On the American swaption in the linear-rational framework

Transcription

1 arxiv: v3 [q-fin.pr] 24 Feb Inroducion On he American swapion in he linear-raional framework Damir Filipović and Yerkin Kiapbayev April 29, 2018 We sudy American swapions in he linear-raional (LR) erm srucure model inroduced in [5]. The American swapion pricing problem boils down o an opimal sopping problem ha is analyically racable. I reduces o a free-boundary problem ha we ackle by he local ime-space calculus of [7]. We characerize he opimal sopping boundary as he unique soluion o a nonlinear inegral equaion ha can be readily solved numerically. We obain he arbirage-free price of he American swapion and he opimal exercise sraegies in erms of swap raes for boh fixed-rae payer and receiver swaps. Finally, we show ha Bermudan swapions can be efficienly priced as well. An ineres rae swap is a conrac beween wo paries who agree o exchange cash flows over a pre-specified ime grid. The holder of a payer (receiver) swap conrac pays a fixed (floaing) rae and receives a floaing (fixed) rae on a noional amoun. The floaing rae is usually ied o he London Inerbank Offered Rae (LIBOR), a daily fixed sandard benchmark for ineres raes in various currencies. A payer (receiver) swapion gives he holder he righ bu no he obligaion o ener a payer (receiver) swap a a pre-specified fixed srike rae. Swapions form an imporan class of derivaives ha allow o price and hedge ineres rae risk. They underlie callable morgage-backed securiies, life insurance producs, and a wide variey of srucured producs. The ousanding noional amoun in he swap marke is in he order of hundreds of rillions US dollars. Swapions can be divided ino hree classes according o heir exercise iming righs. European swapions can be exercised only a he expiraion dae. American swapions allow he holder o ener he swap on any dae ha falls wihin a range of wo daes. Bermudan swapions consiue a simplified varian of American swapions where exercise is only possible on a TheauhorswishohankAndersTrolleforhiscommensandforprovidingheparameersforhenumerical example. We also hank he referee for useful commens ha helped o improve he paper. The research leading o hese resuls has received funding from he European Research Council under he European Unions Sevenh Framework Programme (FP/ ) / ERC Gran Agreemen n POLYTE. EPFL and Swiss Finance Insiue, Lausanne, Swizerland. damir.filipovic@epfl.ch Quesrom School of Business, Boson Universiy, Boson, MA, USA; yerkin@bu.edu Mahemaics Subjec Classificaion Primary 91G20, 60G40. Secondary 60J60, 35R35, 45G10. Key words and phrases: American swapion, swapion, swap, linear-raional erm srucure model, opimal sopping, free-boundary problem, local ime, inegral equaion. 1

2 finie ime grid. The pricing of American swapions is arguably a difficul ask. The sandard approach is o approximae American by Bermudan swapions and price he laer using simple ree models or Mone-Carlo simulaion based mehods (see e.g. [6]). To our knowledge here has no been a model in he lieraure for which American swapions are priced analyically in coninuous ime. A reason why American swapion pricing in coninuous ime was no feasible so far is ha in mos ineres rae models used in he lieraure before [5], he payoff of a swapion is he posiive par of a sum of exponenial-affine funcions in he facor. This canno simply be reduced o an opimal sopping problem for he facor process. In his paper we analyically price American swapions in he one-facor linear-raional erm srucure model inroduced in[5]. The idea of he model and imporan propery ha we exploi is ha discouned bond prices are linear in he facor. In order o fi he erm srucure of European swapions we allow for ime-varying diffusion coefficien of he facor process. Using his we obain ha he American swapion pricing problem boils down o an undiscouned opimal sopping problem for a scalar diffusion process. The laer problem is reduced o a free-boundary problem ha we ackle by he local ime-space calculus of [7]. We characerize he opimal sopping boundary as he unique soluion o a nonlinear inegral equaion ha can readily be solved numerically. Using hese boundaries we obain he American swapion price and he opimal exercise sraegies in erms of he swap rae for boh payer and receiver swapions. We provide he numerical algorihm for solving inegral equaions for he boundaries and compuing he American swapion prices. The numerical example in Secion 8 illusraes his algorihm and we solve he American swapion pricing problem for he calibraed se of parameers. We also show ha he Bermudan swapions can be efficienly priced under he linear-raional erm srucure model (Secion 7). We hen compare he prices of European, American and Bermudan swapions. The discrepancy beween American and Bermudan prices is quie small. Anoher imporan applicaion of his paper is based on he equivalence of he payoffs of a receiver (payer) swapion and a call (pu) opion on a coupon bearing bond. Hence our resuls cover he pricing of American opions on coupon bearing bonds which is iself ineresing pracical problem. Also i is well known ha American swapion can be used o conver noncallable bond o callable bond. Here is an example for illusraion. Suppose a company has issued a bond mauring in 10 years wih annual coupons of 4% on he principal amoun N and wans o add he opion o call (prepay) he bond a par (for N ) a any ime τ before mauriy dae. This opion means ha he company has he righ o prepay he principal N of he bond a ime τ and sop paying coupons aferwards. If he company canno change he original bond, i could buy an American receiver swapion wih srike rae 4% on he swap wih period [0,10]. If he company exercises he swapion a ime τ [0,10], he fixed coupon leg of he swap will hen cancel he fixed coupon paymens of he bond. On he oher hand, paying he floaing rae leg of he swap and he principal N of he bond a mauriy T = 10 is equivalen o paying N a τ, as desired. This is a modified version of Example 2.1 in [4]. The srucure of he paper is as follows. Secion 2 inroduces he linear-raional erm srucure model and ranslaes he American swapion pricing problem o an opimal sopping problem. Secion 3 reduces he opimal sopping problem o a free-boundary problem for American payer swapions. Secion 4 does he same for American receiver swapions. Secion 5 provides an algorihm for numerically solving inegral equaions arising in he soluion of he free-boundary problem. Secion 6 expresses he opimal exercise sraegies for American 2

3 swapions in erms of he underlying swap rae. Secion 7 discusses he pricing of Bermudan opions under he linear-raional framework. Secion 8 presens he numerical resuls. Finally, Secion 9 concludes and provide he agenda for fuure research. 2. Model seup and formulaion of he problem 1. We consider he one-facor linear-raional square-roo diffusion model inroduced in [5]. The facor process is a square-roo diffusion X given by (2.1) dx = κ(θ X )d+σ() X db (X 0 > 0) where B is a sandard Brownian moion sared a zero and κ and θ are posiive parameers, and σ() is ime-varying deerminisic coninuous funcion for > 0. A sufficien condiion for he absence of arbirage opporuniies in a financial marke model is he exisence of a sae price densiy: a posiive adaped process ζ such ha he price Π(,T) a ime of any cash flow C T a ime T is given by (2.2) Π(,T) = 1 ζ E [ζ T C T ]. We specify he sae price densiy as (2.3) ζ = e 0 α(s)ds (1+X ) where he funcion α : [0, ) IR is a deerminisic coninuous funcion. The main feaure of he model (2.1)-(2.3) is ha i provides racable expressions for zerocoupon bond prices P(,T) wih C T = 1 (2.4) P(,T) = E [ζ T ] ζ = e T α(s)ds 1+θ +e κ(t ) (X θ) 1+X where we used he following condiional expecaions (2.5) (2.6) E [X T ] = θ+e κ(t ) (X θ) E [ζ T ] = e T 0 α(s)ds (1+θ+e κ(t ) (X θ)) for 0 T. The funcion α can hen be chosen such ha he model-implied zero-coupon bond prices exacly mach he observed erm srucure a ime = 0 (see Appendix G in [5]). Formula (2.4) explains why his model is called linear-raional in [5]. The shor rae is obained via he relaion r = T logp(,t) T= and is given by (2.7) r = α() κ(θ X ) 1+X. Consequenly, i is bounded from below and above by (2.8) α() κθ r α()+κ. 3

