Research Paper 357 February Less Expensive Pricing and Hedging of Long-Dated Equity Index Options When Interest Rates are Stochastic

Transcription

1 laesquaniaive FINANCE RESEARCH CENRE QUANIAIVE F INANCE RESEARCH CENRE QUANIAIVE FINANCE RESEARCH CENRE Research Paper 357 February 215 Less Expensive Pricing and Hedging of Long-Daed Equiy Index Opions When Ineres Raes are Sochasic Kevin Fergusson and Eckhard Plaen ISSN

2 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED EQUIY INDEX OPIONS WHEN INERES RAES ARE SOCHASIC KEVIN FERGUSSON AND ECKHARD PLAEN Absrac. Many providers of variable annuiies such as pension funds and life insurers seek o hedge heir exposure o embedded guaranees using longdaed derivaives. his paper exends he benchmark approach o price and hedge long-daed equiy index opions using a combinaion of cash, bonds and equiies under a variey of marke models. he resuls show ha when he discouned index is modelled as a squared Bessel process, as in Plaen s minimal marke model, less expensive hedging is achieved irrespecive of he shor rae model. 1. Inroducion Long erm savings producs wih embedded guaranees on capial, such as variable annuiies, are popular among invesors planning for reiremen. Insurers who wrie such producs are ineresed in hedging heir risk exposure eiher hrough reinsurance, derivaive markes or hedging programmes. Several frameworks of accouning sandards such as US GAAP, IASB and IFRS prescribe ha such producs be marked-o-marke and, herefore, hedging hese producs is paramoun for insurers seeking sable earnings and high credi raings. Using he benchmark approach of Plaen [22], Plaen [26] and Plaen and Heah [26], pricing and hedging of long-daed claims on he S&P5 oal Reurn Index, when ineres raes are deerminisic, was demonsraed by Hulley and Plaen [28]. In he curren aricle we exend his work under he benchmark approach o price and hedge long-daed equiy index opions when ineres raes are sochasic. Some pricing and hedging of ineres rae derivaives using he benchmark approach has been done by Fergusson and Plaen [214]. he pricing and hedging of equiy opions when share prices and ineres raes are sochasic has previously been done by Sco [1997] who also incorporaes a jump diffusion componen and sochasic volailiy o he sock price dynamics. Many approaches o pricing equiy opions wih models involving sochasic ineres raes employ inverse Fourier ransforms, as done in Lee [24]. However, in he curren aricle we demonsrae less expensive pricing and hedging of long-daed equiy index opions and provide approximae pricing formulae involving eiher he cumulaive disribuion funcions of he normal disribuion or he non-cenral chi-squared disribuion. Furhermore, we compue he cos of hedging an equiy index pu opion, whose srike price is an exponenial funcion of he spo price, for each of he considered Dae: February 5, Mahemaics Subjec Classificaion. Primary 62P5; Secondary 6G35, 62P2. Key words and phrases. Growh opimal porfolio, benchmark approach, long-daed equiy index opions, minimal marke model. 1

3 2 KEVIN FERGUSSON AND ECKHARD PLAEN marke models and various erms o expiry and idenify he bes performing models as hose involving a discouned GOP being modelled as a squared Bessel process as in Plaen s minimal marke model. In Secion 2 we describe he models of he shor rae and he discouned equiy index used for opion pricing and hedging. In Secion 3 we show how o price various coningen claims such as call, pu, asse-or-nohing and cash-or-nohing opions. Supplemening hese pricing formulae are approximaions given in Appendix A used for he backesing. In Secion 4 we describe how o hedge derivaive securiies and in Secion 5 we ariculae our way of assessing a hedging sraegy. We describe in Secion 6 he marke daa employed and how he models are fied. In Secions 7, 8, 9 and 1 we compare he hedge performances of he deerminisic, Vasicek, CIR and 3/2 shor rae models, respecively, he numerical resuls being shown in Appendix B. Finally, in Secion 11 we menion ha modelling he discouned GOP wih he MMM gives rise o significanly cheaper coss of hedging long-daed equiy index pu opions. 2. Descripion of Models he marke models examined here are specified by he sochasic differenial equaion SDE of he shor rae r and he SDE of he discouned GOP S. he shor rae models considered are he deerminisic shor rae model, where he shor rae is known for all imes, 2.1 r = r, he Vasicek shor rae model described by Vasicek [1977], 2.2 dr = κ r r d + σdz, he Cox-Ingersoll-Ross CIR shor rae model described by Cox e al. [1985], 2.3 dr = κ r r d + σ r dz and he 3/2 shor rae model described by Ahn and Gao [1999], 2.4 dr = pr + qr 2 d + σr 3/2 dz. he discouned GOP models which are considered are he Black-Scholes model, equivalenly he lognormal sock price model, employed by Black and Scholes [1973] 2.5 d S = S θ 2 d + S θdw, and he minimal marke model described by Plaen [21] wih SDE 2.6 d S = ᾱ d + ᾱ Sδ dw, where ᾱ = ᾱ expη. Here Z and W are independen Wiener processes, r is he realised value of he shor rae a ime and r, κ, σ, p, q, θ, ᾱ and η are consans. he cash accoun B is he accumulaed value a ime of $1 deposied a iniial ime zero and we have he formula 2.7 B = exp dr s. he GOP S is obained by muliplying he cash accoun B by he discouned GOP S. he growh rae g of he GOP is equal o he drif erm of he SDE

