of r. The second term on the rde sde adds stochastcty to the process va ε,.e. provdes the dstrbuton around the average growth rate. Importantly, from equaton (1) we can observe that the relatve change dr/r s normally dstrbuted, snce ε s normally dstrbuted. If the relatve change of a varable s normally dstrbuted, t follows that the varable tself s log-normally dstrbuted wth a pdf 1 ln(x) μ 1 - σ x σ π e (3) In the equaton (3), μ and σ are the mean and standard devaton of ln(x) respectvely. Fgure 1 shows a log-normal dstrbuton Fgure 1 Log-normal dstrbuton wth μ = 0 and σ =1. The logarthm of a negatve number s not defned, hence wth the pdf equaton (3), negatve values of nterest rates cannot be modeled. However, as dscussed above, negatve nterest rates do exst n the real fnancal world. Here negatve probabltes come nto play. We wll explan ths wth optons on nterest rates.

12 4. How negatve probabltes allow more adequate nterest rate modelng 4.1. Modelng nterest rate optons Two man types of optons are call optons and put optons. A call opton reflects the rght but not the oblgaton to pay a strke prce and receve an underlyng asset. A put opton reflects the rght but not the oblgaton to receve a strke prce and delver an underlyng asset. Let s derve the equatons used n fnancal practce to value calls and puts. From equaton (), applyng Ito s lemma, we derve the famous 1997 Nobel Prze rewarded work of Black, Scholes and Merton, whch resulted n the PDE 1 D D 1 t D S 1 D 1 S = + S + σ S (4) where D : fnancal dervatves as for example a call opton or a put opton : dscount rate S : modeled varable σ : volatlty of S One equaton that satsfes the PDE (4) s the Black-Scholes-Merton equaton for a call and a put. For a call opton, we have 1 For a proof see For a proof, see

13 C 0 1 T = S N(d ) Ke N(d ) wth d 1 S0 1 ln( ) + T = Ke σ T σ T and d 1 = d σ T (5) where N(x) : cumulatve probablty of a standard normal dstrbuton at x K : Strke,.e. the prce that the buyer may pay at opton maturty T to receve the underlyng asset 4.. Caps and Floors For nterest rate optons, a term structure of nterest rates exsts (.e. market gven nterest rates for dfferent maturtes exst). Ths property s utlzed when valung nterest rate optons n the model Black 1976 model. We wll frst dscuss an nterest rate opton contract on short term nterest rates,.e. Caps and Floors. A Cap conssts of several Caplets. A Caplet s the opton but not the oblgaton to pay an nterest rate r K at opton maturty t x. A Floor conssts of several Floorlets. A Floorlet s the opton but not the oblgaton to receve an nterest rate r K at opton maturty t x. A Cap s typcally used as an nsurance aganst rsng nterest rates: If a borrower pays a floatng nterest rate on hs loan, he can protect hmself aganst rsng nterest rates by buyng a Cap. Conversely, a Floor s typcally used as an nsurance for decreasng nterest rates. If an nvestor receves a floatng nterest rate on a bond, the nvestor can protect hmself aganst decreasng nterest rate by buyng a Floor. Usng forward nterest rates r f, equaton (5) becomes Caplet t s,tl = m e-r l t l {r f N(d 1 ) r k N(d )} wth d 1 = r f ln( ) + rk σ 1 σ t x t x and d = d 1 - σ t l (6)

14 Floorlet ts,tl = m PA e-r l t l {r k N(-d ) r f N(-d 1 )} (7) where d 1 and d are defned as n (6) Caplet : opton on an nterest rate from tme t s to tme t l, t l > t s,.e. the rght but not the oblgaton to pay the rate r K at t l. Floorlet : opton on an nterest rate from tme t s to tme t l, t l > t s,.e. the rght but not the oblgaton to receve the rate r K at opton maturty t l. m : tme between t l and t s, expressed n years t x : opton maturty, t x t s < t l r f : forward nterest rate, derved as r f, ts tl df = df t s t l 1 t l 1 t s where df s a dscount factor,.e. df ty = 1/(1+r y ). r K : strke rate.e. the nterest rate that the Caplet buyer may pay and the Floorlet buyer may receve at opton maturty t x. For more detals, see Messner Applyng negatve probabltes to Caplets and Floorlets Our orgnal problem s that the market appled log-normal dstrbuton, whch s underlyng the valuaton of nterest rate dervatves, cannot model negatve nterest rates. Several solutons to ths problem are possble. a) We can model nterest rates wth an entrely dfferent dstrbuton as for example the normal dstrbuton, whch allows negatve nterest rates. Ths s done by Vascek (1977), Ho and Lee (1986), and Hull and Whte (1990). However, emprcal data shows that nterest rate dstrbuton behaves far more log-normal than normal. Thus, the suggested solutons do not correctly reflect the realty.

