Pricing of Options on two Currencies Libor Rates

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1 Prcng o Optons on two Currences Lbor Rates Fabo Mercuro Fnancal Models, Banca IMI Abstract In ths document we show how to prce optons on two Lbor rates belongng to two derent currences the ormer s domestc, the latter oregn). To ths end, we explctly derve the dynamcs o the oregn rate under the domestc orward measure assocated to the rate maturty. We then consder the undamental case o an opton wrtten on the spread between the two Lbor rates and derve closed orm ormulas or both the up-ront and the n-arrears cases. Explct ormulas are also derved or optons on the product o the two rates as well as or trgger swaps.. Assumptons and Dentons Gven a domestc market and a oregn market, let us assume that the term structures o dscount actors that are observed n the domestc and oregn markets at tme t are respectvely gven by T P t, T ) and T P t, T ) or T t. Let us denote by X t) the exchange rate at tme t between the currences n the two markets, n that unt o the oregn currency equals X t) unts o the domestc currency. Gven the uture tmes T and T, =,..., n, the domestc and oregn orward rates at tme t or the nterval T, T ] are, respectvely, F t) = F t; T, T ) = P t, T ) P t, T ) τ P t, T ) F t) = F t; T, T ) = P t, T ) P t, T ) τ P t, T ) where τ s the year racton between tmes T and T, whch s assumed to be the same n both markets. Denotng by F X t, T ) the orward exchange rate at tme t or maturty T, F X t, T ) = X t) P t, T ) P t, T ),

2 Optons on two Lbor rates and assumng constant proportonal) volatltes, the two orward rates evolve under the domestc orward measure Q accordng to see Brgo and Mercuro, 00, Sectons 6.3 and.4) df t) = σ F t) dw t), df t) = F t) ρσ F X σ dt + σ dw t)], where W and W are two standard Brownan motons wth nstantaneous correlaton ρ, ρ s the nstantaneous correlaton between F X, T ) and F ), and σ F X s the assumed constant proportonal) volatlty o the orward exchange rate F X t, T ): df X t, T ) = σ F X F X t, T ) dw X t), where W X s a standard Brownan moton under Q, wth dw X t)dw t) = ρ dt. Let us consder a dervatve whose payo at tme T s a uncton gf T ), F T )). By ormula.) n Brgo and Mercuro 00), the no-arbtrage value at tme t o such a payo s P t, T )E { gf T ), F T )) F t }, ) where E denotes expectaton under Q and F t s the σ-eld generated by the par F, F ) up to tme t.. Spread Optons A spread opton on the two Lbor rates LT, T ) and L T, T ) s a dervatve payng o at tme T, n domestc currency, τ N ω LT, T ) L T, T ) + )] + = τ N ω F T ) F T ) + )] +, ) where N s the nomnal value, s the contract margn and ω = or a call and ω = or a put. An n-arrears spread opton pays o the same quantty at tme T. Ths s equvalent to payng o at tme T τ N ω F T ) F T ) + )] + + τ F T ) ). 3) The two payos ) and 3) can be summarzed nto τ N ω F T ) F T ) + )] + + ψτ F T ) ), 4) where ψ = or the n-arrears case and ψ = 0 otherwse. Notce that F X s a martngale under Q.

3 Optons on two Lbor rates Proposton.. The no-arbtrage value at tme t o the payo 4) s gven by + )] LSOt, T, T, τ, N,, ω, ψ) = τ NP t, T ) e v + ψτ hv) v) dv, π 5) where v) = ωf t)eµ y+ρ σ y v+ σ y ρ ) Φ ω ln F t) + µ hv) y + ρ σ y v + σy ρ ) σ y ρ +ωhv)φ ω ln F t) + µ hv) y + ρ σ y v {hv)>0} σ y ρ + ω) {hv) 0} hv) + F t)eµ y+ρ σ y v+ σ y ρ ) ] wth A denotng the ndcator uncton o the set A, Φ ) denotng the standard normal cumulatve dstrbuton uncton, and hv) = + F t)e µ x+ v µ x = µ y = ρσ F X σ σ y = σ τ σ y = σ τ τ = T t Proo. By ormula ), the no-arbtrage value at tme t o the payo 4) s Denng τ NP t, T )E { ω F T ) F T ) + )] + + ψτ F T ) ) F t }. 6) X := ln F T ), F t) Y := ln F T ) F t), the jont densty uncton X,Y o X, Y ) under the measure Q s bvarate normal wth mean vector and varance-covarance matrx respectvely gven by ] ] µx σx M X,Y =, V X,Y = ρ σ y ρ σ y µ y 3 σ y