4 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 3 of he logarihm of he GOP, which equals, for models involving he Black-Scholes discouned GOP, g = r θ2, and for models involving he MMM discouned 1 GOP g = r + 2ᾱ/ S. For a given coningen claim H wih mauriy,, i has been shown in Plaen and Heah [26] ha he minimal possible price V δ H for a replicaing hedge porfolio saisfies he real world pricing formula 2.8 V δ H S δ = E S H, where E denoes he real world condiional expecaion under he real world probabiliy measure given he informaion available a ime. Furhermore, he GOP S is aken here as he numéraire or benchmark. he numéraire is he porfolio having maximal growh rae and is approximaed well, see Plaen and Heah [26] and Plaen and Rendek [212], by diversified equiy marke indices such as he S&P/ASX 2 oal Reurn Index or S&P 5 oal Reurn Index. 3. Pricing of Coningen Claims When he claim is H = 1 he real world pricing formula 2.8 gives a ime he price P, of a zero coupon bond ZCB. Furher, when H = S K+ formula 2.8 gives a ime he price c,k, S of an equiy index call opion having srike price K, and when H = K S + formula 2.8 gives he price p,k, S of an equiy index pu opion having srike price K. Because of he following relaion beween payoffs 3.1 S he pu-call pariy relaion 3.2 c,k, S K+ = K S + + S K = p,k, S + S KP, holds. Addiionally, when H = S 1 S >K formula 2.8 gives a ime he price A +,K, S of an asse-or-nohing binary call opion having srike price K and when H = S 1 S K formula 2.8 gives he price A,K, S of an asse-ornohing binary pu opion having srike price K. Finally, when H = 1 S >K formula 2.8 gives he price B +,K, S of a cash-or-nohing binary call opion having srike price K and when H = 1 S K formula 2.8 gives he price B,K, S of a cash-or-nohing binary pu opion having srike price K. Under our considered marke models he real-world pricing formula 2.8 gives he price of a ZCB as 3.3 P, = E B B Sδ E. S For he deerminisic model of he shor rae we have B 3.4 E = exp rsds. B For he Vasicek model of he shor rae we have from Vasicek [1977] B 3.5 E = A, exp r B,, B

5 4 KEVIN FERGUSSON AND ECKHARD PLAEN where 1 exp κ 3.6 B, = κ and 3.7 A, = exp r σ2 σ2 B, + B, 2. 2κ2 4κ For he CIR model of he shor rae we have from Cox e al. [1985] B 3.8 E = A, exp r B,, B where 3.9 A, = 3.1 B, = and h exp 1 2κ κ sinh 1 2 h + h cosh 1 2h 2 sinh 1 2h κ sinh 1 2 h + h cosh 1 2h 3.11 h = κ 2 + 2σ 2. 2κ r/σ 2 For he 3/2 model of he shor rae we have from Ahn and Gao [1999] B 3.12 E = Γγ α1 1 α B Γγ 1 σ 2 Mα 1, γ 1, y, r σ 2 y, r, where 3.13 y, r = r exp p 1 p 1 1 α u = + 2 q σ 2 γ u = 2 α u + 1 q σ 2. 2 q σ 2 Here M is he confluen hypergeomeric funcion given by α n z n 3.14 Mα, γ, z = γ n n! n= 2 + 2u σ 2 and Γx = u x 1 exp u du is he gamma funcion. For he Black-Scholes discouned GOP we have Sδ 3.15 E = 1, S whereas for he MMM discouned GOP we have from Plaen and Heah [26] Sδ 3.16 E S where = 1 exp 1 S /ϕ ϕ ϕ = 1 4ᾱexpη 1/η.,

