Bonn Econ Discussion Papers

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1 Bo Eco Discussio Papers Discussio Paper 7/21 Covertible Bods: Risks ad Optimal Strategies by Haishi Huag April 21 Bo Graduate School of Ecoomics Departmet of Ecoomics Uiversity of Bo Kaiserstrasse 1 D Bo

2 Fiacial support by the Deutsche Forschugsgemeischaft (DFG) through the Bo Graduate School of Ecoomics (BGSE) is gratefully ackowledged. Deutsche Post World Net is a sposor of the BGSE.

3 Covertible Bods: Risks ad Optimal Strategies Haishi Huag Istitute of Social Scieces ad Ecoomics, Uiversity of Bo Abstract Withi the structural approach for credit risk models we discuss the optimal exercise of the callable ad covertible bods. The Vasi cek model is applied to icorporate iterest rate risk ito the firm s value process which follows a geometric Browia motio. Fially, we derive pricig bouds for covertible bods i a ucertai volatility model, i.e. whe the volatility of the firm value process lies betwee two extreme values. Keywords: Covertible bod, game optio, ucertai volatility, iterest rate risk JEL: G12, G33 1 Itroductio Callable ad covertible bods have attracted substatial research attetio due to their exposure to both credit ad market risk ad the correspodig optimal coversio ad call strategies. The bodholder receives coupos plus the retur of pricipal at maturity, give that the issuer (usually the shareholder) does ot default o the obligatios. Moreover, prior to ad at the maturity the bodholder has the right to covert the bod ito a give umber of stocks. O the other had, the bod is also callable by the issuer, i.e. the bodholder ca be eforced to surreder the bod to the issuer for a previously agreed price. I the cotext of the structural model the arbitrage free pricig problem was first treated by Brea ad Schwarz (1977) ad Igersoll (1977). Recet articles of Sirbu, Pilovsky ad Schreve (24) ad Kallse ad Küh (25) treat the optimal behavior of the cotract parters more rigorously. The valuatio of callable ad covertible bod is explicitly related to the game optio. Empirical research idicates that firms that issue covertible bods ofte ted to be highly leveraged, the default risk may play a sigificat role. Moreover, the equity ad default risk caot be treated idepedetly ad their iterplay must be modeled explicitly. Default risk models ca be categorized ito two fudametal classes: firm s value models or structural models, ad reduced-form or default-rate models. I the structural model, oe costructs a stochastic process of the firm s value which idirectly leads to default, while i the reduced-form model the default process is modeled directly. I the structural models default risk depeds maily o the stochastic evolutio of the asset value ad default occurs whe the radom variable describig the firm s value is isufficiet for repaymet of debt. Istead of askig why the firm defaults, i the reduced-form model formulatio, the itesity of the default process is modeled exogeously by usig both market-wide as well as firm-specific factors, such as stock prices ad default itesities. While both approaches have certai shortcomigs, the stregth of the structural approach is that it provides ecoomical explaatio of the capital structure decisio, default triggerig, ifluece of divided paymets ad of the behaviors of debtor ad creditor. 1