15 b) We can add a locaton parameter to the log-normal dstrbuton. Hence equaton (3) x σ 1 ln(x) μ 1 - σ π e becomes (x - α) σ 1 ln(x-α) μ 1 - σ π e, where α s the locaton parameter. For α > 0, the log-normal dstrbuton s shfted to the left. As a result, the probablty dstrbuton acqures negatve values. Ths s mpossble n the conventonal probablty theory but fts well nto extended probablty theory. At the same tme, probablty dstrbutons that take negatve values are mportant for practce, allowng to model negatve nterest rates. c) A further way to model optons on negatve nterest rates s to apply negatve probabltes to equatons (6) and (7). We add a parameter β to equatons (6) and (7) Caplet t s,tl = m e-r l t l {r f [N (d 1 )-β] r k [N(d )-β]} β R (8) Floorlet ts,tl = m e-r l t l {r k [N(-d )-β] r f [N(-d 1 )-β]} (9) Ths also brngs us to negatve probabltes. If the strke r K s smaller than the forward rate s f, for a postve β, negatve probabltes may emerge,.e. N(d 1 )- β, N(d )- β, N(- d )- β and N(-d 1 )- β may become negatve. Thus, appearance of negatve probabltes depends on the value of β and the opton nput parameters r K, r F, r l, and σ. The hgher the value of β, the more lkely t s that negatve probabltes wll emerge. Importantly, applyng negatve probabltes n equatons (8) and (9) decreases the value of the Caplet and ncrease the value of the Floorlet. Ths s an adequate result snce t adjusts the opton prces for the possblty of negatve nterest rates. The magntude of the parameter β, that a trader apples, reflects a trader s opnon on the probablty of negatve rates. A trader wll use more extreme β-values f he/she beleves strongly n the possblty of negatve nterest rates, vce versa Swapoptons

16 Another popular type of nterest rate opton s a Swapopton. Two types exst, Payers and Recevers. A Payers Swapopton s the rght but not the oblgaton to pay a fxed nterest rate,.e. 5%, and receve a floatng nterest rates, e.g. 3-month Lbor 3. A Recevers Swapopton s the rght but not the oblgaton to receve a fxed nterest rate,.e. 5%, and pay a floatng nterest rates, e.g. 3-month Lbor. Smlar to Caps and Floors, Swapoptons are typcally used to protect aganst nterest rate volatlty. For example, a company s bddng on an nvestment project and s concerned about rsng nterest rates untl the bddng s decded. The company can enter nto a Payers Swapopton. If the company wns the bd and nterest rates have rsen, the Payers wll allow the company to pay the lower strke rate, whch was agreed n the Swapopton. A Payers Swapopton can be valued by modfyng equaton (5). Ths gves us an equaton for Payers Swapopton (PSWO): n PA (t t PSWO = = 1 (1 + sr ) where f ) 1 (pt T) e -rt ( sr f N(d 1 ) - r k N(d ) ) (10) PA s the prncpal amount for the perod from t -1 to t sr f s the forward swap rate, for the perod from t -1 to t n. n sr f = (df t - df -1 tn ) / df (t t ) and df s a standard dscount factor,.e. df t 1 l = 1/(1+r l ). = 1 d 1 = sr f ln( ) r k σ + 1 T σ T and d = d 1 - σ T p t s the payment date of the fxed cash flows Smlarly, the Recever Swapopton (RSWO) can be valued as 3 Lbor stands for London Interbank offered rate. It s determned every busness day at 11 AM London tme.