4 In [5] his model has been sudied horoughly boh analyically and numerically, especially from a swap and European swapion pricing poin of view. In [5] i is also shown ha he bounds in (2.8) are no economically binding. In his paper we consider he applicaion of he linear-raional framework o he pricing of he American-syle swapions. 2. Le us now inroduce a fixed versus floaing ineres rae swap which is specified by a enor srucure of rese and paymen daes 0 < T 0 < T 1 <... < T n, where we assume ha T i T i 1 = for i = 1,..,n o be consan, and a pre-deermined fixed rae K. A each dae T i, i = 1,..,n, he fixed leg pays K and he floaing leg pays LIBOR accrued over he preceding ime period. From he perspecive of he fixed-rae payer, he value of he swap a ime T 0 is hen given by (2.9) Π swap = P(,T 0 ) P(,T n ) K P(,T j ). j=1 A payer swapion is an opion o ener ino an ineres rae swap, paying he fixed leg a a pre-deermined rae and receiving he floaing leg. A European payer swapion expiring a T 0 on a swap wih he characerisics described above has a value (payoff) a expiry T 0 (2.10) ( C T0 = (Π swap T 0 ) + = 1 P(T 0,T n ) K +. P(T 0,T j )) j=1 In he linear-raional framework he price of European payer swapion a ime T 0 wih X = x equals (2.11) V E (,x) = 1 ζ E,x [ζ T0 C T0 ] = 1 ζ E,x [p(x T0 ) + ] where he expecaion E,x is aken under condiion ha X = x and p(x) is an explici linear funcion of x. Throughou he paper, we will also use he noaion Xu,x for u when X = x. Remark 2.1. The reason we allow for ime-varying σ in (2.1) is ha we would like o fi he daa of Europeanswapion pricesacurrendae for differenmauriies T 0 > andswapion lenghs T n T 0. I can be seen from (2.4) ha σ() does no effec ZCB prices. Therefore he calibraion can be done as follows: we esimae (α,κ,θ,x 0 ) o fi he se of spo and forward swap raes, and hen calibrae σ() o mach he European swapion prices. Unforunaely, ime-varying σ complicaes he numerical analysis in his paper as he process X becomes ime-inhomogeneous and also he probabiliy densiy funcion is no available explicily anymore unlike in he case of consan σ (CIR process). However, we sill have he Fourier ransform for X and are able o perform numerical analysis. Now we define he American payer swapion as an opion o ener a any ime T beween T 0 and T n ino an ineres rae swap, paying he fixed leg a a pre-deermined rae K and receiving he floaing leg. In he Bermudan version here is finie number of daes when he holder can ener ino he swap. We will formulae he American payer swapion pricing problem 4

5 as an opimal sopping problem. For his we firs noe ha he value of he swap a ime T [T 0,T n ] is (2.12) Π swap T = m=1 ( 1 P(T,T n ) (T m T)KP(T,T m ) K j=m+1 ) P(T,T j ) 1 Tm 1 T<T m where we ake ino accoun he accrual rae beween T and nex paymen dae T m of he swap. According o he definiion of he sae-price densiy, he price of he American swapion a ime T 0 hen can be expressed as he value funcion of he opimal sopping problem (2.13) V A (T 0,x) = 1 ζ T0 [ sup E T0,x ζτ (Π swap τ ) +] T 0 τ T n where he supremum is aken over all sopping imes τ wih respec o X. In his paper we exploi a Markovian approach so ha we inroduce he following exension of (2.13) (2.14) V A (,x) = 1 [ sup E,x ζτ (Π swap τ ) +] ζ τ T n for (,x) [T 0,T n ] (0, ). Once (2.14) is deermined, one can compue he price V A (,x) a ime [0,T 0 ) as (2.15) V A (,x) = 1 ζ E,x [ ζt0 V A (T 0,X T0 ) ] using he known disribuion of X T0. 3. Now using (2.6) and (2.12) le us calculae he payoff in he opimal sopping problem (2.14) when τ [T m 1,T m ) for every m = 1,...,n (2.16) [ ζ τ (Π swap τ ) + = ζ τ E τ [ζ Tn ] (T m τ)ke τ [ζ Tm ] K j=m+1 ] + E τ [ζ Tj ] = [e τ 0 α(s)ds (1+X τ ) e Tn 0 α(s)ds (1+θ+e κ(tn τ) (X τ θ)) (T m τ)ke Tm α(s)ds 0 (1+θ+e κ(tm τ) (X τ θ)) K e T ( j 0 α(s)ds 1+θ+e κ(tj τ) (X τ θ) )] + j=m+1 ] + = [G 1m (τ)x τ +G 2m (τ) where he funcions G 1 m and G 2 m are given on inervals [T m 1,T m ) by (2.17) (2.18) G 1 m () = e 0 α(s)ds c n e κ(tn ) c m (T m )Ke κ(tm ) K j=m+1 c j e κ(t j ) G 2 m() = e 0 α(s)ds c n (1+θ θe κ(tn ) ) c m (T m )K(1+θ θe κ(tm ) ) 5