4 It describes why a firm defaults ad it allows for the descriptio of the strategies of the debtor ad creditor. Especially for complex cotracts where the strategic behaviors of the debtor ad the creditor play a importat role, structural models are well suited for the aalysis of the relative power of shareholders ad creditors. Aother reaso that we work with the structural approach is because it allows for a itegrated model of equity ad default risk through commo depedece o stochastic variables. Callable ad covertible bods are America-style cotract, meaig that coversio is allowed at ay time durig the life of the cotract, ad by existece of a call provisio for the issuer this leads to a problem of optimal stoppig for both bodholder ad issuer. Therefore whe we compute the o-arbitrage price of such a cotract, we have to take ito accout the aspect of strategic optimal behaviors which are the study focus of this paper. Based o the results of Kifer (2) ad Kallse ad Küh (25) we show that the optimal strategy for the bodholder is to select the stoppig time which maximizes the expected payoff give the miimizig strategy of the issuer, while the issuer will choose the stoppig time that miimizes the expected payoff give the maximizig strategy of the bodholder. This max-mi strategy of the bodholder leads to the lower value of the covertible bod, whereas the mi-max strategy of the issuer leads to the upper value of the covertible bod. The assumptio that the call value is always larger tha the coversio value prior to maturity T ad they are the same at maturity T esures that the lower value equals the upper value such that there exists a uique solutio. The o-arbitrage price ca be approximated umerically by meas of backward iductio. I absece of iterest rate risk, the recursio procedure is carried out o the Cox-Ross-Rubistei biomial lattice. To icorporate the ifluece of the iterest rate risk, we use a combiatio of a aalytical approach ad a biomial tree approach developed by Mekveld ad Vorst (1998) where the iterest rate is Gaussia ad correlatio betwee the iterest rate process ad the firm s value process is explicitly modeled. We show that the ifluece of iterest rate risk is small. This ca be explaied by the fact that the volatility of the iterest process is i compariso with that of the firm s value process relatively low, moreover, both parties have the possibility of early exercise. I practice it is ofte a difficult problem to calibrate a give model to the available data. Here oe major drawback of the structural model is that it specifies a certai firm s value process. As the firm s value, however, is ot always observable, e.g. due to icomplete iformatio, determiig the volatility of this process is a o-trivial problem. I this paper, we circumvet this problem by applyig the ucertai volatility model of Avellaeda, Levy ad Parás (1995) ad combiig it with the results of Kallse ad Küh (25) o game optio i icomplete market to derive certai pricig bouds for covertible bods. Hereby we oly kow that the volatility of the firm s value process lies betwee two extreme values. The bodholder selects the stoppig time which maximizes the expected payoff give the miimizig strategy of the issuer, ad the expectatio is take with the most pessimistic estimate from the aspect of the bodholder. Thus the optimal strategy of the bodholder ad his choice of the pricig measure determie the lower boud of the o-arbitrage price. Whereas the issuer chooses the stoppig time that miimizes the expected payoff give the maximizig strategy of the bodholder. This expectatio is also the most pessimistic oe but from the aspect of the issuer, thus the upper boud of the o-arbitrage price ca be derived. Numerically, to make the computatio tractable a costat iterest rate is assumed. The pricig bouds ca be calculated with recursios o a recombiig triomial tree. The remaider of the paper is structured as follows. Sectio 2 itroduces the model framework: market assumptios, dyamics of the iterest rate ad firm s value processes, capital structure ad the default mechaism are established. The cotract feature of the callable ad covertible 2

5 bod is described i Sectio 3. Sectio 4 focuses o the descriptio of the optimal strategies ad the determiatio of the o-arbitrage value of the callable ad covertible bod. The formulatio ad solutio of the optimizatio problem are first preseted with costat iterest rate i Sectio 5, the the iterest rate risk is icorporated i Sectio 6. I Sectio 7 ucertai volatilities of the firm value are itroduced ad pricig bouds are derived. Sectio 8 cocludes the paper. 2 Model Framework 2.1 Model assumptios We adopt a first passage model ad the model assumptios are made maily accordig to Briys ad de Varee (1997) ad Bielecki ad Rutkowski (24) 1, with some slight modificatios. It covers both the firm specific default risk ad the market iterest rate risk ad correlatio of them. The fiacial market is frictioless, which meas there are o trasactios costs, bakruptcy costs ad taxes, ad all securities i the market are arbitrarily divisible. Every idividual ca buy or sell as much of ay security as he wishes without affectig the market price. Risk-free assets ear the istataeous risk-free iterest rate. Oe ca borrow ad led at the same iterest rate ad take short positios i ay securities. The Modigliai-Miller theorem is valid, i.e. the firm s value is idepedet of the capital structure of the firm. I particular, the value of the firm does ot chage at the time of coversio ad is reduced by the amout of the call price paid to the bodholder at the time of the call. Tradig takes place cotiuously. Uder these assumptios, the fiacial market is complete, ad accordig to Harriso ad Kreps (1979) there exists a uique probability measure P uder which the cotiuously discouted price of ay security is a P -martigale. 2.2 Dyamic of the firm s value The Vasi cek model, i its simplest form, a oe-factor mea-revertig model is applied to icorporate iterest rate risk ito the process of the firm s value. The coform short rate follows a Orstei Uhlebeck process dr(t) = (a r b r r(t))dt σ r dw 1 (t), (1) with costat volatility σ r >, ad the short rate is pulled to the log-ru mea a r at a speed b r rate of b r. W1 (t) is a 1 -dimesioal stadard Browia motio uder the martigale measure P. Accordigly, the value of a default free zero coupo bod B(t, T) follows the dyamic db(t, T) = B(t, T)(r t dt b(t, T)dW 1 (t)) (2) where the volatility of the zero coupo bod has the followig form b(t, T) = σ r b r (1 e br(t t) ). The firm s value V is assumed to follow a geometric Browia motio uder the martigale measure P of the form 1 See, Sectio 3.4 of their book. dv t V t = (r t κ)dt σ V (ρdw 1 (t) 1 ρ 2 dw 2 (t)) (3) 3