6 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 5 hus various combinaions of 3.5, 3.8, 3.12, 3.15 and 3.16 insered ino 3.3 give explici formulae for he real world prices of ZCBs under each considered marke model. Under our considered marke models 2.8 gives he price of a call opion as 3.18 c,k, S and he price of a pu opion as 3.19 p,k, S B +,K, S S δ = E S S δ = E S S K+ K S +. he prices of he asse-or-nohing and cash-or-nohing call and pu opions are 3.2 A +,K, S = S E 1 S >K A,K, S = S E 1 S K S δ = E B,K, S S S δ = E S 1 S >K 1 S K Le f S x denoe he probabiliy densiy funcion of he random variable S, and define he random variable R as being relaed o S having he probabiliy densiy funcion 3.21 f R x = S S δ x f S x /E S.. he pricing formulae of he various call and pu opions become 3.22 c,k, S = S 1 F S K P, K 1 F R K p,k, S A +,K A,K = S, S = S, S = S F S K + P, K F R K 1 F S K F S B +,K, S = P, B,K, S K 1 F R K = P, F R K, where F S x and F R x denoe he cumulaive disribuion funcions of he random variables S and R, respecively. he following wo heorems give exac expressions for he cumulaive disribuion funcions F S K and F R K in erms of he cumulaive disribuion funcions

7 6 KEVIN FERGUSSON AND ECKHARD PLAEN of he lognormal disribuion and he noncenral gamma disribuion, a sraighforward generalisaion of he noncenral chi-squared disribuion, hese being y LNy; µ, σ 2 1 = x 2πσ exp log x 2 µ2 dx 2σ2 y α/2 1/2 2γx NCGy; α, γ, λ = γ exp 1 λ 2 λ + 2γx I α 1 2λγx dx, respecively, where I ν x is he modified Bessel funcion of he firs kind wih index ν, given by ν z 1 z 2 j 3.24 I ν z =. 2 j!γj + ν j= In he proof of heorem 2 we make use of an equivalen expression for he cumulaive disribuion funcion of he noncenral gamma disribuion as a Poisson mixure wih a gamma disribuion, analogous o he noncenral chi-squared disribuion being a Poisson mixure wih a chi-squared disribuion, namely 3.25 NCGy; α, γ, λ = λ/2 j exp λ/2gy; α + j, γ, j! j= where Gy; α, γ is he cumulaive disribuion funcion of he gamma disribuion 3.26 Gy; α, γ = γα Γα y x α 1 exp γx dx. Because a lognormal random variable X LNµ, σ 2 is one for which is logarihm is normally disribued, ha is log X Nµ, σ 2, we have ha he cumulaive disribuion funcion of he lognormal disribuion saisfies LNy; µ, σ 2 = N log y µ/σ where Nx denoes he cumulaive disribuion funcion of he sandard normal disribuion. Also, because a noncenral gamma random variable X NCGα, γ, λ is one for which produc wih 2γ is noncenral chi-squared disribued, ha is 2γX χ 2 2α,λ, we have ha he cumulaive disribuion funcion of he noncenral gamma disribuion saisfies NCGy; α, γ, λ = χ 2 2α,λ 2γy where χ 2 ν,λ x denoes he cumulaive disribuion funcion of he noncenral chi-squared disribuion having ν degrees of freedom and noncenraliy parameer λ. heorem 1. For he Black-Scholes discouned GOP logb /B we have S and random variable L = 3.27 F S K = E LNK; log S + L θ2, θ 2 = E N d 1 L F R K = E exp LLNK; log S + L 1 2 θ2, θ 2 /E exp L = E exp LN d 2 L E exp L

8 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 7 where 3.28 d 1 L = L θ2 + log S θ2 K d 2 L = d 1 L θ 2. Proof. We know ha under he Black-Scholes model of he discouned GOP lognormally disribued, ha is 3.29 Sδ LNlog S θ2, θ 2, S is and, herefore, S condiioned on he random variable L = logb /B is also lognormally disribued, ha is 3.3 S which can be rewrien as 3.31 S Hence LNlog B + log 3.32 F S = E E1S S θ2, θ 2, LNL + log S θ2, θ 2. K = E1 S K K L = E LNK; L + log S θ2, θ 2 = EN d 1 L. Also he cumulaive disribuion funcion of he random variable R o be 3.33 F R K = E E S S 1 1 S K L /E is compued E S S 1 L. Bu 3.34 E S S 1 L = S = S = exp L E S 1 L exp L + log S θ θ2 and 3.35 E S = S = S S 1 1 S K K 1 x f S K L Lx dx exp log xf S L x dx.

