Premia 14 Closed Formula Methods

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1 19 pages 1 Premia 14 Closed Formula Methods Routine cf_spm_nig.c Routine cf_hullwhite1d_zbputeuro.c Routine cf_hullwhite1d_zcbond.c Routine cf_hullwhite1d_payerswaption.c Routine cf_hullwhite1d_zbcalleuro.c Routine cf_hullwhite1d_floor.c Routine cf_hullwhite1d_receiverswaption.c Routine cf_hullwhite1d_cap.c Routine cf_libor_affine_cir1d_capfloor_fourier.c Routine cf_libor_affine_cir1d_swaption_fourier.c Routine cf_libor_affine_cir1d_capfloor_direct.c Routine cf_libor_affine_cir1d_swaption_direct.c Routine cf_carr_bns.c Routine cf_attari_bns.c Routine cf_callma.c Routine cf_echange.c Routine cf_putmin.c Routine cf_bharchiarella1d_zcbond.c Routine cf_cgmy_varianceswap.c Routine cf_rogersveraart2.c Routine cf_merhes_varianceswap.c Routine cf_carr_merhes.c Routine cf_attari_merhes.c Routine cf_put_merhes.c Routine cf_call_merhes.c Routine cf_swaptionhw2d.c Routine cf_caphw2d.c Routine cf_zcbondhw2d.c Routine cf_zbputeurohw2d.c Routine cf_floorhw2d.c Routine cf_zbcalleurohw2d.c Routine cf_attari_dps.c Routine cf_carr_dps.c Routine cf_jarrowyildirim1d_yyiis.c Routine cf_jarrowyildirim1d_iicaplet.c

2 19 pages 2 Routine cf_gaussianmapping_cds.c Routine cf_call_merton.c Routine cf_put_merton.c Routine cf_cirpp1d_zcbond.c Routine cf_cirpp1d_payerswaption.c Routine cf_cirpp1d_zbcalleuro.c Routine cf_cirpp1d_cap.c Routine cf_cirpp1d_receiverswaption.c Routine cf_cirpp1d_zbputeuro.c Routine cf_cirpp1d_floor.c Routine cf_hes_varianceswap.c Routine cf_attari_heston.c Routine cf_call_heston.c Routine cf_carr_heston.c Routine cf_put_heston.c Routine cf_libor_affine_gou1d_capfloor_fourier.c Routine cf_libor_affine_gou1d_swaption_fourier.c Routine cf_vasicek1d_cap.c Routine cf_vasicek1d_zcbond.c Routine cf_vasicek1d_floor.c Routine cf_vasicek1d_zbcalleuro.c Routine cf_vasicek1d_receiverswaption.c Routine cf_vasicek1d_zbputeuro.c Routine cf_vasicek1d_payerswaption.c Routine cf_rogersveraart1.c Routine cf_cir1d_zbcalleuro.c Routine cf_cir1d_zcbond.c Routine cf_cir1d_zbputeuro.c Routine cf_varianceswap_dhes.c Routine cf_carr_doubleheston.c Routine cf_callupout.c Routine cf_putdownout.c Routine cf_callupin.c Routine cf_putupout.c Routine cf_putdownin.c Routine cf_calldownin.c Routine cf_putupin.c Routine cf_calldownout.c Routine cf_call.c Routine cf_digit.c Routine cf_callspread.c

3 19 pages 3 Routine cf_put.c Routine cf_putin_kunitomoikeda.c Routine cf_callin_kunitomoikeda.c Routine cf_putout_kunitomoikeda.c Routine cf_callout_kunitomoikeda.c Routine cf_fied_calllookback.c Routine cf_floating_putlookback.c Routine cf_floating_calllookback.c Routine cf_fied_putlookback.c Routine cf_hullwhite1dgeneralized_receiverswaption.c Routine cf_hullwhite1dgeneralized_zbcalleuro.c Routine cf_hullwhite1dgeneralized_zbputeuro.c Routine cf_hullwhite1dgeneralized_floor.c Routine cf_hullwhite1dgeneralized_cap.c Routine cf_hullwhite1dgeneralized_payerswaption.c Routine cf_quadratic1d_zbputeuro.c Routine cf_quadratic1d_cap.c Routine cf_quadratic1d_zbcalleuro.c Routine cf_quadratic1d_floor.c Routine cf_quadratic1d_payerswaption.c Routine cf_quadratic1d_zcbond.c Routine cf_quadratic1d_receiverswaption.c Contents 1 The Black and Scholes model 4 2 Standard European Options Call Options Put Options Call Spread Options Digit Options European Barrier Options Down-Out Call Options Down-Out Put Options Up-Out Call Options Up-Out Put Options Down-In Call Options Down-In Put Options

