We extend to the Heston stochastic volatility framework the parity result of McDonald and Schroder (1998)
for American call and put options

Battauz, Anna

Donno, De

Sbuelz, Alessandro (ORCID:0000-0002-8602-3603)

English

Copyright c © 2014 Battauz, Donno and Sbuelz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. For the American put-call option symmetry in the Heston (1993) model, we provide a new and simple proof that is easily accessible to the general finance readership. We also characterize the link between the free-boundary of the American call and the free boundary of the symmetric American put

Anna Battauz

Marzia De Donno

Alessandro Sbuelz

CiteSeerX

THE PUT-CALL SYMMETRY FOR AMERICAN OPTIONS IN THE HESTON STOCHASTIC VOLATILITY MODEL

For the American put-call option symmetry in the Heston (1993) model, we provide a new and simple proof that is easily accessible to the general finance readership. We also characterize the link between the free-boundary of the American call and the free-boundary of the symmetric American put

A. Battauz

A. Sbuelz

DE DONNO, Marzia

Archivio istituzionale della Ricerca -  Università degli Studi di Parma

The put-call symmetry for American options in the Heston stochastic volatility model

For the American put-call option symmetry in the Heston (1993) model, we provide a new and simple
proof that is easily accessible to the general finance readership. We also characterize the link between the freeboundary
of the American call and the free boundary of the symmetric American pu

BATTAUZ, ANNA

SBUELZ, ALESSANDRO

MARZIA DE DONNO

Archivio istituzionale della Ricerca - Bocconi

We extend to the Heston stochastic volatility framework the parity result of McDonald and Schroder (1998) for American call and put options

We extend to the Heston stochastic volatility framework the parity result of McDonald and Schroder (1998)
for American call and put options