9 8 KEVIN FERGUSSON AND ECKHARD PLAEN Insering he explici expression for he lognormal densiy funcion f S L x gives 3.36 E S = S exp S 1 1 S K { K L 1 x 2πθ 2 1 2θ 2 S log x L + log S δ 1 x 2πθ 2 exp θ2 } 2 + 2θ 2 log x dx and compleing he square in he exponenial in he inegrand above gives K { 3.37 L + log S = S exp 1 log x 2θ θ2 } 2 + 2L + log S δ θ 2 dx exp L + log S { 1 2θ 2 K 1 x 2πθ 2 log x L + log S δ 1 2 θ2 2 } dx herefore, 3.38 = exp L LN K; L + log S 1 2 θ2, θ 2. F R = E K exp L LN K; L + log S 1 2 θ2, θ 2 /E exp L = Eexp L N d 2 L/E exp L, as required. heorem 2. For he MMM discouned GOP logb /B we have 3.39 S and he random variable L = F S K = E NCGK; 2, exp L/2ϕ ϕ B, λ = E χ 2 4,λuL F R K = E exp LNCGK;, exp L/2ϕ ϕ B, λ exp λ/2 1 exp λ/2e exp L where = E exp Lχ 2,λ ul exp λ/2 1 exp λ/2e exp L, 3.4 ul = K B ϕ ϕ expl, 3.41 ϕ = 1 4ᾱexpη 1/η

10 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 9 and 3.42 λ = S ϕ ϕ. Proof. We know ha under he MMM, he discouned GOP gamma disribued, ha is 3.43 Sδ NCG2, 1/2ϕ ϕ, λ, S is noncenral and herefore S condiioned on he random variable L = logb /B is also noncenral gamma disribued, ha is 3.44 S which can be rewrien as NCG2, exp L/2B ϕ ϕ, λ, 3.45 S /explb ϕ ϕ χ 2 4,λ. Hence 3.46 F S = E E1S K = E1 S K K L = E NCGK; 2, exp L/2B ϕ ϕ, λ = Eχ 2 4,λuL. Also he cumulaive disribuion funcion of he random variable R o be 3.47 F R K = E Bu 3.48 and 3.49 S δ E S E S S 1 1 S K L /E E S S 1 L = S exp LE S 1 = exp L1 exp λ/2 = exp LE 1 S K L = exp LE = exp L he random variable densiy funcion Sδ S 3.5 fx = 1 S K/B λ is compued E S S 1 L. S /ϕ ϕ 1 S /ϕ ϕ K exp L/B ϕ ϕ K exp L/Bϕ ϕ λ x f S /ϕ ϕ x dx. S /ϕ ϕ is disribued as χ 2 4,λ, which has probabiliy j= exp λ/2 λ/2j f j! χ 2 4+2j x,

11 1 KEVIN FERGUSSON AND ECKHARD PLAEN where he probabiliy densiy funcion of he chi-squared disribuion having 4 + 2j degrees of freedom has he formula 3.51 f χ 2 4+2j x = 1/22+j Γ2 + j x1+j exp x/2. herefore, he inegrand in he RHS of 3.49 can be wrien as 3.52 λ x f S /ϕ ϕ x = λ x = λ = λ = λ = j= j= j= j= j=1 exp λ/2 λ/2j j! exp λ/2 λ/2j j! exp λ/2 λ/2j j! exp λ/2 λ/2j j! exp λ/2 λ/2j f j! χ 2 2j x = f χ 2,λ x exp λ/21 x=. 1/2 2+j Γ2 + j x1+j exp x/2 1/2 2+j Γ2 + j xj exp x/2 1/2 2+j Γ1 + j Γ2 + j 1/2 1+j f χ 2 x 2+2j 1/2 1 + j f χ 2 2+2j x Hence 3.49 becomes 3.53 K exp L/Bϕ ϕ exp L λ x f S /ϕ ϕ x dx K exp L/Bϕ ϕ = exp L f χ 2,λ x dx exp λ/2 = exp L χ 2,λ K exp L/B ϕ ϕ exp λ/2, which leads o he resul. For a deerminisic shor rae he cumulaive disribuion funcions F S x and F R x are readily compued o be, under a BS discouned GOP, 3.54 F S x = LNx; log S + F R x = LNx; log S + rsds θ2, θ 2 rsds 1 2 θ2, θ 2

12 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 11 and, under he MMM discouned GOP, 3.55 F S x = NCGx; 2, 1/2ϕ ϕb exp F R rsds, λ x = NCGx;, 1/2ϕ ϕb exp rsds, λ exp λ/2. 1 exp λ/2 For a Vasicek model of he shor rae 3.56 L N m,, v, where 3.57 and 3.58 B, = m, = r rb, + r v, = σ2 κ 2 B, 1 2 κb, 2 1 exp κ. κ So for a Vasicek shor rae and BS discouned GOP 3.59 S LN log S θ2 + m,, θ 2 + v, R LN log S 1 2 θ2 + m, v,, θ 2 + v, and he cumulaive disribuion funcions F S x and F R x are readily compued o be 3.6 F S x = LN x; log S θ2 + m,, θ 2 + v, F R x = LN x; log S 1 2 θ2 + m, v,, θ 2 + v,. Also for a Vasicek shor rae and MMM discouned GOP 3.61 F S x = { χ 2 4,λuznzdz F R x = exp m, v, zχ 2,λuznzdz exp λ/2 exp m, + 1 } 2 v, { 1 exp λ/2 exp m, v, } 1, where m, and v, are given in 3.57, uz is given in 3.4 and nz is he probabiliy densiy funcion of he sandard normal disribuion. For he CIR shor rae model and he 3/2 shor rae model he probabiliy densiy funcion of L is compued as he inverse Fourier ransform of he momen

