PRICING and STATIC REPLICATION of FX QUANTO OPTIONS

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1 PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of one uni of foreign currency. r d τ : he (deerminisic) domesic insananeous risk-free rae a ime τ. r f τ : he (deerminisic) foreign insananeous risk-free rae a ime τ. σ τ : he exchange rae (deerminisic) percenage volailiy a ime τ. : a srike price. ω: a flag for call (ω = 1) or pu (ω = 1). T,, T 2 : fuure imes. Q d : he domesic risk-neural measure. E d : expecaion under Q d. Q N : he probabiliy measure associaed wih he numeraire N. E N : expecaion under Q N. F τ : he σ-algebra generaed by S up o ime τ. 1 A : he indicaor funcion of he se A. C(, T, ): price a ime of a (plain-vanilla) call opion wih mauriy T and srike. P(, T, ): price a ime of a (plain-vanilla) pu opion wih mauriy T and srike. AoNC(, T, ): price a ime of an asse-or-nohing call wih mauriy T and srike. AoNP(, T, ): price a ime of an asse-or-nohing pu wih mauriy T and srike. QO(, T,, ω): price a ime of a quano opion wih mauriy T and srike. FSQO(,, T 2, ω): price a ime of a forward-sar quano opion wih forward-sar dae and mauriy T 2. QCq(,, T 2, ω): price a ime of a quano clique opion wih forward-sar dae and mauriy T Assumpions The exchange rae S is assumed o evolve under he domesic risk-neural measure Q d according o: ds τ = S τ [(r d τ r f τ ) dτ + σ τ dw τ 1

r f τ : he (deerminisic) foreign insananeous risk-free rae a ime τ. σ τ : he exchange rae (deerminisic) percenage volailiy a ime τ. : a srike price. ω: a flag for call (ω = 1) or pu (ω = 1).

2 where W is a sandard Brownian moion under Q d. Seing S τ = S τ exp ( τ rf u du ), he dynamics of S under he measure Q S having S as numeraire is ds τ = S τ [(r d τ r f τ + σ 2 τ) dτ + σ τ d W τ (1) where W is a sandard Brownian moion under Q S. 1.3 Pricing The no-arbirage price a ime of he payoff H T a ime T is H = e RT r d u du E d [H T F Using S as numeraire, he ime -price becomes [ H = S E S HT F S T 2 Quano Opions Pricing of a Quano Opion S[ (2) = S e RT ru f du HT E F S T A quano opion pays ou a mauriy T he amoun [ω(s T ) + in foreign currency, which is equivalen o [ω(s T ) + S T in domesic currency: [ω(s T ) + S T T To price he payoff H T = [ω(s T ) + S T i is convenien o use formula (2). In fac QO(, T,, ω) = S e RT r f u du E S[(ωS T ω) + F This expecaion can be easily calculaed under (1), since i is equivalen o an nondiscouned Black-Scholes price for an underlying asse paying a coninuous dividend yield q τ = rτ f στ. 2 We hus obain: QO(, T,, ω) = ωs e RT d = ln S ru f du [S e RT (rd u ru+σ f u) 2 du Φ(ωd ) Φ(ωd 1 ) + T (rd u ru f σ2 u) du T σ u 2 du T (3) d 1 = d 2

3 Pricing The no-arbirage price a ime of he payoff H T a ime T is H = e RT r d u du E d [H T F Using S as numeraire, he ime -price becomes [ H = S E S HT F S T 2 Quano Opions Pricing of a Quano Opion