6 K c j (1+θ θe κ(tj ) ) j=m+1 = θ(ĝ1 m() G 1 m())+ĝ1 m() where we define he coefficiens c i := exp( T i α(s)ds), i = 1,...,n, and 0 Ĝ1 m G 1 m in (2.17) wih κ = 0 (2.19) Ĝ 1 m() = e 0 α(s)ds c n c m (T m )K K j=m+1 c j. are given as Therefore we can formulae he following opimal sopping problem (2.20) V(,x) = sup τ T n E,x [ G(τ,Xτ ) +] for (,x) [T 0,T n ] (0, ) and he funcion G is given by (2.21) G(,x) = ( G 1 m ()x+g 2 m ()) 1 Tm 1 <T m = G 1 ()x+g 2 () m=1 for (,x) [T 0,T n ] (0, ) and where he funcions G 1 and G 2 are given piecewisely on inervals [T m 1,T m ) by G 1 m and G2 m as (2.22) G 1 () = G 1 m ()1 T m 1 <T m and G 2 () = m=1 for [T 0,T n ]. Using (2.14), (2.16) and (2.20) we obain G 2 m ()1 T m 1 <T m m=1 (2.23) V A (,x) = V(,x)/ζ = e 0 α(s)ds V(,x)/(1+X ) for (,x) [T 0,T n ] (0, ) so ha we now focus on he problem (2.23). I is imporan o noe ha G(T n,x) = 0 for all x > 0 and hence i is no opimal o sop in he se where G 0 since wih posiive probabiliy we can ener laer ino he se where G > 0. This observaion allows us o simplify (2.23) by removing he posiive par and formulae he equivalen problem (2.24) V(,x) = sup τ T n E,x [G(τ,X τ )] for (,x) [T 0,T n ] (0, ). 4. Now we urn o he American receiver swapion which is he opion o ener a any ime T beween T 0 and T n ino an ineres rae swap, receiving he fixed leg a a pre-deermined rae K and paying he floaing leg. The value of he swap, from he perspecive of he fixedrae receiver, has he same absolue value as in (2.9) bu he opposie sign. Therefore by doing similar manipulaions as in paragraph 3 above we are delivered he following opimal sopping problem (2.25) Ṽ(,x) = inf τ T n E,x [G(τ,X τ )] 6

7 and he price of American receiver swapion is (2.26) Ṽ A (,x) = e 0 α(s)ds Ṽ(,x)/(1+X ) for (,x) [T 0,T n ] (0, ). Since he (2.24) is a minimizaion problem and G(T n,x) = 0 for all x > 0 i is obvious ha one should no sop in he se where G is posiive. Boh problems (2.24) and (2.25) have he same gain funcion, however he former is a maximizaion problem and he laer is a minimizaion problem. We will analyze (2.24) in he nex secion and will discuss briefly he soluion o (2.25) in Secion Free-boundary problem for fixed-rae payer In his secion we will reduce he problem (2.24) o a free-boundary problem and he laer will be ackled using local ime-space calculus ([7]). Firs using ha he gain funcion G(, x) is coninuous and sandard argumens (see e.g. Corollary 2.9 (Finie horizon) wih Remark 2.10 in [9]) we have ha coninuaion and sopping ses read (3.1) (3.2) C = {(,x) [T 0,T n ) (0, ) : V(,x) > G(,x)} D = {(,x) [T 0,T n ) (0, ) : V(,x) = G(,x)} and he opimal sopping ime in (2.24) is given by (3.3) τ = inf { s T n : (s,x x s ) D }. In view of he bounds (2.8) i follows ha he model implied forward and swap raes are essenially bounded frombelowandaboveby sup >0 α() κθ and inf >0 α()+κ,respecively. More precisely, hese bounds are exac when α() α is consan and close o exac when sup >0 α() is close o inf >0 α(). Hence a payer (receiver) swapion wih srike rae above inf >0 α()+κ (below sup >0 α() κθ ) would rivially have zero value. We hus henceforh assume ha he srike rae K lies in he range { inf >0 α()+κ, for a payer swapion (3.4) K sup >0 α() κθ for a receiver swapion. From 1 (iii) below we will see ha he funcion G 1, he leading erm in (2.21), is posiive on [T 0,T n ) under his condiion. If i does no hold and, say κ+α(t n ) < K, hen G 1 < 0 a leas in some neighborhood of T n and hus G 2 < 0 is negaive as well so ha G < 0 and i is no opimal o exercise he swap a ha period of ime for any value of facor process X. Moreover, he exercise se will have a very complicaed srucure and he problem (2.24) has o be ackled case by case. 1. Below we provide some properies of he funcions G 1 and G 2. (i) I is obvious ha G 1 (T n ) = G 2 (T n ) = 0. From (2.17)-(2.19) and he fac ha G 1 m Ĝ 1 m everywhere, i follows ha G 1 () > G 2 () for all [T 0,T n ). (ii) We show ha G 1 and G 2 are coninuous on [T 0,T n ), however heir derivaives, in general, disconinuous a paymen daes T m, m = 1,...,n 1. From (2.17)-(2.18) we see ha 7

8 G 1 m and G 2 m are smooh on [T m 1,T m ) for m = 1,...,n. We hen observe ha funcions G 1 and G 2 are coninuous a paymen daes T m, m = 1,...,n 1 (3.5) G 1 (T m ) = G 1 m (T m ) = c m c n e κ(tn Tm) K c j e κ(t j T m) j=m+1 = G 1 m+1 (T m) = G 1 (T m +) G 2 (T m ) = G 2 m(t m ) = θ(ĝ1 m(t m ) G 1 m(t m ))+Ĝ1 m(t m ) = θ(ĝ1 m+1 (T m) G 1 m+1 (Tm))+Ĝ1 m+1 (T m) = G 2 m+1 (T m) = G 2 (T m +) a paymen daes T m, m = 1,...,n 1. Now by sraighforward calculaions of he derivaives of G 1 m and G2 m we have (3.6) (3.7) (G 1 m ) () =κ [ c n e κ(tn ) c m (T m )Ke κ(tm ) K α()e 0 α(s)ds +c m Ke κ(tm ) j=m+1 c j e κ(t j ) ] =κ [ G 1 α(s)ds] m () e 0 α()e 0 α(s)ds +c m Ke κ(tm ) ( ) (G 2 m ) () =θ (Ĝ1 m ) () (G 1 m ) () +(Ĝ1 m ) () = (1+θ)(Ĝ1 m ) () θ(g 1 m ) () for [T m 1,T m ) and m = 1,...,n. Then i follows from (2.17)-(2.18) ha he derivaives of G 1 and G 2 generally (excep from specifically chosen se of parameers) are no coninuous a T m (3.8) (3.9) (G 1 ) (T m ) (G 1 ) (T m +) = c m K c m+1 Ke κ > 0 (G 2 ) (T m ) (G 2 ) (T m +) = K (1+θ)K +θke κ < 0 for m = 1,...,n 1. (iii) Here we show ha, due o assumpion (3.4), he funcion G 1 is posiive on [T 0,T n ). From (3.6) we have ha G 1 is increasing a poin [T m 1,T m ) if and only if (3.10) G 1 () > 1 κ [e 0 α(s)ds (κ+α()) c m Ke κ(tm ) ] := π() and hus we need o compare G 1 iself wih he funcion π which is righ-coninuous wih jumps a paymen daes T m, m = 1,...,n 1. The funcion π is posiive (3.11) π() = e 0 α(s)ds [ κ κ+α() Ke Tm ] (κ+α(s))ds > 0 for [T m 1,T m ), m = 1,...,n, by using (3.4). We hen noe ha G 1 (T n ) = 0 and π(t n ) = c n (κ + α(t n ) K)/κ > 0. Therefore we have ha G 1 < π near T n and hus G 1 is decreasing and posiive here. Then he fac ha G 1 is posiive on [T 0,T n ] can be shown by going backward from T n and using wo observaions: a) when G 1 < π he funcion G 1 goes far away from 0 and b) when G 1 > π i dominaes posiive funcion π. 8