13 12 KEVIN FERGUSSON AND ECKHARD PLAEN generaing funcion MGF, ha is 3.62 f L x = exp2πixsmgf L 2πis ds. Here he MGF of L under he CIR shor rae model is h u exp 1 2κ 2κ r/σ MGF L u = κ sinh 1 2 h u + h u cosh 1 2 h u 2 sinh 1 2 exp u h u κ sinh 1 2 h u + h u cosh 1 2 h r, u where h u = κ 2 2uσ 2. From heorem 3 of Carr and Sun [27] he MGF of L under he 3/2 shor rae model is 3.64 MGF L u = Γγ αu u α u 2 2 Γγ u σ 2 Mα u, γ u, y, r σ 2 y, r, where he variables are as in he cumulaive disribuion funcions F S x and F R x under he BS discouned GOP become 3.65 F S x = F R x = N d 1 xf L x dx and, under he MMM discouned GOP, become 3.66 F S x = F R x = where ux is given by 3.67 ux = e x N d 2 xf L x dx MGF L 1 χ 2 4,λuxf L x dx e x χ 2,λ ux e λ/2 f L x dx, MGF L 1 K B expxϕ ϕ, and ϕ = 1 4ᾱexpη 1/η. hus we have demonsraed how he various cumulaive disribuion funcions can be compued and, combined wih 3.22, how prices of various call, pu, asse-ornohing and cash-or-nohing opions can be compued. In Appendix A we provide approximae formulae for he various cumulaive disribuion funcions, which lead o simplified and rapid compuaions. 4. Descripion of Hedging Mehodology Beyond pricing of long-daed pu opions on an equiy index, our aim is o demonsrae cheaper coss of hedging such opions. We focus on hedging a longdaed pu opion expiring a ime, whose srike price K keeps pace wih he level of he equiy index by way of he formula 4.1 K = S exp η + µ r. Here is he ime a which he pu opion is wrien and η = is he ne marke growh rae given in able 1 and µ r = s rs = is he average

14 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 13 of he one year coninuously compounded cash raes over he 141 year period of he daa. In respec of a pu opion, and more generally, for any derivaive securiy, a hedging sraegy is a rading sraegy involving a porfolio of hedge securiies whose value a a prescribed payoff dae is inended o replicae he value of he derivaive securiy. When he marke values of securiies are driven by a deerminisic shor rae and sochasic discouned GOP, hen we have only one random facor in our marke and we can hedge a suiable derivaive securiy using a managed self-financing porfolio π of cash he savings accoun and he GOP. he value of he hedge porfolio can be wrien as 4.2 V π = δ B + δ 1 S, where δ is he number of unis of he cash accoun and δ 1 is he number of unis of he GOP accoun a ime [, ]. he respecive fracions invesed a ime are π = π, π 1 wih π = δ B /V π and π 1 = 1 π = δ 1 S /V π. We have some flexibiliy in our choice of hedge securiies and we could have used insead he savings accoun and fuures on he GOP, for example. When he marke values of securiies are driven by a sochasic shor rae and a sochasic discouned GOP, hen we have wo random facors in our marke and we can hedge any derivaive securiy using a managed porfolio of cash B, he GOP index S and, for insance, a -year zero coupon bond F,. In pracice, because liquidiy is essenial for any hedge sraegy, we would choose o hedge using a managed porfolio of cash, S&P 5 Index Fuures and 1Y US reasury Bonds. he value of he hedge porfolio π can be wrien as 4.3 V π = δ B + δ 1 S + δ 2 F,, where δ and δ 1 describe numbers of unis as before, and δ 2 is he number of unis of he -mauriy zero coupon bond a ime [, ]. he cos C a ime of hedging a derivaive since iniial ime is equal o he cos of he derivaive a ime less any gains from rading he hedge porfolio. We wrie 4.4 C = V δ H where V δ H δ u db u δ u 1 dsu = V δ H is he value of he derivaive a ime and V π hedge porfolio a ime. his equaion can be rewrien as C = V δ H V π = V π + V δ H V π V π dv π u is he value of he and we can see ha he cos of hedging can be expressed alernaively as he cos of he hedge porfolio a ouse, namely V π, plus addiional funds needed a ime o purchase he derivaive in excess of he value of he hedge porfolio. A he payoff dae he cos of hedging is 4.7 C = V δ H dv π u.