3 Saic Replicaion of a Quano Opion In he call opion case, we have (S T ) + S T = S {ST >K} dk = 2 (S T K) + dk + (S T ) + (4) Therefore, a quano call can be saically replicaed by means of asse-or-nohing calls or, equivalenly, plain-vanilla calls as follows: QO(T, T,, 1) = AoNC(T, T, K) dk = 2 In he pu opion case, we have insead ( S T ) + S T = S {K>ST } dk = ( S T ) + 2 C(T, T, K) dk + C(T, T, ) (K S T ) + dk Therefore, a quano pu can be saically replicaed by means of asse-or-nohing pus or, equivalenly, plain-vanilla pus as follows: QO(T, T,, 1) = AoNP(T, T, K) dk = P(T, T, ) 2 P(T, T, K) dk 3 Forward-Sar Quano Opions Pricing of a Forward-Sar Quano Opion A forward-sar quano opion pays ou a mauriy T 2 > he amoun [ω(s T2 S T1 ) + in foreign currency, which is equivalen o [ω(s T2 S T1 ) + S T2 in domesic currency: [ω(s T2 S T1 ) + S T2 T 2 Since we can wrie FSQO(,, T 2, ω) = e R r d u du E d [QO(, T 2, S T1, ω) F using formula (3) and calculaing he (risk-neural) second momen of S T1 condiional on 3

+ C(T, T, ) (K S T ) + dk Therefore, a quano pu can be saically replicaed by means of asse-or-nohing pus or, equivalenly, plain-vanilla pus as follows: QO(T, T,, 1) = AoNP(T, T, K) dk = P(T, T, ) 2

4 F, we obain FSQO(,, T 2, ω) = ωs 2 e R (ru r d u+σ f u) 2 du RT 2 ru f [ert du 2 T (r d 1 u ru+σ f u) 2 du Φ(ωd ) Φ(ωd 1 ) d = T2 (ru d ru f σ2 u) du σu 2 du d 1 = d (5) Saic Replicaion of a Forward-Sar Quano Opion The saic replicaion of he value a ime of a forward-sar quano opion boils down o he saic replicaion of S 2, boh in he call and pu cases. We hen use (4), wih = and T =, hus obaining S 2 = S T1 1 {ST1 >K} dk = 2 (S T1 K) + dk Therefore, he squared exchange rae can be saically replicaed by means of asse-ornohing calls or, equivalenly, plain-vanilla calls as follows: S 2 = AoNC(,, K) dk = 2 C(,, K) dk Remark 3.1. If he evaluaion ime lies, insead, in he inerval (, T 2 ), a forward-sar quano opion is equivalen o a quano opion wih a given srike (he previously se S T1 ). We hen refer o he previous secion for is pricing and replicaion. 4 Quano Cliques Pricing of a Quano Clique A quano clique opion pays ou a mauriy T 2 > he amoun [ω(s T2 S T1 )/S T1 + in foreign currency, which is equivalen o [ω(s T2 S T1 )/S T1 + S T2 in domesic currency: [ω S T 2 S T1 S T1 + S T2 T 2 4

We hen use (4), wih = and T =, hus obaining S 2 = S T1 1 {ST1 >K} dk = 2 (S T1 K) + dk Therefore, he squared exchange rae can be saically replicaed by means of asse-ornohing calls or, equivalenly,

5 Since he ime T 2 -payoff of a quano clique is equal o ha of he corresponding forwardsar quano opion divided by S T1, he same applies o he corresponding values a ime : QCq(,, T 2, ω) = FSQO(,, T 2, ω) S T1 By (5), he calculaion of he ime -price boils down o he calculaion of he (risk-neural) expecaion of S T1 condiional on F. We obain QCq(,, T 2, ω) = ωs e RT 2 ru f [ert du 2 T (r d 1 u rf u+σu 2) du Φ(ωd ) Φ(ωd 1 ) d = T2 (ru d ru f σ2 u) du (6) d 1 = d Saic Replicaion of a Quano Clique The quano clique value a ime is linear in S T1. A saic replicaion is hen achieved by buying a proper amoun of foreign currency S. Remark 4.1. If he evaluaion ime lies, insead, in he inerval (, T 2 ), a quano clique is equivalen o a consan by a quano opion wih a given srike, where he inverse of he consan and he srike are equal o he known value of S T1. We hen refer o he relaed secion for is pricing and replicaion. 5

We obain QCq(,, T 2, ω) = ωs e RT 2 ru f [ert du 2 T (r d 1 u rf u+σu 2) du Φ(ωd ) Φ(ωd 1 ) d = T2 (ru d ru f + 3 2 σ2 u) du (6) d 1 = d Saic Replicaion of a Quano Clique The quano clique value a ime

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