9 (iv) Now we consider he limis of G 1 and G 2 near = T n. By L Hospial s rule we figure ou ha (3.12) (3.13) G 1 () lim T n T n = c n(α(t n )+κ K) > 0 G 2 () lim T n T n = c n( θκ+α(t n ) K). 3. In order o ge some insighs ino he srucure of sopping region D we firs need o calculae he funcion (3.14) H(,x) = (G + L X G)(,x) for (,x) [T 0,T n ) (0, ) where L X = κ(θ x)d/dx+(σ 2 /2)xd 2 /dx 2 is he infiniesimal generaor of X. The funcion H has he meaning of he insananeous benefi of waiing o exercise. By sraighforward calculaions we have ha (3.15) ( H(,x) = H 1 m ()x+hm() ) 2 1 Tm 1 <T m = H 1 ()x+h 2 () m=1 for (,x) [T 0,T n ) (0, ) where (3.16) (3.17) Hm 1 () = (κ+α())e 0 α(s)ds +c m Ke κ(tm ) = κπ() ( ) Hm 2 () = θh1 m ()+(1+θ) c m K α()e 0 α(s)ds for [T m 1,T m ), m = 1,...,n, and (3.18) H 1 () = Hm 1 ()1 T m 1 <T m and H 2 () = Hm 2 ()1 T m 1 <T m m=1 m=1 for [T 0,T n ). Therefore we have ha H(,x) is righ-coninuous wih jumps a T m, m = 1,...,n 1, for each x > 0 fixed. We also observe from (3.16) ha he funcion H 1 < 0 on [T 0,T n ) as we proved above ha he funcion π > 0 on [T 0,T n ). Since he funcion G is no C 1 a paymen daes T m, m = 1,...,n 1 wih respec o ime variable, we have o use Iô-Tanaka formula o ge E [ G(τ,Xτ,x ) ] [ τ ] (3.19) = G(,x)+E H(s,Xs.x )1 {s T m,m=1,..,n 1}ds [ τ ] = G(,x)+E H(s,Xs.x )ds for (,x) [T 0,T n ) (0, ) where he inegral erm wih respec o he local ime is no presen since he underlying process, ime, is of bounded variaion, and in he las inegral we omied he indicaor of Lebesgue-measure null se. I is obvious ha he expression (3.19) indicaes ha he se {(,x) [T 0,T n ) (0, ) : H(,x) > 0} belongs o coninuaion se C (for his one can make use of he firs exi ime from a sufficienly small ime-space ball cenred a he poin where H > 0). 9

10 4. Nex we show up-connecedness of he sopping se D. For his, le us ake wo poins (,x) and (,y) such ha [T 0,T n ) and y > x > 0, hen le us denoe by τ = τ (,y) he opimal sopping ime for V(,y) so ha using (3.19) we have (3.20) V(,x) V(,y) E [ G(τ,Xτ,x ) ] E [ G(τ,Xτ,y ) ] [ τ ( = G(,x) G(,y)+E H(s,X,x s ) H(s,Xs,y ) ) ] ds [ τ ] = G(,x) G(,y)+E H 1 (s)(xs,x Xs,y )ds G(,x) G(,y) whereinhelasinequaliy weusedfacshahefuncion H 1 isnegaiveand X,y X,x P - a.s. by he comparison principle for SDEs. Now if we ake (,x) D, i.e. V(,x) = G(,x), we have ha V(,y) = G(,y) and hus (,y) D. Therefore here exiss a funcion b : [T 0,T n ) (0, ) such ha (3.21) D = {(,x) [T 0,T n ) (0, ) : x b()}. A direc examinaion of funcions G and H in (2.21) and (3.15) imply ha here exis real-valued curves g and h on [T 0,T n ] defined as (3.22) G(,g()) = 0 and H(,h()) = 0 for [T 0,T n ) (see Figure 1) such ha G(,x) > 0 for x > g() and G(,x) < 0 for x < g(), H(,x) > 0 for x < h() and H(,x) < 0 for x > h() when [T 0,T n ) is given and fixed. I is no opimal o sop when G < 0 or H > 0 so ha we have b > g 0 and b > h 0 on [T 0,T n ) as X is always posiive. I is clear ha if x > h() 0 and < T n is sufficienly close o T n hen i is opimal o sop immediaely (since he profi obained from being below h canno compensae he cos of geing here due o he lack of remaining ime). This shows ha b(t n ) = h(t n ) 0 where (3.23) h(t n ) = H2 (T n ) H 1 (T n ) = θκ α(t n)+k α(t n )+κ K as is easily seen from (3.16)-(3.17). We also noice ha (3.24) g(t n ) = G2 (T n ) G 1 (T n ) = θκ α(t n)+k α(t n )+κ K = h(t n ) by (3.12)-(3.13). 4. Sandard Markovian argumens lead o he following free-boundary problem (for he value funcion V = V(,x) and he opimal sopping boundary b = b() o be deermined): (3.25) (3.26) (3.27) V + L X V = 0 in C V(,b()) = G(,b()) for [T 0,T n ) V x (,b()) = G x (,b()) for [T 0,T n ) 10

11 x Figure 1. A compuer drawing of he funcions g (red line) and h (blue line) defined in (3.22). The parameer se is T 0 = 1 year, = 0.5 year, n = 4, K = 0.05, θ = 2.55, κ = 0.03, α θκ = Coefficien σ is imedependen funcion and calibraed from European swapion prices. (3.28) (3.29) V(,x) > G(,x) V(,x) = G(,x) in C in D where he coninuaion se C and he sopping se D are given by (3.30) (3.31) C = {(,x) [T 0,T n ) (0, ) : x < b()} D = {(,x) [T 0,T n ) (0, ) : x b()}. I can be shown ha his free-boundary problem has a unique soluion V and b which coincide wih he value funcion (2.24) and he opimal sopping boundary respecively (cf. [9]). Compleed deails of he analysis above go beyond our goals in his paper and for his reason will be omied. I should be noed however ha one of he main issues which makes his analysis quie complicaed (in comparison wih e.g. he American pu opion problem) is ha b seems no monoone funcion of ime. This fac is suppored by numerical analysis (see Figure 2 near = 0 ). The sandard probabilisic inuiion and proof of monooniciy requires ha H < 0 however sraighforward differeniaion in (3.15) shows ha i is no rue. Proof of he coninuiy of he free boundary wihou having is monooniciy is open and challenging problem, which can help o ackle oher opimal sopping problems. In he nex secion we will derive simpler equaions which characerize V and b uniquely and can be used for financial analysis of American swapions. 5. We now provide he early exercise premium represenaion formula for he value funcion V which decomposes i ino he sum of he expeced payoff wih exercise a T n (which is zero) and early exercise premium which depends on he boundary b. The opimal sopping boundary b will be obained as he unique soluion o he nonlinear inegral equaion of Volerra ype. We denoe he following funcion (3.32) L(,u,x,z) = E,x [H(u,X u )I(X u z)] 11