15 14 KEVIN FERGUSSON AND ECKHARD PLAEN Because we are ineresed in he real world price of hedging, as given in 2.8, we consider he benchmarked cos of hedging, compued as 4.8 Ĉ = C S = ˆV δ H π π d ˆV u = ˆV + ˆV δ H ˆV π. According o 2.8 he average of he benchmarked coss of hedging performed over a large number of backess ough o approximae he real world price of he derivaive wih payoff H. Given a fully specified model wih known parameers, we backes hedging of he derivaive over he ime inerval [, ] by seing he n 1 rebalancing imes 1 < 2 <... < n 1 saisfying = < 1 and n 1 < n =. he hedge porfolio V π is adjused a he rebalancing imes and is compued ieraively using he formula 4.9 V π i = δ i 1 B i + δ 1 i 1 S i + δ 2 i 1 F i, for i = 1, 2,..., n wih iniial condiion 4.1 V π = V δ H, where, for i = 1, 2,... n 1, he numbers of unis held in he GOP and he ZCB a ime i are compued as 4.11 δ 1 i = Ss δ 2 i = V δ H s r s V δ H s r s, S i s s=i δ 2 S s r s, S s s=i / r s F s, s=i F s, s=i and he number of unis held in he cash accoun a ime i is compued as 4.12 δ i = V π i δ 1 S i δ 2 i F i, /B i. i 5. Assessing he Performance of a Hedging Sraegy A perfec hedge sraegy is one for which 5.1 C = V π for all imes [, ]. ha is o say, he hedge porfolio replicaes he value of he derivaive over he life of he hedging sraegy. However, perfec hedging is no possible for many reasons and we are ineresed in sraegies which generae he payoff a expiry dae, wih minimum cos. herefore, for a given marke model, a given daa se and a given erm o expiry we compue he benchmarked coss of hedging a pu opion a expiry over all possible periods wihin he daa se. From his he p-h percenile of he se of benchmarked coss is compued. he bes hedge sraegy is derived from he marke model which gives he minimum percenile benchmarked cos of hedging. Consequenly, our ask in his aricle is o compare he percenile benchmarked coss of hedging across all menioned marke models.

16 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 15 able 1. Maximum likelihood esimaes of model parameers fied o US daa Model Parameers Sandard Errors Log Likelihood Vasicek r = κ = σ = CIR r = κ = σ = /2 p = q = σ = Black-Scholes θ = MMM ᾱ = η = Marke Daa and Fiing he Models he daa se used for backesing has been he annual series of US 1Y deposi raes, 1Y reasury bond yields and S&P Composie Sock Index from 1871 o 212, shown in Chaper 26 of Shiller [1989] and subsequenly updaed on Shiller s websie hp://aida.wss.yale.edu/ shiller/daa/chap26.xls. he 141 year lengh of his daa series makes i a mos useful series for analysing he hedging of long-daed ZCBs because we are able o backes any given hedge sraegy over he large erm o mauriy of he ZCB. Also, because here are 1Y bond yields accompanying he 1Y deposi raes and sock index values we are able o consruc and backes a hedge porfolio which immunises agains movemens in boh he sock index and shor rae. Each shor rae model and discouned GOP model has an explici formula for he ransiion densiy funcion and his has allowed us o fi he models o he hisorical daa using maximum likelihood esimaion MLE. he maximum likelihood esimaes MLEs of he parameers of all models fied o US daa are shown in able 1. he backess of he hedging sraegies were performed using in-sample esimaion of parameers. Of course in realiy one would backes a hedge sraegy using ou-of-sample parameer esimaes bu by employing in-sample esimaes any poorly performing model is readily falsified. 7. Hedging Coss under a Deerminisic Shor Rae We presen he coss of hedging GOP opions under marke models having a deerminisic shor rae. In able 2 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he deerminisic shor rae and Black-Scholes discouned GOP model.