PubliCatt

The put-call symmetry for American optionsin the Heston stochastic volatility modelAnna BattauzDepartment of Finance,and IGIER,Bocconi University,Milan, ItalyMarzia De DonnoDepartment of Economics,University of Parma,ItalyAlessandro SbuelzDepartment of Mathematics, Quantitative Finance,and Econometrics,Catholic University of Milan,Milan, ItalyJuly 2014AbstractWe extend to the Heston stochastic volatility framework the parity result of McDonald and Schroder (1998)for American call and put options.Keywords: Heston model, American options, put-symmetry, free-boundary.AMS Code: 60G40, 60G46, 91G20. JEL Code: G12, G13.1 IntroductionSeveral authors have studied American options within the Heston model (see for instance Broadie and Kaya(2006), Andersen (2008), Vellekoop and Nieuwenhuis (2009) and the references therein). This paper con-tributes to the literature on American options in the Heston model by providing the link between Americanput options and American call options in this framework. In the European case, the put-call parity relatesthe prices of European call and put options on the same underlying asset, with the same maturity and thesame strike via the law of one price. Violations of the put-call parity lead to arbitrages that are eagerlyCorresponding author. E-mail: anna.battauz@unibocconi.it.1exploited by investors. When the options are American, early exercise is possible before maturity. Buy-and-holding an American option therefore is not anymore a self-nancing strategy. The put-call parity fails inthe American case. However it is possible to derive a symmetry relation (see Carr and Chesney (1996) andSchroder (1999)) that is very signicant, since American options are vastly traded. Moreover, the symmetryrelation is useful for the analysis of optimal decision making for real option holders (see for instance Battauzand alii, 2012 and 2014). In the standard Black-Scholes model the American put-call symmetry relates theprice of an American call option to the price of an American put option by swapping the initial underlyingprice with the strike price and the dividend yield with the interest rate. Similar symmetry results have beenobtained by Fajardo and Mordecki (2008) when the underlying asset follows a Levy process. In this paperwe provide a simple proof of the symmetry relation between American call and put options in the Hestonstochastic volatility model. Our result is obtained by applying the change of numeraire (see Geman, et alii,(1995) for a discussion on the change of numeraire technique and A. Battauz (2002) for appilications toAmerican options). See also Meyer (2013) for an alternative proof based on partial di¤erential equations.2 The American put-call symmetry in the Heston model.In the Heston model (see Heston (1993)) the stock price S is described by the following stochastic di¤erentialequation with respect to the risk-neutral measure QdS (s)S (s)= (r   q) ds+pv (s)dW1 (s) ; S (0) = S0 for any s  0 (1)dv (s) = k (v   v (s)) ds+ pv (s)dW1 (s) +p1  2dW2 (s)v (0) = v0 (2)where W1 and W2 are two independent standard Brownian motions under the risk neutral measure Q andthe ltration F ; r is the riskless interest rate; q is the dividend yield of the stock; pv (s) is the stochasticvolatility of S at time s; v is the long variance; k is the speed of mean reversion of v towards v;  is thevol of vol ;  is the correlation between S and v: We assume that 2kv > 2; to ensure that the volatility isalways positive.We denote by B (t) = ert the riskless bond at date t:Consider now an American call option on S: Its no-arbitrage price isc (t) = ess suptTEhe r( t) (S() K)+Fti (3)for any t 2 [0; T ] ; where E [] denotes the (conditional) risk neutral expectation, and  denotes a stoppingtime with respect to the ltration F . It can be shown that c (t) is a deterministic function of t; S (t) and2current levels of volatilitypv (t): With a small abuse of notations we writec (t) = c (t; S (t) ; v (t)) :The function c depends on the values of the fundamental parameters. We denote such dependence by writingc (t) = c (t; S (t) ; v (t) ; r; q; v; k; ; ;K) :The no-arbitrage price of the American put option on S isp (t) = ess suptTEhe r( t) (K   S())+Fti (4)for any t 2 [0; T ] : It can be shown that p (t) is a deterministic function of t; S (t) and current levels ofvolatilitypv (t): With a small abuse of notations we writep (t) = p (t; S (t) ; v (t)) = p (t; S (t) ; v (t) ; r; q; v; k; ; ;K) :As