6 where W2 (t) is a 1 -dimesioal stadard Browia motio, idepedet of W 1 (t) ad ρ [ 1, 1] is the correlatio coefficiet betwee the iterest rate ad the firm s value. The volatility σ V > ad the payout rate κ are assumed to be costat. The amout κv t dt is used to pay coupos ad divideds. Uder the martigale measure P the o-arbitrage price of a cotiget claim is derived as expected discouted payoff, but i the case of stochastic discout factor the calculatio ca be quite complicated. The calculatio ca be simplified if the T-forward risk adjusted martigale measure P T is applied. Defiitio 2.1. A T-forward risk adjusted martigale measure P T o (Ω, F T ) is equivalet to P ad the Rado-Nikodým derivative is give by the formula dp T dp = E P T exp{ [ r(u)du} exp{ ] = T r(u)du} ad whe restricted to the σ field F t, [ dp T T exp{ dp F t := E r(u)du} ] P F t B(,T) exp{ T = exp{ t r(u)du} B(,T) r(u)du}b(t, T). B(,T) Especially for Gaussia term structure model, whe the zero bod price is give by Equatio (2), a explicit desity fuctio exists. Namely, Furthermore, dp T { dp F t = exp 1 2 t b 2 (u, T)du W T 1 (t) = W 1 (t) t t } b(u, T)dW1 (u). follows a stadard Browia motio uder the forward measure P T., b(u, T)du (4) Thus the forward price of the firm s value F V (t, T) := V t /B(t, T) satisfies the followig dyamics uder the T-forward risk adjusted martigale measure P T 2, df V (t, T) F V (t, T) = κdt (ρσ V b(t, T))dW T 1 (t) σ V 1 ρ 2 dw 2 (t) = κdt σ F (t, T)dW T (t), (5) where W1 T (t) is give by Equatio (4) ad σ 2 F(t, T) = t ( ) σv 2 2ρσ V b(u, T) b 2 (u,t) du, (6) ad W T (t) is a 1-dimesioal stadard Browia motio that arises from the idepedet Browia motios W T 1 (t) ad W 2 (t) 3 by the followig equality i law aw T 1 (t) bw 2 (t) a 2 b 2 W T (t), where a, b are costat. Thus the auxiliary process F κ V (t, T) := F V (t, T)e κt (7) 2 The dyamic of the forward firm value is derived by applicatio of Itô s Lemma. 3 The idepedece of W T 1 (t) ad W 2 (t) is due to the assumptio that W 1 (t) ad W 2 (t) are idepedet ad this property remais after the chage of measure acted o W 1 (t). 4

7 is a martigale uder P T ad is log-ormally distributed. Specifically, we have df κ V (t, T) = F κ V (t, T) σ F (t, T)dW T (t). (8) Accordig to Equatio (3) a costat payout rate of κ is assumed, ad κv t dt is the sum of the cotiuous coupo ad divided paymets. Thus the firm s value F V (t, T) is ot a martigale uder the T-forward risk adjusted martigale measure P T, but after compesated with the payout, the auxiliary process F κ V (t, T) is a martigale uder P T. 2.3 Capital structure ad default mechaism The equity price may drop at time of coversio, as the equity-holders may ow a smaller portio of the equity after bodholders covert their holdigs ad become ew equity-holders. To capture this effect, we assume that util time of coversio, at time t, the firm s asset cosists of m idetical stocks with value S t ad of idetical callable ad covertible bods with value CCB t, thus V t = m S t CCB t. Especially, at time t =, the iitial firm s value satisfies V = m S CCB. (9) Moreover, we set the pricipal that the firm must pay back at maturity T to be L for each bod ad assume that bodholders are protected by a safety coveat that allows them to trigger early default. The firm defaults as soo as its value hits a prescribed barrier ad the default time τ is defied i a stadard way by 3 Cotract Feature τ = if {t > : V t ν t }. (1) I the followig we assume that the bod matures at time T R. The coupos are paid out cotiuously with a costat rate of c, give that the firm s value is above the level η t. The cotract termiates either at maturity T or, i case of premature default, at the default time τ, which is the first hittig time of the barrier ν t by the firm s value. Moreover, the cotract stops also by coversio or call. The bodholder ca stop ad covert the bod ito equities accordig to the prescribed coversio ratio γ. The coversio time of the bodholder is deoted as τ b [,τ]. The shareholder ca stop ad buy back the bod at a price give by the maximum of the determiistic call level H t ad the curret coversio price. This esures that the payoff by call is ever lower tha the coversio payoff. This assumptio makes the aspect of game optio relevat ad iterestig for the valuatio of callable covertible bods. The call time of the seller is deoted as τ s [,τ]. 3.1 Discouted payoff First, we itroduce the otatio β(s,t) = exp{ t s r(u)du} which is the discout factor, where r(t) is the istataeous risk-free iterest rate. The discouted payoff of a callable ad covertible bod ca be distiguished i four cases. (i) Let τ b < τ s T, such that the cotract begis at time ad is stopped ad coverted by the bodholder. I this case, the discouted payoff cov() of the callable ad covertible 5