17 16 KEVIN FERGUSSON AND ECKHARD PLAEN In able 3 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he deerminisic shor rae and MMM discouned GOP model. For hedging GOP opions wih erms o expiry up o 1 years he BS discouned GOP model and MMM discouned GOP model perform similarly. Beyond GOP opion erms o expiry of 1 years he MMM discouned GOP model ouperforms he BS discouned GOP model. For example, hedging a 5Y GOP opion a he 99% probabiliy level incurs a cos of under he MMM discouned GOP model, which is significanly less han he corresponding cos of under he BS discouned GOP model. 8. Hedging Coss under a Vasicek Shor Rae We presen he coss of hedging GOP opions under Vasicek shor rae models. In able 4 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he Vasicek shor rae and Black-Scholes discouned GOP model. In able 5 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he Vasicek shor rae and MMM discouned GOP model. For hedging GOP opions wih erms o expiry up o 15 years he BS discouned GOP model and MMM discouned GOP model perform similarly. However, beyond 15 years he MMM discouned GOP model ouperforms he BS discouned GOP model. In paricular, he cos of hedging a 5Y GOP opion a he 99% probabiliy level is under he MMM discouned GOP model, which is significanly less han he corresponding cos of under he BS discouned GOP model. 9. Hedging Coss under a CIR Shor Rae We presen he coss of hedging GOP opions under CIR shor rae models. In able 6 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he CIR shor rae and Black-Scholes discouned GOP model. In able 7 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he CIR shor rae and MMM discouned GOP model. For GOP opion erms o expiry up o 15 years he BS discouned GOP model provides a significanly lower cos of hedging han under he MMM discouned GOP model. However, beyond a GOP opion erm o expiry of 15 years he MMM discouned GOP model ouperforms he BS discouned GOP model. For example, he cos of hedging a GOP opion is significanly reduced for a 5 year erm o expiry a he 99% probabiliy level, he cos being under he MMM discouned GOP model, which is significanly less han he corresponding cos of under he BS discouned GOP model. 1. Hedging Coss under a 3/2 Shor Rae We presen he coss of hedging GOP opions under 3/2 shor rae models. In able 8 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he 3/2 shor rae model and Black-Scholes discouned GOP model.

18 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 17 In able 9 he percenile benchmarked coss of hedging GOP opions of various erms o expiry and levels of probabiliy are shown for he 3/2 shor rae model and MMM discouned GOP model. For GOP opion erms o expiry shorer han 15 years he BS discouned GOP model is abou he same as or lower han he MMM discouned GOP model. Beyond a GOP opion erm o expiry of 15 years he MMM discouned GOP model ouperforms he BS discouned GOP model. For example, he cos of hedging a 5Y GOP opion a he 99% probabiliy level is under he MMM discouned GOP model, which is significanly less han he corresponding cos of under he BS discouned GOP model. 11. Conclusions on he Hedge Performances In Figure 1 he percenile coss of hedging GOP opions of varying erms o expiry are graphed. Each model for which he discouned GOP is modelled by he MMM has significanly cheaper coss of hedging long-daed GOP opions. In paricular, we find ha among he models having a sochasic shor rae, he Vasicek shor rae and MMM discouned GOP model provides he cheapes hedging sraegy for longdaed GOP pu opions. We remark on he effec of sochasic versus deerminsiic ineres raes ha Jensen s Inequaliy gives 11.1 E exp r s ds exp E r s ds, since he funcion fx = exp x is convex. his indicaes wha we have also seen and we see ha sochasic ineres raes will give rise o higher derivaive prices han hose from deerminisic ineres raes if everyhing else is modelled analogously. References D.H. Ahn and B. Gao. A parameric nonlinear model of erm srucure dynamics. Review of Financial Sudies, 12: , F. Black and M. Scholes. he pricing of opions and corporae liabiliies. Journal of Poliical Economy, 81: , P. Carr and J. Sun. A new approach for opion pricing under sochasic volailiy. Review of Derivaives Research, 12:87 5, 27. J. C. Cox, J. E. Ingersoll, and S. A. Ross. A heory of he erm srucure of ineres raes. Economerica, 532:385 47, K. Fergusson and E. Plaen. Hedging long-daed ineres rae derivaives for Ausralian pension funds and life insurers. Ausralian Journal of Acuarial Pracice, 1:29 44, 214. H. Hulley and E. Plaen. Hedging for he long run. Research Paper Series 214, Quaniaive Finance Research Cenre, Universiy of echnology, 28. R. W. Lee. Opion pricing by ransform mehods: Exensions, unificaion and error conrol. Journal of Compuaional Finance, 73:51 86, 24. E. Plaen. A minimal financial marke model. In rends in Mahemaics, pages Birkhäuser, 21. E. Plaen. Benchmark model for financial markes. Working Paper, Universiy of echnology, Sydney, 22.