4 19 pages Up-In Call Options Up-In Put Options Double Barrier European Option Knock-Out Call Options Knock-Out Put Options Knock-In Call Options Knock-In Put Options Lookback European Options Fied Lookback Call Options Fied Lookback Put Options Floating Lookback Call Options Floating Lookback Put Options Standard 2D European Options Call On the Maimum Put On the Minimum Echange Options Best Of Option The Black and Scholes model Suppose the underlying asset price (S t, t 0) evolves according to the Black and Scholes model with continuous yield δ, that is ds s = S s ((r δ)ds + db s ), S t = From now on, we denote by T = maturity date (T t) θ = T t b = r δ K = strike price In the following, we state the closed form solutions for the price of options whose payoff is given by f(s T ), being f a suitable function, i.e. for the quantity F(t, ) = E [ e rθ f(s T ()) ].

5 19 pages 5 2 Standard European Options We have the general version of the Black-Sholes Formula [4] to price European options on stocks paying a continuos dividend yieald. In this section, we set: d 1 = log ( ) ( ) K + b + 2 d 2 = d 1 and N as the cumulative normal distribution function: 2.1 Call Options N(d) = 1 2π d Payoff C T = (S T K) + e 2 /2 d. Price C(t, ; K) = e δθ N(d 1 ) Ke rθ N(d 2 ) Routine cf_call.c C(t, ; K) = e δθ N(d 1 ) 2.2 Put Options Payoff P T = (K S T ) + Price P (t, ; K) = Ke rθ N( d 2 ) e δθ N( d 1 ) Routine cf_put.c P (t, ) = e δθ N( d 1 ) 2.3 Call Spread Options Payoff CS T = (S T K 1 ) + (S T K 2 ) + Price CS(t, ) = C(t, : K 1 ) C(t, ; K 2 ) CS(t, ) Routine cf_callspread.c = C(t, : K 1) C(t, : K 2)

6 19 pages Digit Options Payoff : C T = { K if ST K, 0 Price : C(t, ) = Ke rθ N(d 1 ) : Routine cf_digit.c C(t, ) = e rθ K e d2 2 /2 2πθ 3 European Barrier Options Barrier options are known as knock-out if the value of the option nullifies if the underlying asset price reaches a fied barrier before the maturity date and knock-in if it does not. We use the names up-out and up-in call/put options, or down-out and down-in call/put options, to stress if the considered barrier is an upper or lower one. Reiner-Rubinstein [3] have developed formulas for pricing standard barrier options with cash rebate R. We set here: A = φe δθ N(φ 1 ) φke rθ N(φ 1 φ) B = φe δθ N(φ 2 ) φke rθ N(φ 2 φ) C = φ( L )2(µ+1) e δθ N(ηy 1 ) φke rθ ( L )2µ N(ηy 1 η) D = φ( L )2(µ+1) e δθ N(ηy 2 ) φke rθ ( L )2µ N(ηy 2 η) E = Re rθ [ N(η 2 η) ( L )2µ N(ηy 2 η) ] F = R( L )µ+λ N(ηz) + ( L )µ λ N(ηz 2ηλ))

7 19 pages 7 and da = φe δθ N(φ 1 ) db = φe δθ N(φ 2 ) + e δθ n( 2 ) (1 k) θ l dc = 1 φ2µ [ ) 2µ ( L e δθ L 2 N(ηy 2 1 ) Ke rθ N(ηy 1 η) φ ( ) 2(µ+1) L e bθ N(ηy 1 )] [ ) dd = 2µ φ 2µ ( L se δθ L 2 N(ηy 2 2 ) Ke rθ N(ηy 2 ) ( N ηy 2 η φ ( ) 2µ+2 L e δθ) N(ηy 2 ) φη ( ) 2µ+2 L e δθ] n(y ( ) 2) 1 k l [ de = 2 R e rθ ( L )2µ N(ηy 2 η)µ + η n(ηy 2 ] θ) where df = R ( L )µ+λ [ (µ + λ)n(ηz) + (µ λ)( L )2λ] 2ηR( L )µ+λ n(z) 1 = log(/k) y 1 = log(l2 /K) z = log(l/) + (1 + µ) 2 = log(/l) + (1 + µ) y 2 = log(l/) + λ µ = b 2 /2 2 λ = µ 2 + 2r 2 + (1 + µ) + (1 + µ) and η, φ are suitable numbers belonging to { 1, 1} (see net sections for details). Let M t,t = sup S τ and m t,t = inf S τ t τ T t τ T