8 bod at time is composed of the accumulated coupo paymets ad the payoff through coversio τb τ cov() = c β(,s)1 {Vs>ηs}ds ν τ β(,τ)1 {τ τ b } β(,τ b )1 {τb <τ}( γvτb m γ ). (11) (ii) Let τ s < τ b T, such that the cotract is bought back by the shareholder before the bodholder coverts. I this case, the discouted payoff call() of the callable ad covertible bod at time is composed of the accumulated coupo paymets ad the payoff through call, τs τ call() = c β(,s)1 {Vs>ηs}ds ν τ { β(,τ s )1 {τs<τ} max H τs, β(,τ)1 {τ τ s} γv τs m γ }. (12) (iii) If τ s = τ b < T the discouted payoff of the bod equals the smaller value, i.e. the discouted payoff with coversio. (iv) For τ b T ad τ s T, the discouted payoff of a callable ad covertible bod at time is τ T term() = c β(,s)1 {Vs>ηs}ds ν τ β(,τ)1 {τ T } β(,t)1 {T<τ} max { γvt m γ,mi { VT,L }}. Note that V T > γv T m γ sice, m N ad γ R. Hece i the case V T L the bodholder would ot covert ad { { }} γvt 1 {VT L} max m γ, mi VT,L = V T. Thus, i the case (iv), we ca rewrite the discouted payoff term() as τ T term() = c β(,s)1 {Vs>ηs}ds ν τ β(,τ)1 {τ T } { } γvt β(,t)1 {T<τ,VT >L} max m γ,l T β(,t)1 {T<τ,VT L}V. (13) Deote the miimum of coversio ad call time by ζ = τ s τ b. The, all i all, the discouted payoff of a callable ad covertible bod ccb() is give as the sum of the payoffs i the former 6

9 four cases ad amouts to ζ T { } γv ζ cbb() = 1 {ζ<τ} (c β(,s)1 {Vs>ηs}ds 1 {ζ=τs<τb T }β(,ζ) max H ζ, m γ { } ) γv ζ 1 {ζ=τb <τ s<t }β(,ζ) m γ 1 γvt {ζ=t }β(,ζ) max m γ,l τ T 1 {τ ζ} (c β(,s)1 {Vs>ηs}ds 1 {τ T } β(,τ) ν τ { } ) VT 1 {T<τ} β(,t) mi,l. (14) 3.2 Decompositio of the payoff The callable ad covertible bod ca be decomposed ito a straight bod compoet ad a optio compoet. The decompositio eables us to ivestigate the pure effect caused by the coversio ad call rights. Theorem 3.1. The payoff of a callable ad covertible bod ca be decomposed ito a straight bod d() ad a defaultable game optio compoet g(). with ad ccb() = d() g() (15) τ T d() := c β(,s)1 {Vs>ηs}ds 1 {τ T } β(,τ) ν { } τ 1 VT {T<τ}β(,T) mi,l, g() := 1 {ζ<τ} β(,ζ) { 1 {ζ=τs<τ b T } 1 {ζ=τb <τ s<t } ( ) γvζ m γ φ ζ ( { } ) ( ) } γv ζ γvt max H ζ, φ ζ 1 m γ {ζ=t } m γ L, where φ ζ := c τ T ζ β(,s)1 {Vs>ηs}ds 1 {τ T } β(ζ, τ) ν { } τ 1 VT {T<τ}β(ζ, T) mi,l (16) is the discouted value (discouted to time ζ ) of the sum of the remaiig coupo paymets ad the pricipal paymet of a straight coupo bod give that it has ot defaulted till time ζ. 7