12 for u 0 and x,z > 0. If he probabiliy densiy funcion p( x;u,x,) of X u under E,x is known (when σ() σ so ha X has non-cenral chi-squared disribuion, see [1]), hen L can be compued as as follows (3.33) L(,u,x,z) = z H(u, x)p( x; u, x, )d x. by univariae numerical inegraion. Oherwise, in he case of ime-dependen σ we can obain he Fourier ransform q(z; u, x, ) for X u (3.34) q(w;u,x,) = E,x [ e wx u ] = e ϕ(w,u,)+xφ(w,u,) for w C, u 0 and x > 0, where ϕ and φ are obained by solving corresponding Ricai equaions (see e.g. [3]) (3.35) (3.36) u e κ(v ) w ϕ(w,u,) = κθ 1 w v e 2 κ(v s) σ 2 (s)ds dv e κ(u ) w φ(w,u,) = 1 w u e 2 κ(u s) σ 2 (s)ds. We can hen recover he probabiliy densiy funcion p of X u as follows p( x;u,x,) = 1 (3.37) e iw x q(iw;u,x,)dw 2π for x,x 0 and u 0. Moreover, using Theorem 4 in [5] we have ha [ E,x (Xu z) +] = 1 [ ] q(µ+iλ;u,x,) (3.38) Re dλ π e (µ+iλ)z (µ+iλ) 2 IR 0 for u 0 and x,z > 0, where µ > 0 is chosen such ha q(µ;u,x,) <. Hence, we can compue L in he following way (3.39) L(,u,x,z) =E,x [H(u,X u )I(X u z)] =H 1 (u)e,x [X u I(X u z)]+h 2 (u)p,x (X u z) [ =H 1 (u)e,x (Xu z) +] +(H 2 (u)+h 1 (u)z)p,x (X u z) =H 1 (u) 1 [ ] q(µ+iλ;u,x,) Re dλ π 0 e (µ+iλ)z (µ+iλ) ( 2 1 +(H 2 (u)+h 1 (u)z) ) Im[e iλz q(iλ;u,x,)] dλ π λ 0 where in he las equaliy we used he formula for he cumulaive disribuion funcion via Fourier ransform. Thus he compuaion of L in his case requires univariae inegraion as in (3.33). The main resul of his secion may now be saed as follows. 12

13 x Figure 2. A compuer drawing of he opimal sopping boundary b() for he problem (2.24). The parameer se is T 0 = 1 year, = 0.5 year, n = 4, K = 0.05, θ = 2.55, κ = 0.03, α θκ = Coefficien σ is imedependen funcion and calibraed from European swapion prices. Theorem 3.1. The value funcion V of (2.24) has he following represenaion (3.40) V(,x) = Tn L(,u,x,b(u))du for [T 0,T n ) and x (0, ). The opimal sopping boundary b in (2.24) (see Figure 2) can be characerized as he unique soluion o he nonlinear inegral equaion (3.41) G(,b()) = Tn L(, u, b(), b(u))du for [T 0,T n ) in he class of coninuous funcions b() wih b(t n ) = θκ α(tn)+k α(t n)+κ K 0. Proof. (A) By applying he local ime-space formula on curves [7] for V(s,X s ) we have ha (3.42) V(s,X s ) = V(,x)+M s s (V + L X V)(u,X u )I(X u b(u))du s = V(,x)+M s + = V(,x)+M s + ( Vx (u,x u +) V x (u,x u ) ) I ( X u = b(u) ) dl b u (X) s s (G + L X G)(u,X u )I(X u b(u))du H(u,X u )I(X u b(u))du where we used (3.25), he definiion of H (3.14), he smooh-fi condiion (3.27) and where M = (M u ) u is he maringale erm, (l b u (X)) u is he local ime process of X spending a 13

14 he boundary b. Now upon leing s = T n, aking he expecaion E,x, using he opional sampling heorem for M, rearranging erms and noing ha V(T n,x) = G(T n,x) = 0 for all x > 0, we ge (3.40). The inegral equaion (3.41) is obained by insering x = b() ino (3.40) and using (3.26). (B) Now we show ha b is he unique soluion o he equaion (3.41) in he class of coninuous funcions b(). The proof is divided in few seps and i is based on argumens originally derived in [8]. (B.1) Le c : [T 0,T n ] IR be a soluion o he equaion (3.41) such ha c is coninuous. We will show ha hese c mus be equal o he opimal sopping boundary b. Now le us consider he funcion U c : [T 0,T n ) (0, ) IR defined as follows (3.43) U c (,x) = Tn L(,u,x,c(u))du for (,x) [T 0,T n ] (0, ). Observe he fac ha c solves he equaion (3.41) means exacly ha U c (,c()) = G(,c()) for all [T 0,T n ]. We will moreover show ha U c (,x) = G(,x) for x [c(), ) wih [T 0,T n ]. This can be derived using maringale propery as follows; he Markov propery of X implies ha (3.44) U c (s,x s ) s H(u,X u )I(X u c(u))du = U c (,x)+n s where (N s ) s Tn is a maringale under P,x. On he oher hand, we know from (3.19) (3.45) G(s,X s ) = G(,x)+ s H(u,X u )du+m s where (M s ) s Tn is a coninuous maringale under P,x. For x [c(), ) wih [T 0,T n ] given and fixed, consider he sopping ime (3.46) σ c = inf { s T n : X s c(s) } under P,x. Using ha U c (,c()) = G(,c()) for all [T 0,T n ] and U c (T n,x) = G(T n,x) = 0 for all x > 0, we see ha U c (σ c,x σc ) = G(σ c,x σc ). Hence from (3.44), (3.45) and (3.46) using he opional sampling heorem we find: [ σc ] (3.47) U c (,x) = E,x [U c (σ c,x σc )] E,x H(u,XG u )I(XG u c(u))du [ σc ] = E,x [G(σ c,x σc )] E,x H(u,X u )du = G(,x) since X u (c(+u), ) for all u [0,σ c ). This proves ha U c (,x) = G(,x) for x [c(), ) wih [T 0,T n ] as claimed. (B.2) We show ha U c (,x) V(,x) for all (,x) [T 0,T n ] (0, ). For his consider he sopping ime (3.48) τ c = inf { s T n : X s c(s) } 14