19 Benchmarked Cos of Hedging 18 KEVIN FERGUSSON AND ECKHARD PLAEN Figure 1. Percenile coss of hedging GOP pu opions of varying erms o expiry Deerminisic-SR & BS-DGOP Deerminisic-SR & MMM-DGOP Vasicek-SR & BS-DGOP Vasicek-SR & MMM-DGOP CIR-SR & BS-DGOP CIR-SR & MMM-DGOP 3/2-SR & BS-DGOP 3/2-SR & MMM-DGOP erm o Expiry of Opion in Years E. Plaen. A benchmark approach o asse managemen. Journal of Asse Managemen, 66:39 45, 26. E. Plaen and D. Heah. A Benchmark Approach o Quaniaive Finance. Springer Finance, 26. E. Plaen and R. Rendek. Approximaing he numeraire porfolio by naive diversificaion. Journal of Asse Managemen, 131:34 5, 212. L. Sco. Pricing sock opions in a jump-diffusion model wih sochasic volailiy and ineres raes: Applicaions of Fourier inversion mehods. Mahemaical Finance, 74: , R. Shiller. Marke Volailiy. he MI Press, Cambridge, Massachuses, O. A. Vasicek. An equilibrium characerizaion of he erm srucure. he Journal of Financial Economics, 5: , Appendix A: Approximae Pricing of Equiy Index Opions he calculaion of inverse Fourier ransforms is compuaionally inensive on a compuer and a faser compuaional mehod approximaes he disribuion of he GOP wih a probabiliy disribuion having he same momens up o he second or hird order and having he same form as he disribuion of he discouned GOP. So for a model involving a BS discouned GOP he disribuion of he GOP is approximaed by a lognormal disribuion ha maches he firs wo momens of he GOP. Also, for a model involving a MMM discouned GOP he disribuion of he GOP is approximaed by a noncenral gamma disribuion ha maches he firs hree momens of he GOP.

20 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 19 Because of he independence of he driving Wiener processes Z and W of he shor rae and he discouned GOP, respecively, he momens of he GOP S are he produc of he corresponding momens of he savings accoun B and discouned GOP S. Also, he k-h momen of he relaed random variable R is S E S S k 11.2 = S P, E S k 1 E S S and herefore can be compued from he k 1-h momen of S. he k-h momen of B is compued as 11.3 B k MGF L k. When he discouned GOP obeys he BS model, he k-h momen of 11.4 S k exp k 2 θ2 + k2 2 θ2. When he discouned GOP obeys he MMM model, he firs, second and hird momens of S, which is disribued as NCG2, 1/2ϕ ϕ, λ, are E Sδ 11.5 = ϕ ϕ 4 + λ E S 2 = ϕ ϕ λ E S 3 = ϕ ϕ λ. Having compued he momens of S as he produc of corresponding momens of B and S he lognormal approximaions o he disribuions of S and R are S 11.6 S LNm, v R LNm v, v, where 11.7 v = log 1 + V AR S m = log E S 1 2 v. /E S δ 2 Also, he noncenral gamma approximaions o he disribuions of S are S 11.8 NCGα, γ, λ R NCGα, γ, λ, where α, γ, λ are given by γ = 2 V AR S SKEW + V AR S δ S SKEW 2 S α = 2γE S γ 2 V AR S λ = 2γE S + 2γ 2 V AR S E S SKEW S and α, γ, λ have he corresponding formulae in erms of momens of R. is and R

21 2 KEVIN FERGUSSON AND ECKHARD PLAEN Using hese approximaions we can compue prices of various opions sraighforwardly from Appendix B: ables of Percenile Coss of Hedging able 2. Percenile coss of hedging pu opions under a deerminisic shor rae & Black-Scholes discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y able 3. Percenile coss of hedging pu opions under a deerminisic shor rae & MMM discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y

22 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 21 Curren address, Kevin Fergusson: Curin Universiy GPO Box U1987 address, Kevin Fergusson: URL: hp:// Curren address, Eckhard Plaen: Universiy of echnology, Sydney PO Box 123, Broadway NSW 27, Ausralia address, Eckhard Plaen: URL: hp:// able 4. Percenile coss of hedging pu opions under a Vasicek shor rae & Black-Scholes discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y able 5. Percenile coss of hedging pu opions under a Vasicek shor rae & MMM discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y

23 22 KEVIN FERGUSSON AND ECKHARD PLAEN able 6. Percenile coss of hedging pu opions under a CIR shor rae & Black-Scholes discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y able 7. Percenile coss of hedging pu opions under a CIR shor rae & MMM discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y

24 LESS EXPENSIVE PRICING AND HEDGING OF LONG-DAED OPIONS 23 able 8. Percenile coss of hedging pu opions under a 3/2 shor rae & Black-Scholes discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y able 9. Percenile coss of hedging pu opions under a 3/2 shor rae & MMM discouned GOP based on US daa erm o Expiry 99-h 95-h 9-h 85-h 8-h of Pu Opion Percenile Percenile Percenile Percenile Percenile 1Y Y Y Y Y Y Y Y Y Y Y Y Y