10 Proof 3.2. We ca reformulate ccb() i Equatio 14 as follows ( γv ζ ccb() = 1 {ζ<τ} β(,ζ) 1 {ζ=τb <τ s<t } m γ 1 {ζ=τ s<τ b T } max { } ) γvt 1 {ζ=t } max m γ,l { H ζ, } γv ζ m γ ( τ T c β(,s)1 {Vs>ηs}ds 1 {τ T } β(,τ) ν { }) τ 1 VT {T<τ}β(,T) mi,l }{{} =d() τ T 1 {ζ<τ} (c β(,s)1 {Vs>ηs}ds 1 {τ T } β(,τ) ν { } ) τ ζ 1 VT {T<τ}β(,T) mi,l. }{{} :=β(,ζ)φ ζ Sice V T L if ζ T, otherwise the bodholder would ot make use of his coversio right. 4 Optimal Strategies After the iceptio of the cotract, the bodholder s aim is to maximize the value of the bod by meas of optimal exercise of the coversio right. The icetive of the issuer to call a bod is to limit the bodholder s participatio i risig stock prices. The embedded optio rights owed by both of the bodholder ad issuer ca be treated with the well developed theories o the game optio. 4.1 Game optio I this sectio we summarize the valuatio problem of game optios ad highlight some importat results derived by Kifer (2).. Defiitio 4.1. Let T R. Cosider a filtered probability space (Ω, F,(F t ) t [,T],P). A game optio is a cotract betwee a seller A ad a buyer B which eables A to termiate it ad B to exercise it at ay time t [,T] up to the maturity date T. If B exercises at time t, he obtais from A the paymet X t. If A termiates the cotract at time t before it is exercised by B, the he has to pay B the amout Y t, where X t ad Y t are two stochastic processes which are adapted to the filtratio (F t ) t [,T], ad satisfy the followig coditio X t Y t, for t [,T], ad X T = Y T. (17) Moreover, if the seller A termiates ad the buyer B exercises at the same time, A oly has to pay the lower value X t. Loosely speakig, the seller must pay certai pealty if he termiates the cotract before the buyer exercises it. Game optios iclude both America ad Europea optios as special cases. Formally, if we set Y t = for t [,T), the we obtai a America optio. A Europea optio is obtaied by settig X t = for t [,T) ad X T is a oegative F T -measurable radom variable. 8

11 If the seller A selects a stoppig time τ A as termiatio time ad the buyer B chooses a stoppig time τ B as exercise time, the A promises to pay B at time τ A τ B the amout which deotes the payoff of a game optio. g(τ A,τ B ) := X τb 1 {τb τ A } Y τa 1 {τa <τ B }, (18) The aim of the buyer B is to maximize the payoff g(τ A,τ B ), while the seller A teds to miimize the payoff. The optimal strategy for the buyer is therefore to select the stoppig time which maximizes his expected discouted payoff give the miimizig strategy of the seller, while the seller will choose the stoppig time that miimizes the expected discouted payoff give the maximizig strategy of the buyer. This max-mi strategy of the buyer leads to the lower value of the game optio, whereas the mi-max strategy of the seller leads to the upper value of the game optio. I a complete market the coditio described by Equatio (17) esures that the lower value equals the upper value such that there exists a solutio for the pricig problem of a game optio. The existece ad uiqueess of the o-arbitrage price i a complete market where the filtratio {F u } u T is geerated by a stadard oe-dimesioal Browia motio is proved i Kifer (2), Theorem 3.1. The o-arbitrage price of a game optio equals G(), G() = sup if E P [e r(τ A τ B ) g(τ A,τ B )] τ B F T τ A F T = if sup E P [e r(τ A τ B ) g(τ A,τ B )] (19) τ A F T τ B F T where F T is the set of stoppig times with respect to the filtratio {F u } u T with values i [, T]. After the iceptio of the cotract, the value process G(t), t (, T] satisfies G(t) = esssup τb F tt essif τa F tt E P [e r(τ A τ B ) g(τ A,τ B ) F t ] (2) = essif τa F tt esssup τb F tt E P [e r(τ A τ B ) g(τ A,τ B ) F t ]. Where F tt is the set of stoppig times with values i [t, T]. Further, the optimal stoppig times for the seller A ad buyer B respectively are τ A = if{t [,T] e rt Y t G(t)} τ B = if{t [,T] e rt X t G(t)}. (21) It is optimal for the seller A to buy back the optio as soo as the curret exercise value e rt Y t is equal to or smaller tha the value fuctio G(t), while the optimal strategy for the buyer B is to exercise the optio as soo as the curret exercise value e rt X t is equal to or greater tha the value fuctio G(t). 4.2 Optimal stoppig ad o-arbitrage value of callable ad covertible bod The discouted coversio value of the callable ad covertible bod, described with Equatio (11), cotais expressios about default times. But i the structural approach, the default time is a predictable stoppig time, ad adapted to the filtratio (F t ) t [,T] geerated by the firm s value. Thus the discouted coversio value is adapted to the filtratio (F t ) t [,T]. Ad the same is valid for the discouted call value ad the discouted termial payoff, described with Equatios (12) ad (13) respectively. Moreover, the call value is always larger tha the coversio value for t < T, ad they coicide at maturity T. Hece, the payoffs i the case 9