15 under P,x wih (,x) [T 0,T n ] (0, ) given and fixed. The same argumens as hose following (3.46) above show ha U c (τ c,x τc ) = G(τ c,x τc ). Insering τ c insead of s in (3.44) and using he opional sampling heorem, we ge: (3.49) U c (,x) = E,x [U c (τ c,x τc )] = E,x [G(τ c,x τc )] V(,x) proving he claim. (B.3) We show ha c b on [T 0,T n ]. For his, suppose ha here exiss [T 0,T n ) such ha b() < c() and choose a poin x [c(), ) and consider he sopping ime (3.50) σ = inf { s T n : b(s) X s } under P,x. Insering σ insead of s in (3.42) and (3.44) and using he opional sampling heorem, we ge: [ σ ] (3.51) E,x [V(σ,X σ )] = V(,x)+E,x H(u,X u )du [ σ E,x [U c (σ,x σ )] = U c (,x)+e,x H(u,X u )I ( X u c(u)) ) ] (3.52) du. Since U c V and V(,x) = U c (,x) = G(,x) for x [c(), ) wih [T 0,T n ], i follows from (3.51) and (3.52) ha: [ σ E,x H(u,X u )I ( X u c(u) ) ] (3.53) du 0. Due o he fac ha H is sricly negaive above b we see by he coninuiy of b and c ha (3.53) is no possible so ha we arrive a a conradicion. Hence we can conclude ha b() c() for all [T 0,T n ]. (B.4) We show ha c mus be equal o b. For his, le us assume ha here exiss [T 0,T n ) such ha c() < b(). Choose an arbirary poin x (c(),b()) and consider he opimal sopping ime τ from (2.24) under P,x. Insering τ insead of s in (3.42) and (3.44), and using he opional sampling heorem, we ge: (3.54) (3.55) E,x [G(τ,X τ )] = V(,x) E,x [G(τ,X τ )] = U c (,x)+e,x [ τ H(u,X u )I ( X u c(u) ) ] du where we use ha V(τ,X τ ) = G(τ,X τ ) = U c (τ,x τ ) upon recalling ha c b and U c = G eiher above c or a T n. Since U c V we have from (3.54) and (3.55) ha: [ τ E,x H(u,X u )I ( X u c(u) ) ] (3.56) du 0. Due o he fac ha H is sricly negaive above b we see from (3.56) by coninuiy of b and c ha such a poin (,x) canno exis. Thus c mus be equal o b and he proof of he heorem is complee. 15

16 Remark 3.2. I can be seen from (3.33) and (3.39) ha he cases of consan and ime-varying σ have similar numerical complexiy for solving he inegral equaion and compuing he value funcion V(T 0, ). The difference arises when one is ineresed in he American swapion price V A (,x) a < T 0, which can be compued using (2.15). Indeed, when only Fourier ransform is available, we have o firs inver i (see (3.37)) o obain he probabiliy densiy funcion p and hen perform he inegraion in (2.15). 4. Pricing problem for fixed-rae receiver In his secion we will discuss he problem (2.25) corresponding o he fixed-rae receiver. Since minimizaion problem (2.25) has he same payoff funcion G, we will only highligh differences beween problems and sae he main resul (Theorem 4.1). As in he previous secion we assume he condiion (3.4). 1. The coninuaion and sopping ses are now following (4.1) C = {(,x) [T 0,T n ) (0, ) : Ṽ(,x) < G(,x)} D = {(,x) [T0,Tn) (0, ) : (4.2) Ṽ(,x) = G(,x)} and he opimal sopping ime in (2.25) is given by (4.3) τ b = inf { s T n : (s,x s ) D }. Then we can prove he following inequaliy as in (3.20) (4.4) Ṽ(,x) Ṽ(,y) G(,x) G(,y) for y > x > 0 using he same argumens apar from ha τ = τ (,x) is now opimal sopping ime for Ṽ(,x). Now if we ake (,y) D, i.e. Ṽ(,y) = G(,y), we have ha Ṽ(,x) = G(,x) and hus (,x) D. Therefore we showed ha here is a funcion b : [T0,T n ) (0, ) such ha (4.5) D = {(,x) [T 0,T n ) (0, ) : x b()}. Since he problem (2.25) is he minimizaion one, we should no sop when G > 0 or H < 0, i.e. we have ha b < g and b < h on [T 0,T n ), where g and h from (3.22). The erminal value of b is b(t n ) = g(t n ) 0 = h(t n ) 0 = ( (θκ α(t n )+K)/(α(T n )+ακ K) )+ = b(t n ). 2. Sandard Markovian argumens lead o he following free-boundary problem (for he value funcion Ṽ = Ṽ(,x) and he opimal sopping boundary b = b() o be deermined): (4.6) (4.7) Ṽ + L X Ṽ = 0 in C Ṽ(, b()) = G(, b()) for [T 0,T n ) 16

17 (4.8) (4.9) (4.10) Ṽ x (, b()) = G x (, b()) for [T 0,T n ) Ṽ(,x) < G(,x) in C Ṽ(,x) = G(,x) in D where he coninuaion se C and he sopping se D are given by (4.11) (4.12) C = {(,x) [T 0,T n ) (0, ) : x > b()} D = {(,x) [T 0,T n ) (0, ) : x b()}. Icanbeshownhahisfree-boundaryproblemhasauniquesoluion Ṽ and b which coincide wih he value funcion (2.25) and he opimal sopping boundary respecively. As for he problem (2.24) in previous secion, numerical drawings show ha b is no monoone funcion of ime (see Figure 3). I can be inuiively explained as follows: H > 0 on inervals (T m 1,T m ) (which is sufficien condiion for exhibiing increasing boundary) however a paymen daes T m he funcion H exhibis jumps down (which makes he boundary decreasing on he lef-side neighborhoods of T m ). 3. We now provide he early exercise premium represenaion formula for he value funcion Ṽ which decomposes i ino he sum of he expeced payoff if we do no exercise unil T n (which is zero) and early exercise premium depending on b. The opimal sopping boundary b will be obained as he unique soluion o he nonlinear inegral equaion of Volerra ype. We will make use of he following funcion in Theorem 4.1 below (4.13) L(,u,x,z) = E,x [H(u,X u )I(X u z)] for u 0, x,z > 0, and if X u has he known probabiliy densiy funcion p, hen (4.14) L(,u,x,z) = z 0 H(u, x)p( x; u, x, )d x. Oherwise, we exploi Fourier ransform as in he previous secion. The main resul of his secion is saed below and provided wihou proof since i is very similar o Theorem 3.1. Theorem 4.1. The value funcion Ṽ of (2.25) has he following represenaion (4.15) Ṽ(,x) = Tn L(,u,x, b(u))du for [T 0,T n ) and x (0, ). The opimal sopping boundary b in (2.25) (see Figure 3) can be characerized as he unique soluion o he nonlinear inegral equaion (4.16) G(, b()) = Tn L(,u, b(), b(u))du for [T 0,T n ) in he class of coninuous funcions wih b(t n ) = θκ α(tn)+k α(t n)+κ K 0. 17