we already anticipated, in the American it is possible to write c (t) in terms of a symmetric Americanput option,whose denition in the stochastic volatility setting is provided here follows:Denition 1 (The symmetric put option) The symmetric American put option associated to the Amer-ican call option (3) is the American put option on a Heston underlying Sput driven by the following equationsfor s  tdSput (s)Sput (s)= putds+qvput (s)dW1 (s) ; (5)dvput (s) = kput (vput   vput (s)) ds+ putqvput (s)putdW1 (s) +q1  2putdW2 (s)(6)where the values for the fundamental parameters are: Sput (t) = K; put = q r; vput (t) = v (t) ; vput = kvk  ;kput = (k   ) ; put = ; put =  ; rput = q; and Kput = S (t) :In the next theorem we provide the fundamental symmetry result that relates the time t price of theAmerican call option c (t) to the time t price of the symmetric American put option.Theorem 2 (American put-call symmetry) Consider the American call option dened in (3) whosevalue at time t 2 [0;T ] is denoted with c (t) = c (t; S (t) ; v (t) ; r; q; v; k; ; ;K).Consider the symmetric American put option dened in Denition 1, whose value at time t 2 [0;T ] isdenoted withp (t) = p (t; Sput (t) ; vput (t) ; rput; qput; vput; kput; put; put;Kput) :3The value of the American call coincides with the value of the symmetric American put as dened in De-nition 1. More precisely, for any 0  t  T we havec (t; S (t) ; v (t) ; r; q; v; k; ; ;K) = p (t; Sput (t) ; vput (t) ; rput; qput; vput; kput; put; put;Kput) (7)Moreover, given x = S (t) ; K and v = v (t) ; for any bxput; bKput such that xK = bKputbxput we have thatc (t; x; v; r; q; v; k; ; ;K) =pxKpt; bxput; vput; rput; qput; vput; kput; put; put; bKputqbxput bKput ; (8)where bxput replaces Sput (t) and bKput replaces Kput in Denition 1.Proof. Dene the numeraire (see Battauz (2002)) N (t) = S (t) e (r q)t;which is a Q martingale, sincedN(t)N(t) =pv (t)dW1 (t) : The numeraire N is associated to the equivalent martingale measure QN whosedensity with respect to Q is L (T ) = dQNdQ =N(T )N(0) : Girsanov theorem ensures thatdWN1 (t) =  pv (t) dt + dW1 (t) ; dWN2 (t) = dW2 (t) (9)are the di¤erentials of two standard independent QN Brownian motions.We apply the change of numeraire to c (t) in (3).To evaluate the American call option at any t; we consider a generic stopping time t    T and com-pute Ee r( t) (S() K)+Ft = EQN h 1L(T ) e r( t)(S() K)+FtiEQNh1L(T )Fti =EQNhEQNh1L(T )e r( t)(S() K)+FiFtiEQNh1L(T )Fti ;where the rst equation follows from Bayes theorem, and the second from the law of iterated condi-tional expectation. Since e r( t) (S() K)+ is F measurable and 1L(t) is a QN martingale we getEe r( t) (S() K)+Ft = EQN he r( t)(S() K)+ 1L() Fti1L(t)= L (t)EQNhe r( t) (S() K)+ 1L()Fti :Recalling the denition of L we obtain Ee r( t) (S() K)+Ft = S(t)e (r q)tS(0) EQN e r( t) (S() K)+ S(0)S()e (r q)Fti = S(t)S(0)EQN e q( t) S (0)  S(0)KS() +Ft = EQN e q( t) S (t)  S(t)KS() +Ft :Passing to the essential supremum over all stopping times t    T we get thatc (t) = ess suptTEQN"e q( t)S (t)  S (t)KS ()+Ft#(10)The argument of the Ft expectation under QN in Equation (10) is the payo¤ at   t of an Americanput option with maturity T; interest rate rput = q; strike Kput = S (t) = x on the asset Sput (s) = xKS(s) :Applying Ito formula we derive the stochastic di¤erential of Sput for any s  t : dSput (s) = xK  d1S(s)=xKS(s)   (r   q) ds pv (s)dW1 (s) + v (s) ds = Sput (s)    (r   q) ds pv (s)dW1 (s) + v (s) ds. FromEquation (9) we substitute dW1 (s) =pv (s) ds + dWN1 (s) and getdSput(s)Sput(s)=   (r   q) ds   pv (s)4pv (s) ds + dWN1 (s)+v (s) ds = (q   r) ds pv (s)dWN1 (s) : Therefore the underlying of the Americanput option is driven under the evaluation measure QN by the Heston dynamicsof type (1)dSput (s)Sput (s)= (q   r) ds pv (s)dWN1 (s) ;with rput = q and qput = r. We verify now that the volatility term follows a dynamics of the same type ofEquation (2) : By Girsanov theorem (9) ; v (s) is driven by dv (s) = k (v   v (s)) ds+ pv (s) (dW1 (s) +p1  2dW2 (s)= (k   )kvk    v (s)ds+pv (s)dWN1 (s) +p1  2dWN2 (s):Since dcWN1 (s) = dWN1 (s) denes a standard QN Brownian motion that is QN independent of WN2 we have thatdSput (s)Sput (s)= (q   r) ds+pv (s)dcWN1 (s) anddv (s) = (k   )kvk      v (s)ds+ pv (s)( ) dcWN1 (s) +p1  2dWN2 (s) :Therefore under QN the underlying of the put option Sput follows an Heston dynamics with Sput (t) = K;vput =kvk  ; kput = (k   ) ; put = ; and put =  ; as in Denition 1. We conclude that Equation(10) can be rewritten as c (t) = ess suptT EQN e q( t) (Kput   Sput ())+Ft = p (t; Sput (t) ; vput (t) ;rput; qput; vput; kput; put; put;Kput), which is (4) :To prove (8) ; take a  > 0 such that bKput = x ; is an unconstrained strike for the put option, and letbxput = Sput(t) = K : The remaining parameters for the symmetric put are rput; qput; vput; kput; put; put;Kputas before: for simplicity we omit them. By formula (7) c (t; x; :::;K) = p (t;K; :::; x) = pt; K ; :::;x=  pt; bxput; :::; bKput ; where the second equality follows from the homogeneity property of the put option.Since  = xbKput = Kbxput ; writing  =p   =qxbKput  Kbxput ; we arrive at (8). In the constant volatility framework, the optimal exercise policy for an American call option is the rsttime the underlying asset exceeds the critical price. The critical price is time-varying, and its graph inthe plane (t; S) separating the continuation region from the immediate exercise region is called the freeboundary. In the Heston model, the free boundary is a surface in the space (t; S; v): The free boundaryof the American call option is linked to the free boundary of the symmetric American put option via thefollowing theorem:Theorem 3 (The free boundary) Consider the American call option dened in (3) whose value at timet 2 [0;T ] is denoted with c (t) = c (t; S (t) ; v (t) ; r; q; v; k; ; ;K) = c(t; x; v; :::;K). The free boundary forthe American call option at t and v = v (t) isfb(t; v) = infx  0 : c(t; x; v; :::;K) = (x K)+	 :5Let bKput = 1 and consider the symmetric American put option where bxput replaces Sput (t) and bKput =1 replaces Kput in Denition 1 as for (8). The free boundary of the symmetric American put optionvput (t; bxput; vput; rput; qput; vput; kput; put; put; 1) = vput (t; bxput; vput; :::; 1) isfbput(t; vput) = supbxput  0 : vput (t; bxput; vput; :::; 1) = (1  bxput)+	 :Thenfb(t; v) = K  fbput(t; vput)Proof. The parameters x;K ; and bxput are constrained by the equality xK = 1bxput : It follows that fb(t; v) =infKbxput  0 : pxK p(t;bxput;vput;:::;1)pbxput bKput =Kbxput  K+=K supbxput  0 : pxK vput(t;bxputvput;:::;1)pbxput = Kbxput (1  bxput)+= K supbxput  0 :q KbxputK vput(t;bxput;vput;:::;1)pbxput = Kbxput (1  bxput)+; since x = Kbxput : Therefore fb(t; v) =K  supbxput  0 : vput (t; bxput; vput; :::; 1) = (1  bxput)+	 = K  fbput (t; bxput; vput; :::; 1) :3 ConclusionsIn this paper we provide a simple proof for the symmetry between American call and put options in theHeston stochastic volatility framework, relying on the change of numeraire technique. We supply also thelink between the free boundaries of the symmetric American options.References[1] L. Andersen (2008). Simple and e¢ cient simulation of the Heston stochastic volatility model, TheJournal of Computational Finance 11(3), 142.[2] A. Battauz (2002). Change of numeraire and American options, Stochastic Analysis and Applications,20 (04), 709-730.[3] Battauz, A., M. De Donno, A. Sbuelz (2012). Real options with a double continuation region, Quanti-tative Finance, 12(3), 465-475.[4] A. Battauz, M. De Donno, A. Sbuelz (2014). Real options and American derivatives: The doublecontinuation region, Management Science, http://dx.doi.org/10.1287/mnsc.2013.1891.6[5] M. Broadie, O. Kaya (2006). Exact simulation of stochastic volatility and other a¢ ne jump di¤usionprocesses, Operations Research 54, 217231.[6] P. Carr, M. Chesney (1996). American Put Call Symmetry, downloadable athttp://www.math.nyu.edu/research/carrp/papers/pdf/apcs2.pdf.[7] J. Fajardo, E. Mordecki (2008). Duality and Symmetry with Time-Changed Levy Processes, BrazilianReview of Econometrics 28 (1).[8] H. Geman, N. El Karoui, J.C. Rochet (1995). Changes of numeraire, changes of probability measureand option pricing, Journal of Applied Probability 32, 443-458.[9] S. L. Heston (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applicationsto Bond and Currency Options, The Review of Financial Studies, Volume 6(2), 327343.[10] R.L. McDonald and M. Schroder (1998). A parity result for American options, Journal of ComputationalFinance, 1 (3).[11] G. H. Meyer (2013). Pricing Options and Bonds with the Method of Lines,http://people.math.gatech.edu/~meyer/MOL-notes/BookGHM.pdf.[12] M. Vellekoop, H. Nieuwenhuis (2009). A tree-based method to price American options in the Hestonmodel, The Journal of Computational Finance Volume 13(1), 121.7