4 Optons on two Lbor rates that s X,Y x, y) = exp π σ y ρ ) x µ x x µ x y µ y ρ σ y + y µy ρ ) σ y ). It s well known that where Y X x, y) = X x) = X,Y x, y) = Y X x, y) X x), exp σ y π ρ π exp y µy σ y ) ] x µx. ρ x µ x ρ ) ) The expectaton n 6) can thus be wrtten as + + ψτ F t)e x) + ] + ωf t)e x ωf t)ey + ω) Y X x, y) dy X x) dx The expresson between square brackets can be calculated analytcally by dstngushng two cases:. F t)e x + 0. I ω =, the expresson s equal to 0 the postve part o a negatve number s zero). I ω =, nstead, ] + = F t)e x + F t) e y Y X x, y) dy. F t)e x + > 0. = F t)e x + F t)eµ y+ρ σ x µx y σx + σ y ρ ) Set := F t)e x + and ω := ω. Then ] + + = ωf t)ey ω) Y X x, y) dy = ωf t)eµ y+ρ σ x µx y σx + σ y ρ ) Φ ω ln F t) + µ x µ F t)e x + y + ρ y + σy ρ ) σ y ρ ωf t)e x + )Φ ω ln F t) + µ x µ F t)e x + y + ρ y σ x σ y ρ by ormula B.) n Appendx B o Brgo and Mercuro 00). Fnally, to obtan 5), we smply have to set v := x µ x )/. 4 7)

5 .3 Optons on the Product Optons on two Lbor rates The second example we consder s that o an opton wrtten on the product o the two Lbor rates LT, T ) and L T, T ), whose payo at tme T, n domestc currency, s τ N ω LT, T )L T, T ) )] + = τ N ω F T )F T ) )] +, 8) where N s the nomnal value, s the strke prce and ω = or a call and ω = or a put. Proposton.. The no-arbtrage value at tme t o the payo 8) s gven by LPt, T, T, τ, N,, ω) = τ NP t, T ) ωf t)f t)e ρσ F X σ +ρ σ σ ]τ Φ ωφ ω ln F t)f Proo. Snce t) ω ln F t)f t) ρσ F X σ + σ + σ )] τ σ + σ ) + ρ σ σ ]τ + ρ σ σ ρσ F X σ + σ + σ )] τ σ + σ ) + ρ σ σ ]τ F T )F T ) = F t)f t)e ρσ F X σ + σ + σ ) ]τ+σ W T ) W t)]+σ W T ) W t)], we have that, under Q, ln F T )F T ] ) F t N M, V ), M = ln F t)f t)] ρσ F X σ + σ + σ ) ]τ, V = σ + σ ) + ρ σ σ ]τ. To obtan 9), we smply have to remember ) and apply ormula B.) n Appendx B o Brgo and Mercuro 00)..4 Trgger swaps The nal example we consder s that o a swap where, n one leg, derent payments are trggered by derent levels o ether the domestc or the oregn Lbor rates. In ormulas, a leg o the trgger swap pays o at tme T, n domestc currency, ether af τ N T ) + bf T ) + c ) {ωf T ) ω}] + ψτ F T ) ), 0) or, n case the payment s trggered by the oregn rate, af τ N T ) + bf T ) + c ) ] {ωf T ) ω} + ψτ F T ) ), ) 5 9)

6 Optons on two Lbor rates where N s the nomnal value, a, b, c are real constants speced by the contract, ω s ether or, ψ = or the n-arrears case and ψ = 0 otherwse. Proposton.3. The no-arbtrage value at tme t o the payo 0) s gven by TSDt, T, T, τ, N,, ω, ψ) = τ NP t, T ) a + cψτ )F t)φ ω ln F t) + ) σ σ + aψτ F t)e σ Φ ω ln F t) + ) 3 σ + cφ ω ln F t) ) σ σ σ + bf t)e ρσ F X σ Φ ω ln F t) + ρ σ σ ] ) σ σ +bψτ F t)f t)e ρσ F X σ +ρ σ σ ]τ Φ ω ln F t) + ρ σ σ + σ ] τ σ τ The no-arbtrage value at tme t o the payo ) s nstead gven by TSFt, T, T, τ, N,, ω, ψ) = τ NP t, T ) cφ ω ln F t) ] ρσ F X + σ σ σ + a + cψτ )F t)φ ω ln F t) ρσ F X + σ ρ ] σ σ σ + aψτ F t)e σ Φ ω ln F t) ρσ F X + σ ρ ] σ σ σ + bf t)e ρσ F X σ Φ ω ln F t) + ] ρσ F X + σ σ σ +bψτ F t)f t)e ρσ F X σ +ρ σ σ ]τ Φ ω ln F t) + ρσ F X + σ + ρ ] σ σ σ. )]. ) 3) Proo. The proo s qute smlar n sprt to that o Proposton. and s thereore omtted. The only derence s that here the outer ntegral, n both cases, can be explctly calculated, too. Reerences ] D. Brgo and F. Mercuro 00). Interest Rate Models: Theory and Practce. Sprnger Fnance, Hedelberg. 6

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