12 of coversio ad call satisfy the requiremets o the payoffs of the game optio. Furthermore, the market i our structural approach is assumed to be complete. Therefore the theory o game optio developed by Kifer (2) ca be applied to derive the uique o-arbitrage value ad the optimal strategies. Propositio 4.2. Pluggig the payoff fuctios ccb() i Equatio (19), the uique oarbitrage price CCB() at time t = of the callable ad covertible bod is give by CCB() = sup τ b F T if E P [ccb()] = if τ s F T τ s F T After the iceptio of the cotract, the value process CCB(t) satisfies sup E P [ccb()]. (22) τ b F T CCB(t) = esssup τb F tt essif τs F tt E P [ccb() F t ] (23) = essif τs F tt esssup τb F tt E P [ccb() F t ]. The optimal strategy for the bodholder is to select the stoppig time which maximizes the expected payoff give the miimizig strategy of the issuer, while the issuer will choose the stoppig time that miimizes the expected payoff give the maximizig strategy of the bodholder. This max-mi strategy of the bodholder leads to the lower value of the covertible bod, whereas the mi-max strategy of the issuer leads to the upper value of the covertible bod. The assumptio that the call value is always larger tha the coversio value prior to the maturity ad they are the same at maturity T esures that the lower value equals the upper value such that there exists a uique solutio. Furthermore, the optimal stoppig times for the equity holder ad bodholder respectively are τ b = if{t [,T] cov() CCB(t)} τ s = if{t [,T] call() CCB(t)}. (24) It is optimal to covert as soo as the curret coversio value is equal to or larger tha the value fuctio CBB(t), while the optimal strategy for the issuer is to call the bod as soo as the curret call value is equal to or smaller tha the value fuctio CBB(t). Remark 4.3. The o-arbitrage value of the callable ad covertible bod ad the optimal stoppig times described by Equatio (22) ad (24) icorporate also the case of stochastic iterest rate. Kifer (2) assumes that the iterest rate is costat, but this assumptio is ot ecessary, because the game optio is essetially a zero-sum Dyki stoppig game ad the mi-max ad max-mi strategies are also valid for the stochastic discout factor. For details, see e.g. Kifer (2) ad Cvitaić ad Karatzas (1996). I sectio 3.2 it has bee show that the callable ad covertible bod ca be decomposed ito a straight bod ad a game optio compoet ccb() = d() g(). Therefore the o-arbitrage price of the callable ad covertible bod ca also be derived i the followig way CCB() = E P [d()] E P [g()]. The o-arbitrage price of the game optio compoet G() equals G() := E P [g()] = sup τ b F T if E P [g()] = if τ s F T τ s F T sup E P [g()]. (25) τ b F T 1