For the American put-call option symmetry in the Heston (1993) model, we provide a new and simple proof that is easily accessible to the general finance readership. We also characterize the link between the freeboundary of the American call and the free boundary of the symmetric American put

De Donno, Marzia (ORCID:0000-0003-0188-4437)

Available online at http://scik.orgMath. Finance Lett. 2014, 2014:7ISSN 2051-2929THE PUT-CALL SYMMETRY FOR AMERICAN OPTIONS IN THE HESTONSTOCHASTIC VOLATILITY MODELANNA BATTAUZ1,∗, MARZIA DE DONNO2, ALESSANDRO SBUELZ31Department of Finance and IGIER, Bocconi University, Milan, Italy2Department of Economics, University of Parma, Milan, Italy3Department of Mathematics, Quantitative Finance, and Econometrics, Catholic University of Milan, Milan, ItalyAbstract. For the American put-call option symmetry in the Heston (1993) model, we provide a new and simpleproof that is easily accessible to the general finance readership. We also characterize the link between the free-boundary of the American call and the free boundary of the symmetric American put.Keywords: American options; stochastic volatility; put-call symmetry; free-boundary; change of numeraire.2010 AMS Subject Classification: 60G40, 60G46, 91G20.1. IntroductionSeveral authors have studied American options within the Heston (1993) model (e.g. Broadieand Kaya (2006), Andersen (2008), and Vellekoop and Nieuwenhuis (2009)). We contribute byproviding a proof of the pricing parity between the American call and its symmetric Americanput in the Heston (1993) model that is easily accessible to a general finance audience. In theEuropean case, the put-call parity relates the prices of European call and put options on thesame underlying asset, with the same maturity and the same strike via the law of one price.In the American case the put-call parity fails, but it is possible to derive a put-call symmetryrelation. Such a relation is important for pricing purposes given the size of American derivatives∗Corresponding authorReceived July 9, 201412 ANNA BATTAUZ, MARZIA DE DONNO ALESSANDRO SBUELZmarkets. Moreover such a relation is useful for the analysis of optimal decision making forreal option holders (see for instance Battauz et alii (2012) and (2014)). The American put-call symmetry relation equates the price of an American call to the price of an American put byswapping the initial underlying price with the strike price and the dividend yield with the interestrate. American put-call symmetry results1 have been obtained by Carr and Chesney (1996)and McDonald and Schroder (1998) in the absence of stochastic volatility, and by Fajardo andMordecki (2008) in a Levy process framework. We provide a change-of-numeraire-based proofof the American put-call symmetry in the Heston (1993) stochastic volatility model (see Gemanet alii (1995) for a discussion on the change of numeraire technique and Battauz (2002) forapplications to American options). We also characterize the link between the free-boundary ofthe American call and that of the symmetric American put. Meyer (2013) offers an alternativeproof based on partial differential equations.2. The American put-call symmetry in the Heston (1993) modelIn the Heston (1993) model the stock price S is described by the following stochastic differ-ential equation with respect to the risk-neutral measure QdS (s)S (s)= (r−q)ds+√v(s)dW1 (s) , S (0) = S0 for any s≥ 0(1)dv(s) = k (v− v(s))ds+ξ√v(s)(ρdW1 (s)+√1−ρ2dW2 (s))v(0) = v0,(2)where W1 and W2 are two independent standard Brownian motions under the risk neutral mea-sure Q and the filtration