8 19 pages Down-Out Call Options and take Payoff C T = Price C(t, ) = C(t, ) Routine cf_calldownout.c = (S T K) + if m t,t L, R A C + F if KL B D + F da dc + df if KL db dd + df η = 1 φ = Down-Out Put Options Payoff P T = Price P (t, ) = and take P (t, ) = Routine cf_putdownout.c (K S T ) + if m t,t L, R A B + C D + F if KL, F da db + dc dd + df if KL, df η = 1 φ = 1

9 19 pages Up-Out Call Options Payoff C T = Price C(t, ) = and take C(t, ) = Routine cf_callupout.c (S T K) + if M t,t < L, R. F if KL, A B + C D + F df if KL, da db + dc dd + df η = 1 φ = Up-Out Put Options and take Payoff P T = Price P (t, ) = Routine cf_putupout.c P (t, ) = η = 1 (K S T ) + if M t,t < L, R. B D + F if KL, A C + F db dd + df if KL, da dc + df φ = Down-In Call Options Payoff C T = Price C(t, ) = P (t, ) = R if m t,t L, (S T K) +. C + E if KL, A B + D + E dc + de if KL, da db + dd + de

10 19 pages 10 and take η = 1 φ = 1 Routine cf_calldownin.c 3.6 Down-In Put Options and take Payoff P T = Price P (t, ) = P (t, ) Routine cf_putdownin.c = η = 1 R if m t,t L, (K S T ) +. B C + D + E if KL, A + E db dc + dd + de if KL, da + de φ = Up-In Call Options and take Payoff C T = Price P (t, ) = C(t, ) = R if M t,t < L, (S T K) +. A + E if KL, B C + D + E da + de if KL, db dc + dd + de η = 1 φ = 1 Routine cf_callupin.c

11 19 pages Up-In Put Options and take Payoff P T = Price P (t, ) = C(t, ) = η = 1 R if M t,t < L, (K S T ) +. A B + D + E if KL, C + E da db + dd + de if KL, dc + de φ = 1 Routine cf_putupin.c 4 Double Barrier European Option A double barrier option is knock-in or knock-out if the underlying price reaches or not a lower and/or an upper boundary prior to epiration. The eact value for double barrier call/put knock-out options is given by the Ikeda-Kunitomo formula [5], which allows to compute eactly the price when the boundaries suitably depend on the time variable t. More precisely, set U(s) = Ue δ 1s L(s) = Le δ 2s where the constants U, L, δ 1, δ 2 in are such that L(s) < U(s), for every s [t, T]. The functions U(s) and L(s) play the role of upper and lower barrier respectively. δ 1 and δ 2 determine the curvature and the case of δ 1 = 0 and δ 2 = 0 corresponds to two flat boundaries. The numerical studies suggest that in most cases it suffices to calculate the leading five terms of the series giving the price of the knock-out and knock-in double barrier call options. Let τ stand for the first time at which the underlying asset price S reaches at least one barrier, i.e. τ = inf{st ; S s L(s) or S s U(s)}.

12 19 pages 12 We define the following coefficients: µ 1 = 2 b δ 2 n(δ 1 δ 2 ) µ 2 = 2 n(δ 1 δ 2 ) 2 µ 3 = 2 b δ 2 + n(δ 1 δ 2 ) Knock-Out Call Options (S T K) + if τt Payoff C T = 0 + [ Price C(t, ) = e δθ (U n )µ1 ( L n= L n ( L n+1 U n [ (U n where F = Ue δ 1θ and Ke rθ + n= ) µ2 ( N(d + 1 ) N(d+ 2 )) ) µ3 ( N(d + 3 ) N(d+ 4 )) )µ1 2 ( ) L µ2 ( N(d L n 1 ) N(d 2 ) ) ( ) L n+1 µ3 2 ( N(d U n 3 ) N(d 4 ) ) ] d ± 1 = log(u 2n /KL 2n ) + ( ) b ± 2 d ± 3 = log(l2n+2 /KU 2n ) + ( ) b ± 2 d ± 2 = log(u 2n /FL 2n ) + ( ) b ± 2 d ± 4 = log(l2n+2 /FU 2n ) + ( b ± 2 2 ) θ Routine cf_callout_kunitomoikeda.c