13 5 Determiistic Iterest Rates I geeral, closed-form solutios of the optimizatio problems stated i Equatios (22) ad (25) are ot available. Oe alterative solutio is to approximate the cotiuous time problem with a discrete time oe. The o-arbitrage value of the callable ad covertible bod ca the be derived by a recursio formula. I order to focus o the recursio procedure, we assume i the first step that the iterest rate is costat. Theorem 2.1 of Kifer (2) illustrates the recursio method for the game optio ad the optimal stoppig strategies of both couterparts. The discretizatio method ad its covergece is proved i Propositio 3.2 of the same paper. We will apply ad adapt this recursio method to determie the o-arbitrage value ad optimal stoppig times of the callable ad covertible bod. 5.1 Discretizatio ad recursio schema The time iterval [,T] is discretized ito N equidistat time steps = t < t 1 <... < t N = T, with t i t i 1 =. Assume that the bodholder does ot receive the coupo for the period i which the bod is coverted, while receives the divideds for the coverted shares, though. If the bod is called, coupo will be paid. CCB(t ), the recursio value of the callable ad covertible bod at time t, ca be derived by meas of the max-mi or mi-max recursio, illustrated i Figure 1 ad 2. Note that i complete markets the max-mi strategy leads to the same value as the mi-max strategy. Hece it does ot matter whether we carry out the recursio accordig to the strategy of the bodholder or that of the shareholder. I the recursio schema V t is the firm s value just before payout ad ν t is the default barrier. The discretized coupo c t equals c, ad will oly be paid out if the firm s value is above certai level, i.e. V t > η t, therefore c t is path-depedet. For =,1,...,N 1, { { γv } mi e rt t max H c t,, m γ } { CCB(t ) = γv } max e rt t m γ, E P [CCB(t 1) F t ] e rt c t V e rt t ad CCB(T) = { } e rt γvt max m γ, L c t N e rt V T if if if V T > (L c tn ) if V T (L c tn ) V t > ν t V t ν t (26) (27) Figure 1: Mi-max recursio callable ad covertible bod, strategy of the issuer 11

14 For =,1,...,N 1, { max CCB(t ) = e rt γv { { t m γ,mi γv } e rt t max H c t,, m γ E P [CCB(t 1 ) F t ] e rt c t } } if V t > ν t (28) e rt V t if V t ν t ad CCB(T) = { } e rt γvt max m γ, L c t N e rt V T if V T > (L c tn ) if V T (L c tn ) (29) Figure 2: Max-mi recursio callable ad covertible bod, strategy of the bodholder Furthermore, for each i =,1,...,N 1, the ratioal coversio time after time t i equals { τb (t i) = mi t k {t i,...,t N 1 } e rt k γv t k m γ = CCB(t k) }, the ratioal call time after time t i equals { { τs (t i ) = mi t k {t i,...,t N 1 } e rt k max H c tk, } γv t } k = CCB(t k ). m γ Therefore at time t k, it is optimal to covert or call whe the curret coversio or call value equals the payoff fuctio CCB(t k ). Remark 5.1. For coveiece of otatio, the call value H is assumed be costat, but the same recursio formulas also hold i the case of a determiistic ad time depedet call level H(t). I that case H has to be replaced by H(t ) i the above formulas. Aalogously, the o-arbitrage value of the pure game optio compoet G(t ) at time t ca be derived through the recursio show i Figure 3 with φ t as discretized value defied by Equatio (16). 5.2 Implemetatio with biomial tree As the firm s value i our structural model follows a geometric Browia motio, i absece of iterest rate risk, it ca be approximated by the Cox-Ross-Rubistei model. The time iterval [,T] is divided i N subitervals of equal legths, the distace betwee two periods is = T/N. The stochastic evolutio of the firm s value is the modeled by ad V (i,j) = V ()u j d i jˆκ i, for all j =,...,i, i = 1,...,N, (3) u = e σ V, d = e σ V, ˆκ = e κ, where V (i, j) deotes the firm s value at time t i after j up movemets, ad less the amout to be paid out. Ad accordig to Equatio (3), the firm s value just before the paymet equals V (i,j) ˆκ, ad the total amout to be paid out at time t i is V (i,j). We see that u, d ˆκ 1 ˆκ 12

15 For =,1,...,N 1, mi G(t ) = ( { {e rt max H c t, γv t m γ } φ t ), { ( ) γvt max e rt m γ φ t, E P [G(t 1 ) F t ]} } if V t > ν t if V t ν t or G(t ) = { ( ) γvt max e rt m γ φ t, ( { mi {e rt max H c t, γv } ) t φ t, m γ ad E P [G(t 1 ) F t ]} } if V t > ν t if V t ν t { } e rt γvt max G(T) = m γ L c N, if V T > (L c N ) if V T (L c N ) Figure 3: Max-mi ad mi-max recursio game optio compoet ad ˆκ are time ad state idepedet. The equivalet martigale measure P exists if the periodical discout factor d < 1 ˆr = e r < u. The trasitio probability is give by p := 1 ˆr d u d. Cocretely, the recursio procedure of the mi-max strategy 4 of the issuer of a callable ad covertible bod, described by Equatios (26) ad (27), ca be implemeted withi the Cox- Ross-Rubistei model with Algorithm I (Figure 4). The o-arbitrage price of the callable ad covertible bod is the give by CCB(,). The first loop i Algorithm I (Figure 4) determies the optimal strategy ad thus the optimal termial value CCB(N, j). While the secod loop determies the value of CCB(i, j) accordig to the mi-max strategy at ode (i,j) of the tree. The value of each CCB(i,j) is stored i a data matrix, ad the evet of coversio, call or cotiuatio of the cotract is recorded for each ode (i,j). The give the developmet, i.e. the path of the firm s value V (i,j), the bodholder ad issuer ca determie their optimal stoppig times by movig forward alogside the tree. At the time the cotract is termiated, i.e. coverted, called or default at the ode (i,j), CCB(i, j) is the the discouted payoff of the callable ad covertible bod for this realizatio of the firm s value. 5.3 Iflueces of model parameters illustrated with a umerical example The o-arbitrage value of the callable ad covertible bod is affected by the radomess of the firm s value, ad the radomess of the termiatio time. It is a complex cotract ad 4 The algorithm of max-mi strategy ad recursio of the best strategy of the game optio compoet ca be writte i the similar way, therefore we omit these cases. 13