F ; r is the riskless interest rate; q is the dividend yield of the stock;√v(s) is the stochastic volatility of S at time s; v is the long variance; k is the speed of meanreversion of v towards v; ξ is the vol of vol; ρ is the correlation between S and v. We assumethat 2kv > ξ 2, to ensure that the volatility is always positive.We denote by B(t) = ert the riskless bond at date t.Consider now an American call option on S. Its no-arbitrage price is(3) c(t) = ess supt≤τ≤TE[e−r(τ−t) (S(τ)−K)+∣∣∣Ft]1For a general treatment of the European put-call symmetry see Carr and Lee (2009).THE PUT-CALL SYMMETRY FOR AMERICAN OPTIONS 3for any t ∈ [0,T ] , where E [·] denotes the (conditional) risk neutral expectation, and τ denotesa stopping time with respect to the filtration F . It can be shown that c(t) is a deterministicfunction of t, S (t) and current levels of volatility√v(t). With a small abuse of notations wewritec(t) = c(t,S (t) ,v(t)) .The function c depends on the values of the fundamental parameters. We denote such depen-dence by writingc(t) = c(t,S (t) ,v(t) ;r,q,v,k,ξ ,ρ,K) .The no-arbitrage price of the American put option on S is(4) p(t) = ess supt≤τ≤TE[e−r(τ−t) (K−S(τ))+∣∣∣Ft]for any t ∈ [0,T ] . It can be shown that p(t) is a deterministic function of t, S (t) and currentlevels of volatility√v(t). With a small abuse of notations we writep(t) = p(t,S (t) ,v(t)) = p(t,S (t) ,v(t) ;r,q,v,k,ξ ,ρ,K) .As we already anticipated, in the American case it is possible to write c(t) in terms of a sym-metric American put option, whose definition in the stochastic volatility setting is provided herefollows:Definition 2.1. (The symmetric put option) The symmetric American put option associatedto the American call option (3) is the American put option on a Heston (1993) underlying Sputdriven by the following equations for s≥ tdSput (s)Sput (s)= µputds+√vput (s)dW1 (s) ,(5)dvput (s) = kput (vput− vput (s))ds+ξput√vput (s)(ρputdW1 (s)+√1−ρ2putdW2 (s)),(6)where the values for the fundamental parameters are: Sput (t) = K, µput = q− r, vput (t) = v(t) ,vput = kvk−ξρ , kput = (k−ξρ) , ξput = ξ , ρput =−ρ, rput = q, and Kput = S (t) .In the next theorem, we provide the fundamental symmetry result that relates the time−t priceof the American call option c(t) to the time−t price of the symmetric American put option.4 ANNA BATTAUZ, MARZIA DE DONNO ALESSANDRO SBUELZTheorem 2.2. (American put-call symmetry) Consider the American call option defined in (3)whose value at time t ∈ [0;T ] is denoted with c(t) = c(t,S (t) ,v(t) ;r,q,v,k,ξ ,ρ,K).Consider the symmetric American put option defined in Definition 2.1, whose value at timet ∈ [0;T ] is denoted withp(t) = p(t,Sput (t) ,vput (t) ;rput ,qput ,vput ,kput ,ξput ,ρput ,Kput) .The value of the American call coincides with the value of the symmetric American put asdefined in Definition 2.1. More precisely, for any 0≤ t ≤ T we have(7)c(t,S (t) ,v(t) ;r,q,v,k,ξ ,ρ,K) = p(t,Sput (t) ,vput (t) ;rput ,qput ,vput ,kput ,ξput ,ρput ,Kput) .Moreover, given x = S (t) , K and v = v(t) , for any x̂put , K̂put such that xK =K̂putx̂putwe have that(8) c(t,x,v;r,q,v,k,ξ ,ρ,K) =√xKp(t, x̂put ,vput ;rput ,qput ,vput ,kput ,ξput ,ρput , K̂put)√x̂putK̂put,where x̂put replaces Sput (t) and K̂put replaces Kput in Definition 2.1.Proof. Define the numeraire (see Battauz (2002)) N (t)= S (t)e−(r−q)t ,which is a Q−martingale,since dN(t)N(t) =√v(t)dW1 (t) . The numeraire N is associated to the equivalent martingale mea-sure QN whose density with respect to Q is L(T ) = dQNdQ =N(T )N(0) . Girsanov theorem ensuresthat(9) dW N1 (t) =−√v(t) dt + dW1 (t) , dW