13 19 pages Knock-Out Put Options (K S T ) + if τt Payoff P T = 0 + [ Price P (t, ) = Ke rθ (U n )µ1 2 ( ) L µ2 ( N(y n= L n 1 ) N(y 2 )) ( ) L n+1 µ3 2 ( N(y U n 3 ) N(y 4 )) e δθ + n= [ (U n where E = Le δ2θ and y 1 ± = log(u 2n /EL 2n ) + ( ) b ± 2 y 3 ± = log(l2n+2 /EU 2n ) + ( ) b ± 2 Routine cf_putout_kunitomoikeda.c )µ1 ( L L n ( L n+1 U n ) µ2 ( N(y + 1 ) N(y + 2 ) ) ) µ3 ( N(y + 3 ) N(y + 4 ) ) y 2 ± = log(u 2n /KL 2n ) + ( ) b ± 2 y 4 ± = log(l2n+2 /KU 2n ) + ( b ± 2 2 ] ) θ 4.3 Knock-In Call Options The double barrier knock-in call option is priced via the no-arbitrage relationship between knock-out and knock-in option: European IN + European OUT = European Standard Routine cf_callin_kunitomoikeda.c 4.4 Knock-In Put Options The double barrier knock-in call option is priced via the no-arbitrage relationship between knock-out and knock-in option: European IN + European OUT = European Standard

14 19 pages 14 Routine cf_putin_kunitomoikeda.c 5 Lookback European Options Floating strike lookback options can be priced using Goldman-Sosin-Gatto formula [2] while fied strike lookback options can be priced using Conze- Viswanathen formula[7]. We set, as 0 u v T, M u,v = sup S τ and m u,v = inf S τ u τ v u τ v and d 1 = log ( ) ( ) K + b + 2 e 1 = log ( ) ( ) M 0,t + b + 2 f 1 = log ( ) ( ) M 0,t + b + 2 d 2 = d 1 e 2 = e 1 f 2 = f Fied Lookback Call Options Payoff C T = (M t,t K) + Both price and delta depend on K and M 0,t. if K > M 0,t then Price C(t, ) = e δθ N(d 1 ) Ke rθ N(d 2 ) [ ( ) rθ 2 ( +e 2 N d 1 ) ] θ + e bθ N(d 1 ) K ( ) C(t, ) = e δθ N(d 1 ) e δθ n(d 1) K n(d 2 ) e rθ ( ) +e rθ 2 N ( ) ( ) d 1 t 1 2 K

15 19 pages 15 if K M 0,t then Price C(t, ) = e rθ (M 0,t K) + e δθ N(e 1 ) M 0,t e rθ N(e 2 ) ( ) ( rθ 2 +e 2 N e 1 ) θ + e bθ N(e 1 ) C(t, ) M 0,t = e δθ N(e 1 )(1 + 2 ) + n(e 1) e δθ M e rθ 0,t ) ( ) +e rθ M 0,t Routine cf_fied_calllookback.c 2 N ( e 1 t ) ( 1 2 n(d 2 ) 5.2 Fied Lookback Put Options Payoff P T = (K m t,t ) + Both price and delta depend on K and m 0,t. if K < m 0,t then Price P (t, ) = Ke rθ N( d 2 ) e δθ N( d 1 ) [ ( ) rθ 2 ( +e 2 N d 1 + ) ] θ e bθ N( d 1 ) K ( ) P (t, ) = e δθ N(d 1 ) e δθ n(d 1) K n(d 2 ) e rθ if K m 0,t then +e rθ ( K ) 2 N ( d 1 + t ) ( 2 1) e δθ ( 2 Price P (t, ) = e rθ (K m 0,t ) e δθ N( f 1 ) + m 0,t e rθ N( f 2 ) ( ) ( rθ 2 +e 2 N f 1 + ) θ e bθ N( f 1 ) P (t, ) m 0,t ( = e δθ ) ( ) +e rθ M 0,t 1 (N(f 1 ) 1) + e δθ n(f 1) M e rθ 0,t 2 N ( ) ( f ) t 1 ) n(d 2 )