18 x Figure 3. A compuer drawing of he opimal sopping boundary b() for he problem (2.25). The parameer se is T 0 = 1 year, = 0.5 year, n = 4, K = 0.05, θ = 2.55, κ = 0.03, α θκ = Coefficien σ is imedependen funcion and calibraed from European swapion prices. 5. Numerical soluion o inegral equaions In his secion we provide an algorihm for numerical soluion of he inegral equaions (3.41) and (4.16), and compuing he swapion prices (3.40) and (4.15). In order o obain he prices of American swapions (3.40) and (4.15) we need o solve numerically inegral equaions of Volerra ype (3.41) and (4.16). We proved above ha b and b are unique soluions o he equaions (3.41) and (4.16), respecively. These equaions canno be solved analyically bu can be ackled numerically in an efficien way. The following simple mehod can be used o illusrae he laer (see e.g. Chaper 8 in [2]). Se k = kh for k = 0,1,...,N where h = (T n T 0 )/N so ha he following discree approximaions of he inegral equaions (3.41) and (4.16), respecively, are valid: (5.1) (5.2) N 1 G( k,b( k )) = h L ( k, l+1,b( k ),b( l+1 ) ) l=k N 1 ( ) G( k, b( k )) = h L k, l+1, b( k ), b( l+1 ) l=k for k = 0,1,...,N 1. Seing k = N 1 and b( N ) = b( N ) = θκ α(tn)+k α(t n)+κ K 0 we can solve he equaions (5.1) and (5.2) numerically and ge numbers b( N 1 ) and b( N 1 ), respecively. Seing k = N 2 and using he values b( N 1 ), b( N ) and b( N 1 ), b(n ), we can solve (5.1) and (5.2) numerically and ge numbers b( N 2 ) and b( N 2 ), respecively. Coninuing he recursion we obain b( N ),b( N 1 ),...,b( 1 ),b( 0 ) and b( N ), b( N 1 ),..., b( 1 ), b( 0 ) as approximaions of he opimal boundaries b and b, respecively, a he poins T n,t n h,...,t 0 +h,t 0 (see Figures 2 and 3 above). We noe ha we solve separaely he equaions for he boundaries b and b. 18

19 (5.3) (5.4) Finally, he prices of American swapions (3.40) and (4.15) can be approximaed as follows: for k = 0,1,...,N 1 and x > 0. N 1 V( k,x) = h L ( k, l+1,x,b( l+1 ) ) l=k N 1 Ṽ( k,x) = h L ( k, l+1,x, b( l+1 ) ) l=k 6. Opimal exercise boundaries for swap raes The formulas (3.40) and (4.15) provide he prices of American swapions for floaing-rae receiver and fixed-rae receiver, respecively. However, he opimal sopping boundaries b and b in (3.41) and (4.16) provide he opimal exercise sraegies in erms of he laen facor process X ha is no direcly observable in he financial marke in general. Therefore our goal now is o connec he process X wih some observable financial objec and he naural choice is he swap rae of he underlying swap conrac. Le [T 0,T n ] such ha T m 1 < T m for some m = 1,...,n, hen le us consider he swap wih fuure paymens a T m,t m+1,...,t n. The swap rae S is he fixed rae which makes Π swap = 0 and hence (6.1) S = 1 P(,T n ) (T m )P(,T m )+ n j=m+1 P(,T j) and by recalling and insering (2.4) we ge he following relaionship beween S and X : (6.2) where S = f 1(,X ) f 2 (,X ) =: f(,x ) (6.3) (6.4) [ f 1 (,X ) =X 1 e ] Tn (κ+α(s))ds +1 e Tn [ f 2 (,X ) =X (T m )e Tm (κ+α(s))ds + j=m+1 +(T m )e ( Tm α(s)ds 1+θ θe κ(tm )) + e T ( j α(s)ds 1+θ θe ) κ(t j ) j=m+1 α(s)ds (1+θ)+θe Tn (κ+α(s))ds e T j (κ+α(s))ds ] for [T 0,T n ]. Now if we look ino he map X P(,T;X ) in (2.4) we see by direc differeniaion ha i is sricly decreasing in X and herefore using (6.1) we have ha X f(,x ) is sricly increasing. Thus here is one-o-one relaionship beween S and X and we have ha (6.5) x b() f(,x) f(,b()) 19

20 (6.6) x b() f(,x) f(, b()) for [T 0,T n ) so ha he opimal exercise sraegies for fixed-rae payer and fixed-rae receiver in erms of swap rae S, respecively, are given as follows (6.7) (6.8) τ = inf { T 0 s T n : S s R(s) } τ = inf { T 0 s T n : S s R(s) } where he opimal exercise boundaries R and R are given as (6.9) R() = f(,b()) R() = f(, b()) for [T 0,T n ]. I can be seen ha R(T n ) = R(T n ) = K. We also noe ha alernaively one can work wih he exercise boundaries relaed o he shor ineres rae r using (2.7) as here is one-o-one relaionship beween r and X as well. 7. Bermudan swapion In his secion, we discuss how he Bermudan swapions can be priced under he linearraional framework (2.1). The case of he fixed-rae receiver is symmeric and can be analyzed in he same way. In conras wih American swapions, Bermudan conracs can be exercised only a cerain number of daes. Usually, hese swapions can be execued a he mauriy T 0 of opion or a he paymen daes of underlying swap, i.e., a T i, i = 1,...,n. To keep he valuaion as general as possible, we assume ha se of possible exercise daes is represened by finie se T B = { j,j = 0,...,m} wih 0 = T 0 and m = T n. Therefore he Bermudan fixed-rae payer solves he following discree ime opimal sopping problem (7.1) V B (,x) = 1 ζ sup τ T B E,x [G(τ,X τ )] for (,x) T (0, ), where he supremum is aken over all X -sopping imes τ wih values in he se T B, and he funcion G is given in (2.21). Once (7.1) is solved, he price a ime [0,T 0 ) can be hen compued as V B (7.2) V B (,x) = 1 ζ E,x [ ζt0 V B (T 0,X T0 ) ] using he known disribuion of X T0. Obviously, V E (, ) V B (, ) V A (, ) for any [0,T 0 ]. There are a leas wo sandard ways o ackle he discree ime opimal sopping problems: Mone-Carlo simulaion and backward inducion. The former mehod is more appropriae when he model is no quie racable or dimension is high. The laer approach is accurae and efficien in low dimensions and when he marginal disribuions are given. I can be seen below han he linear-raional model allows us o use he backward inducion mehod. According 20