17 n PA (t t RSWO = (pt = 1 (1 + sr ) f ) 1 T) e -rt ( r k N(-d ) - sr f N(-d 1 ) ) (11) For more detals see Messner, We once more encounter a problem because the market appled log-normal dstrbuton, whch s underlyng the valuaton of nterest rate dervatves, cannot model negatve nterest rates. To solve ths problem, we apply the approach used for Caps and Floors to Payers and Recevers Optons. In ths case, t s agan possble as n (a), to use an entrely dfferent dstrbuton as the normal dstrbuton or as n (b), to add a locaton parameter to the log-normal dstrbuton. A thrd possblty s to apply negatve probabltes once more. We can add a parameter γ to equatons (10) and (11): n PA (t t PSWO = = 1 (1 + sr ) f ) 1 (p t T) e -rt ( sr f [N(d 1 )-γ]- r k [N(d )-γ] ) (1) and n PA (t t RSWO = (p = 1 (1 + sr ) t f ) 1 T) e -rt ( r k [N(-d )- γ] - sr f [N(-d 1 )- γ] ) (13) We derve equvalent results as we dd for Caps and Floors. If the strke r K s smaller than the forward rate sr f, for a postve γ, negatve probabltes may emerge,.e. N(d 1 )-γ, N(d )-γ, N(- d )- γ and N(-d 1 )- γ may become negatve. Thus, appearance of negatve probabltes depends on the value of γ and the opton nput parameters r K, sr f, r, T and σ. The hgher the value of γ, the more lkely t s that negatve probabltes wll emerge. Importantly, applyng negatve probabltes decreases the value of the Payers Swapopton and ncreases the value of the Recevers Swapopton. Ths s the desred result snce t adjusts the opton prces for the possblty of negatve nterest rates. The magntude of parameter γ that a

18 trader apples, reflects a trader s opnon on the probablty of negatve rates. A trader wll use more extreme γ-values f he/she beleves strongly n the possblty of negatve nterest rates, vce versa. 5. Concludng Summary We have defned extended probabltes, whch nclude negatve probabltes, and derved ther general propertes. Then we have appled extended probabltes to fnancal modelng. We have shown that negatve nomnal nterest rates have occurred several tmes n the past n fnancal practce, as n the 008/009 global fnancal crss. Ths s nconsstent wth the conventonal theoretcal models of nterest rates, whch typcally apply a log-normal dstrbuton. In partcular, when Caps, Floors and Swapopton are valued n a log-normal Black-Scholes- Merton framework, then the probablty of negatve nterest rates s zero. Here negatve probabltes come nto play. We have shown that ntegratng negatve probabltes n the Black- Scholes-Merton framework allows to consstently model negatve nomnal nterest rates, whch exst n fnancal practce. References: Bartlett, M. S. (1945). "Negatve Probablty". Math Proc Camb Phl Soc 41: Black F. and M. Scholes, The Prcng of Optons and Corporate Labltes, The Journal of Poltcal Economy, May-June 1973, p Black, F., The Prcng of Commodty Contracts, Journal of Fnancal Economcs, 3 (March 1976), Drac, P.A.M. (194) The Physcal Interpretaton of Quantum Mechancs, Proc. Roy. Soc. London, (A 180), pp Duffe D., and K. Sngleton, Modelng Term Structures of Defaultable Bonds, Revew of Fnancal Studes, 1 (1999), p Ho T., and S. Lee, Term Structure Movements and Prcng Interest Rate Contngent Clams, Journal of Fnance, 41, 1986, p

19 Haug, E. G. (007): Dervatves Models on Models, John Wley & Sons, New York Hull J., and A. Whte, Prcng Interest Rate Dervatves Securtes, Revew of Fnancal Studes, 3, no.4 (1990), p Jarrow R., and S. Turnbull, Prcng Dervatves on Fnancal Securtes Subject to Credt Rsk, Journal of Fnance, Vol L, No. 1, March 1995, p Khrennkov, A. Interpretatons of Probablty, Walter de Gruyter, Berln/New York, 009 Kolmogorov, A. N. (1933) Grundbegrffe der Wahrschenlchketrechnung, Ergebnsse der Mathematk (Englsh translaton: (1950) Foundatons of the Theory of Probablty, Chelsea P.C.) A.N. Kolmogorov and S.V. Fomn, Elements of Functon Theory and Functonal Analyss, Nauka, Moscow, 1989 (n Russan) Kuratowsk, K. and Mostowsk, A. Set Theory, North Holland P.C., Amsterdam, 1967 Messner, G., Tradng Fnancal Dervatves, Pearson 1998 Messner. G., Arbtrage Opportuntes n Fxed Income Markets Dervatves Quarterly, Aprl 1999 Merton R., Theory of Ratonal Opton Prcng, Bell Journal of Economcs and Management Scence, Sprng 1973, 4(1), p Vascek, O., An Equlbrum Characterzaton of the Term Structure, Journal of Economcs, 5, 1977, p Wgner, E. P., (193) On the quantum correcton for thermodynamc equlbrum, Phys. Rev. v. 40, pp