16 19 pages 16 Routine cf_fied_putlookback.c 5.3 Floating Lookback Call Options Payoff C T = S T m t,t Price C(t, ) = e δθ N(f 1 ) m 0,t e rθ N(f 2 ) ( ) rθ 2 +e 2 N ( ) f 1 + θ e bθ N( f 1 ) C(t, ) m 0,t ( = e δθ N(f 1 ) rθ 2 e + e rθ ) e δθ n(a 1) ( ) ( 2 N f 1 + m 0,t Routine cf_floating_calllookback.c t ) ( 2 1 ) 5.4 Floating Lookback Put Options Payoff P T = M t,t S T Price P (t, ) = M 0,t e rθ N( e 2 ) e δθ N( e 1 ) ( ) rθ 2 +e 2 N ( ) e 1 θ + e bθ N(b 1 ) P (t, ) M 0,t ( ) ( = e δθ N(e 1 ) e rθ ( ) ( ) e rθ n(b1 ) M 0,t 1 Routine cf_floating_putlookback.c M 0,t ) ( 2 N + e δθ ( n(b1 ) 1 ) e 1 ) ) t (1 2 ]

17 19 pages 17 6 Standard 2D European Options Consider the pair of processes S t = (St 1, S2 t ) solution to ds 1 t = S1 t ((r δ 1)dt + 1 dw 1 t ), S1 0 = 1 ds 2 t = S2 t ((r δ 2)dt + 2 dw 2 t ), S2 0 = 2. where (Wt 1 2, t 0) and (Wt, t 0) denote two real valued Brownian motions with istantaneous correlation ρ. The price of option with payoff f is: F(t, 1, 2 ) = E [ e rt f ( S 1 T (1 ), S 2 T (, 2 ) )]. Here, closed formulas due to Johnson and Stulz are presented [1],[6]. We set d = log ( ) δ 2 δ theta = ρ 1 2 log ( ) ( ) i K + r δ i + 2 i 2 θ d i =, i = 1, 2 1 θ ρ 1 = 1 ρ 2 ρ 2 = 2 ρ 1 and M as the cumulative bivariate normal distribution function: M(a, b; ρ) = 6.1 Call On the Maimum Payoff C T = (ma(s 1 T, S2 T ) K) + 1 a b 2π e 2 2ρy+y 2 1 ρ 2 2(1 ρ 2 ) ddy. Price C(t, 1, 2 ) = 1 e δ 1θ M(d 1, d; ρ 1 ) + 2 e δ 2θ M(d 2, d + ; ρ 2 ) Ke rθ ( 1 M( d θ, d2 + 2 θ; ρ) ) C(t, 1, 2 ) 1 = e δ 1θ M(d 1, d; ρ 1 ) C(t, 1, 2 ) 2 = e δ 2θ M(d 2, d + ; ρ 2 )

18 19 pages 18 Routine cf_callma.c 6.2 Put On the Minimum Payoff P T = (K min(s 1 T, S 2 T)) + Price P (t, 1, 2 ) = Ke rθ c 0 + c 1 where P (t, 1, 2 ) 1 = e δ 1θ (1 N(d)) + e δ 1θ M(d 1, d; ρ 1 ) P (t, 1, 2 ) 2 = e δ 2θ N(d ) + e δ 2θ M(d 2, d ; ρ 2 ) c 0 = 1 e δ 1θ (1 N(d)) + 2 e δ 2θ N(d ) c 1 = 1 e δ 1θ M(d 1, d, ρ 1 ) + 2 e δ 2θ M(d 2, d ; ρ 2 ) Routine cf_putmin.c Ke rθ M(d 1 1 θ, d2 2 θ; ρ) 6.3 Echange Options Payoff E T = (S 1 T λs 2 T)) + where Price E(t, 1, 2 ) = 1 e δ 1θ N( d 1 ) λ 2 e δ 2θ N( d 2 ) d 1 = log ( ) ( ) 1 λ 2 + δ2 δ 1 + 2, d 2 = ˆd 1 θ Routine cf_echange.c 6.4 Best Of Option The payoff is B T = (ma(s 1 T K 1, S 2 T K 2 ))) +.

19 19 pages 19 References [1] H.JOHNSON. Options on the maimum ot the minimum of several assets. J.Of Finance and Quantitative Analysis, 22: , [2] B.M.GOLDMAN H.B.SOSIN M.A.GATTO. Path dependent options: buy at low, sell at high. J. of Finance, 34: , [3] E.REINER M.RUBINSTEIN. Breaking down the barriers. Risk, 4:28 35, [4] F.BLACK M.SCHOLES. The pricing of Options and Corporate Liabilities. Journal of Political Economy, 81: , [5] N.KUNIMOTO N.IKEDA. Pricing options with curved boundaries. Mathematical finance, 2: , [6] R.STULZ. Options on the minimum or the maimum of two risky assets. J. of Finance, 10: , [7] A.CONZE R.VISWANATHAN. Path dependent options: the case of lookback options. J. of Finance, 46: ,