21 o he sandard procedure of backward inducion(see Chaper 1 in [9]), he sequence of value funcions can be obained as follows (7.3) V B ( j,x) = max ( [ G( j,x),e j,x V B ( j+1,x )]) j+1 for j = 0,...,m 1 and x > 0 saring from V B ( m, ) = V B (T n, ) = G(T n, ) = 0. The opimal exercise sraegy in (7.1) saring from T 0 is given as (7.4) τ B = τb (T 0,x) = inf{ j T B : V B ( j,x j ) = G( j,x j )}. The recursive formulas (7.3) do no allow for closed form expressions. However, one can perform he following simple numerical scheme. Firs, we runcae he sae space of X by he inerval [0, X] and discreize i 0 = x 0 < x 1 <... < x N = X for some N > 0 and h X = x n x n 1 for n = 1,...,N. Then we approximae he formula (7.3) as ( N ) (7.5) V B ( j,x n ) max G( j,x n ),h X V B ( j+1,x i )p(x i ; j+1,x n, j ) for j = 0,...,m 1 and n = 1,...,N. Sandard argumens can be applied o show convergence resuls. If we hink of V B (j) V B ( j, ) as he N -dimensional vecor and P(j) p( ; j+1,, j ) as he N N -marix for j = 0,...,,m, hen we can rewrie (7.5) in he vecorized form. Thus (7.5) becomes he sequence of recursive vecor equaions. We noe ha in he case of consan σ, we have a single marix P as he process X is ime-homogeneous and corresponding probabiliies can be obained from he known densiy funcion of non-cenral chi-square disribuion. Then, he algorihm works exremely fas. Oherwise, if σ is ime-varying, hen he compuaions becomes more involved. Firs, we have m disinc marices of ransiional probabiliies due o ime-inhomogeneiy of X and secondly we have o inver he Fourier ransform. However, if he number m of possible exercise daes is no so large (as we menioned above, in pracice he Bermudan swapion can be exercised wice a year on he same daes as he underlying swap paymens occur), hen his mehod i sill feasible. Once he family of marices is compued, he recursive formulas (7.5) can be easily implemened. In he nex secion, we provide he numerical resuls for he case of ime-varying σ() which we calibrae from he European swapions daa. i=1 8. Numerical resuls As an applicaion of Theorems 3.1 and 4.1 wih he numerical algorihm described above, we use he parameers (α,κ,θ,σ()) calibraed o he Euribor swap and swapions marke. Specifically, we se θ = 2.55, κ = 0.03, α θκ = so ha ineres raes are bounded below by zero and assumpion (3.4) holds. The coefficien σ is a ime-dependen funcion and calibraed from European swapion prices. The underlying swap sars a T 0 = 1 and has four subsequen semiannual paymens, i.e. = 0.5, n = 4, T n = 3. The nominal value of he swap is 1 million and he fixed rae is K = 0.05, We are ineresed in he prices of American, European and Bermudan swapions a ime = 0. We obain he opimal sopping boundaries b and b on [T 0,T n ] (see Figures 2 and 3) as soluions o (3.41) and (4.16) using he algorihm above. We consider he a-he-money swap, 21

22 x Figure 4. A compuer drawing of he opimal exercise boundaries r (upper line) and r (lower line) in erms of he swap rae S. The parameer se is T 0 = 1 year, = 0.5 year, n = 4, K = 0.05, θ = 2.55, κ = 0.03, α θκ = Coefficien σ is ime-dependen funcion and calibraed from European swapion prices. i.e. S 0 = K = 0.05 a = 0, hen we solve f(0,x 0 ) = S 0 so ha he iniial facor value is X 0 = and hen using (2.23) wih (5.3) and (2.26) wih (5.4) we ge he American swapion prices V A (0,X 0 ) = and Ṽ A (0,X 0 ) = , respecively, a ime = 0. Figure 4 shows a compuer drawing of he opimal exercise boundaries R and R for swap rae based on curves b and b from Figures 2 and 3. We observe ha he lower boundaries b and R are no monoone in. I is remarkable ha he boundaries b and b are no smooh a he paymen daes T m, m = 1,...,n 1, which is a very rare siuaion in he opimal sopping heory. This fac is caused by he disconinuiy of H(,x) a T m, m = 1,...,n 1, for fixed x > 0. Finally, we price he fixed-rae payer European swapion a = 0 using (2.11). We hen consider he fixed-rae payer Bermudan swapion ha can be exercised only a underlying swap paymen daes. Figure 5 shows he values of he fixed-rae payer European, Bermudan and American swapions a = 0 as he funcions of x. As expeced, American price is upper bound for Bermudan swapion price. The prices are close o each oher. 9. Conclusion The modeling of he sae-price densiy in he linear-raional framework allows us o formulae he American swapion problem as he undiscouned opimal sopping problem for a one-dimensional square-roo diffusion process. We characerize he opimal sopping boundaries b and b as he unique soluion o nonlinear inegral equaions and using his we obain he arbirage-free prices of he American swapions (see (3.40) and (4.15)) and he opimal exercise sraegies in erms of swap raes (see (6.7)-(6.8) and Figure 4). The opimal sopping boundaries are no differeniable a paymen daes according o numerical soluions, which is a rare siuaion in he lieraure on he opimal sopping heory. 22

23 V(x) x Figure 5. The swapion prices of American ype V A (black solid), Bermudan ype V B (blue line) and European ype V E (dashed) a = 0. Bermudan swapion can be exercised a swap paymen daes T m only. The parameer se is T 0 = 1 year, = 0.5 year, n = 4, K = 0.05, θ = 2.55, κ = 0.03, α θκ = Coefficien σ is ime-dependen funcion and calibraed from European swapion prices. Muli-facor facor models end o empirically ouperform one-facor models (see [5] for deails). If one considers LRSQ(m,n) specificaion wih (m+n) -dimensional facor process X, hen he corresponding pricing problem is reduced o he muli-dimensional sopping problem (9.1) V(,x) = sup τ T n E,x [G(τ,X τ )] for (,x) [T 0,T n ] (0, ) m+n. The funcion G is affine in facor and is given by (9.2) G(,x) = G 1 ()x G m+n ()x m+n +G 0 () for (,x) [T 0,T n ] (0, ) m+n for some known funcions G i, i = 0,1,...,m+n. We can ackle (9.1) numerically using, e.g., inegral equaion approach (for low dimensions), and backward inducion(see Secion 7) or Mone-Carlo mehods(for higher dimensions). Therefore his leads o an exensive program of research of American swapions which we aim o presen in subsequen publicaions. References [1] Cox, J., Ingersoll, J. and Ross, S. (1985). A heory of he erm srucure of ineres raes. Economerica 53 ( ). [2] Deemple, J. (2006). American-Syle Derivaives. Chapman & Hall/CRC. [3] Filipović, D. (2005). Time-inhomogeneous affine processes sochasic processes and heir applicaions. Sochasic Process. Appl. 115 ( ). 23

24 [4] Filipović, D. (2009). Term-Srucure Models. Springer-Verlag, Berlin. [5] Filipović, D., Larsson, M. and Trolle A. (2017). Linear-raional erm srucure models. J. Finance. 72 ( ) [6] Longsaff, F.A., Sana-Clara, P. and Schwarz, E.S. (2001). Throwing away a billion dollars: he cos of sub-opimal exercise sraegies in he swapion marke. J. Financ. Econ. 62 (39 66). [7] Peskir, G. (2005). A change-of-variable formula wih local ime on curves. J. Theore. Probab. 18 ( ). [8] Peskir, G. (2005). On he American opion problem. Mah. Finance 15 ( ). [9] Peskir, G. and Shiryaev, A. N. (2006). Opimal Sopping and Free-Boundary Problems. Lecures in Mahemaics, ETH Zürich, Birkhäuser. 24