N2 (t) = dW2 (t)are the differentials of two standard independent QN Brownian motions.We apply the change of numeraire to c(t) in (3).To evaluate the American call option at any t, we consider a generic stopping time t ≤ τ ≤ Tand computeE[e−r(τ−t) (S(τ)−K)+∣∣∣Ft]= EQN [ 1L(T )e−r(τ−t)(S(τ)−K)+∣∣∣Ft]EQN [ 1L(T )∣∣∣Ft]=EQN[EQN[1L(T )e−r(τ−t)(S(τ)−K)+∣∣∣Fτ]∣∣∣Ft]EQN [ 1L(T )∣∣∣Ft] , where the first equation follows from Bayes theorem,and the second from the law of iterated conditional expectation. Since e−r(τ−t) (S(τ)−K)+ isFτ−measurable and 1L(t) is a QN−martingale we getTHE PUT-CALL SYMMETRY FOR AMERICAN OPTIONS 5E[e−r(τ−t) (S(τ)−K)+∣∣∣Ft]= EQN [e−r(τ−t)(S(τ)−K)+ 1L(τ) ∣∣∣Ft]1L(t)= L(t)EQN[e−r(τ−t) (S(τ)−K)+ 1L(τ)∣∣∣Ft] .Recalling the definition of L we obtainE[e−r(τ−t) (S(τ)−K)+∣∣∣Ft]= S(t)e−(r−q)tS(0) EQN [e−r(τ−t) (S(τ)−K)+ · S(0)S(τ)e−(r−q)τ ∣∣∣Ft]= S(t)S(0)EQN[e−q(τ−t)(S (0)− S(0)KS(τ))+∣∣∣∣Ft] = EQN [e−q(τ−t)(S (t)− S(t)KS(τ) )+∣∣∣∣Ft] .Passing to the essential supremum over all stopping times t ≤ τ ≤ T we get that(10) c(t) = ess supt≤τ≤TEQN[e−q(τ−t)(S (t)− S (t)KS (τ))+∣∣∣∣∣Ft].The argument of the Ft−expectation under QN in Equation (10) is the payoff at τ ≥ t ofan American put option with maturity T, interest rate rput = q, strike Kput = S (t) = x onthe asset Sput (s) = xKS(s) . Applying Ito formula we derive the stochastic differential of Sputfor any s ≥ t : dSput (s) = xK · d(1S(s))= xKS(s) ·(−(r−q)ds−√v(s)dW1 (s)+ v(s)ds) =Sput (s) ·(−(r−q)ds−√v(s)dW1 (s)+ v(s)ds). From Equation (9) we substitute dW1 (s) =√v(s) ds+ dW N1 (s) and getdSput(s)Sput(s)=−(r−q)ds−√v(s)· (√v(s) ds + dW N1 (s))+v(s)ds= (q− r)ds−√v(s)dW N1 (s) . Therefore the underlying of the American put option is drivenunder the evaluation measure QN by the “Heston (1993) dynamics” of type (1)dSput (s)Sput (s)= (q− r)ds−√v(s)dW N1 (s) ,with rput = q and qput = r. We verify now that the volatility term follows a dynamics of thesame type of Equation (2) . By Girsanov theorem (9) , v(s) is driven bydv(s)= k (v− v(s))ds+ ξ√v(s)(ρdW1 (s) +√1−ρ2dW2 (s))= (k−ξρ)( kvk−ξρ − v(s))ds+ξ√v(s)(ρdW N1 (s)+√1−ρ2dW N2 (s)). Since dŴ N1 (s)=−dW N1 (s) defines a standardQN−Brownian motion that is QN−independent of W N2 we have thatdSput (s)Sput (s)= (q− r)ds+√v(s)dŴ N1 (s) anddv(s) = (k−ξρ)(kvk−ξρ − v(s))ds+ξ√v(s)((−ρ)dŴ N1 (s)+√1−ρ2dW N2 (s)).6 ANNA BATTAUZ, MARZIA DE DONNO ALESSANDRO SBUELZTherefore under QN the underlying of the put option Sput follows an Heston (1993) dynamicswith Sput (t) = K, vput = kvk−ξρ , kput = (k−ξρ) , ξput = ξ , and ρput =−ρ, as in Definition 2.1.We conclude that Equation (10) can be rewritten asc(t) = esssupt≤τ≤T EQN[e−q(τ−t) (Kput−Sput (τ))+∣∣∣Ft]= p(t,Sput (t) ,vput (t) ; rput ,qput ,vput ,kput ,ξput ,ρput ,Kput), which is (4) .To prove (8) , take a β > 0 such that K̂put = xβ , is an unconstrained strike for the put option,and let x̂put =Sput(t)β =Kβ . The remaining parameters for the symmetric put arerput ,qput ,vput ,kput ,ξput ,ρput ,Kput as before: for simplicity we omit them. By formula (7)c(t,x, ...,K) = p(t,K, ...,x) = β p(t, Kβ , ...,xβ)= β · p(t, x̂put , ..., K̂put), where the second e-quality follows from the homogeneity property of the put option. Since β = xK̂put= Kx̂put , writingβ =√β ·β =√xK̂put· Kx̂put , we arrive at (8).In the constant volatility framework, the optimal exercise policy for an American call optionis the first time the underlying asset exceeds the critical price. The critical price is time-varying,and its graph in the plane (t,S) separating the continuation region from the immediate exerciseregion is called the free boundary. In the Heston (1993) model, the free boundary is a surface inthe space (t,S,v). The free boundary of the American call option is linked to the free boundaryof the symmetric American put option via the following theorem.Theorem 2.3. (The free boundary) Consider the American call option defined in (3) whosevalue at time t ∈ [0;T ] is denoted with c(t) = c(t,S (t) ,v(t) ;r,q,v,k,ξ ,ρ,K) = c(t,x,v; ...,K).The free boundary for the American call option at t and v = v(t) isf b(t,v) = inf{x≥ 0 : c(t,x,v; ...,K) = (x−K)+} .Let K̂put = 1 and consider the symmetric American put option where x̂put replaces Sput (t)and K̂put = 1 replaces Kput in Definition 2.1 as for (8). The free boundary of the symmetricAmerican put option vput (t, x̂put ,vput ;rput ,qput ,vput ,kput ,ξput ,ρput ,1) = vput (t, x̂put ,vput ; ...,1)isf bput(t,vput) = sup{x̂put ≥ 0 : vput (t, x̂put ,vput ; ...,1) = (1− x̂put)+}.Thenf b(t,v) = K · f bput(t,vput).THE PUT-CALL SYMMETRY FOR AMERICAN OPTIONS 7Proof. The parameters x,K , and x̂put are constrained by the equality xK =1x̂put. It follows thatf b(t,v) = inf{Kx̂put≥ 0 :√xK p(t,x̂put ,vput ;...,1)√x̂put K̂put=(Kx̂put−K)+}= K sup{x̂put ≥ 0 :√xKvput(t,x̂putvput ;...,1)√x̂put= Kx̂put (1− x̂put)+}= K sup{x̂put ≥ 0 :√Kx̂putKvput(t,x̂put ,vput ;...,1)√x̂put= Kx̂put (1− x̂put)+}, since x = Kx̂put . Thereforef b(t,v)=K ·sup{x̂put ≥ 0 : vput (t, x̂put ,vput ; ...,1) = (1− x̂put)+}=K · f bput (t, x̂put ,vput ; ...,1) .3. ConclusionsWe employ a change-of-numeraire technique to provide a new and easy proof of the pricingparity between the American call and its symmetric American put in the Heston (1993) stochas-tic volatility model. We work out in detail the link between the free boundaries of the symmetricAmerican options.Conflict of InterestsThe authors declare that there is no conflict of interests.REFERENCES[1] L. Andersen, Simple and efficient simulation of the Heston stochastic volatility model, J. Comput. Finance11 (2008), 1–42.[2] A. Battauz, Change of numeraire and American options, Stochastic Anal. Appl. 20 (2002), 709-730.[3] A. Battauz, M. De Donno, A. Sbuelz, Real options with a double continuation region, Quantitative Finance,12 (2012), 465-475.[4] A. Battauz, M. De Donno, A. Sbuelz, Real options and American derivatives: The double continuation region,Management Science, in press, (2014).[5] M. Broadie, O. Kaya, Exact simulation of stochastic volatility and other affine jump diffusion processes,Operations Res. 54 (2006), 217–231.[6] P. Carr, M. Chesney, American Put Call Symmetry, (1996).[7] P. Carr, R. Lee, Put-call symmetry. Extensions and applications, Math. Finance, 19 (2009), 523–560.[8] J. Fajardo, E. Mordecki, Duality and Symmetry with Time-Changed Levy Processes, Brazilian Rev. Econo-metrics (2008).8 ANNA BATTAUZ, MARZIA DE DONNO ALESSANDRO SBUELZ[9] H. Geman, N. El Karoui, J.C. Rochet, Changes of numeraire, changes of probability measure and optionpricing, J. Appl. Probability 32 (1995), 443-458.[10] S. L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond andCurrency Options, The Review of Financial Studies, Volume 6 (1993), 327–343.[11] R.L. McDonald and M. Schroder, A parity result for American options, J. Comput. Fiance 1 (1998), 5-13.[12] G. H. Meyer, Pricing Options and Bonds with the Method of Lines, (2013).[13] M. Vellekoop, H. Nieuwenhuis, A tree-based method to price American options in the Heston model, J.Comput. Finance 13 (2009), 1–21.

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The put-call symmetry for American options in the Heston stochastic volatility model

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