16 for j =,1,...,N, V (N,j) if > L c N,j, ˆκ { γ the CCB(N,j) = max m γ V (N,j) }, L c N,j ˆκ else, CCB(N,j) = V (N,j) ˆκ for i = N 1,...,, for j = i,...,, if V (i,j) > K, the { CCB(i, j) = mi [ γ max m γ V (i,j), ˆκ [ max H c i,j, (1 p ) CCB(i 1,j 1) else, CCB(i, j) = V (i, j) ˆκ γ m γ V (i,j) ], ˆκ 1 ( p CCB(i 1,j) 1 ˆr ) c i,j ] }, Figure 4: Algorithm I: Mi-max recursio America-style callable ad covertible bod iflueced by a umber of parameters: e.g. the value of coupo ad pricipal, default barrier, volatility of the firm s value, coversio ratio, call level, maturity, etc. The firm s value i total follows a diffusio process, while the bod ad equity value are results of a strategic game, which are ot simple diffusio processes. Chage of oe parameter causes simultaeous chages of the value of bod ad equity. For example, ituitively, the icremet of the coversio ratio causes the rise of coversio value, thus the rise of the bod price, but at the same time the reductio of the equity value, ad cosequetly the declie of the coversio value. The directio ad quatity of the total effect caot be determied without umerical evaluatio. Moreover, to desig a meaigful callable ad covertible bod, the parameters should i accordace with each other. The situatio such that the bod will be coverted or called immediately after the start of the cotract, should ot happe. I the followig, we will illustrate the iflueces of the model parameters ad their iteractios with a close study of a umerical example. Example 5.2. As a explicit umerical example we choose the followig parameters: T = 8, σ V =.2, r =.6, V () = 1, K = 4, ω = 13, L = 1, γ = 1.5, m = 1, = 8, H = 12. The results i Table 1 are derived for differet payout ratios κ ad coupos c 5. They illustrate first that the value of the game optio compoet decreases whe coupo paymet rises. The reaso is that the value of the remaiig coupo ad pricipal paymet defied by Equatio (16) ca be thought as the strike of the game optio, which is a icreasig fuctio of coupo rate c, ad the value of the game optio compoet decreases i strike. The large price differece of game optio compoet G() i the case κ =, c =, to the case κ =.4, c = 2 is due to 5 The coupos are to be paid if the firm s value is above η t = ω e r(t t) e κt, The default barrier is ν t = K e r(t t) e κt 14

17 σ V =.2 σ V =.4 κ c SB() G() CCB() S() SB() G() CCB() S() Table 1: Ifluece of the volatility of the firm s value ad coupos o the o-arbitrage price of the callable ad covertible bod (384 steps) the icremet of payout ratio ad coupo rate. Both factors together result i a large drop of the value of G(). The secod effect show by Table 1 is that the more volatile the firm s value, the larger the default probability, hece the smaller the value of straight bod SB(). But o the other side the game optio compoet G() becomes more valuable. I our example, the value of the callable ad covertible bod CCB() which is the sum of the both compoets decreases i volatility 6. σ V =.2 σ V =.4 SB() CCB() G() SB() CCB() G() / / / / Table 2: Stability of the recursio Remark 5.3. The stability of the recursio is demostrated with Table 2. The recursios are carried out alogside trees with differet steps for σ V =.2 ad σ V =.4. We ca see that the umerical results stabilized at = 1/48. Further refiemets ( = 1/1 ad = 1/25 ) of the tree do ot chage the umerical results cosiderably while much more time are eeded for the calculatio. Therefore, for the further calculatios i this example is always set to be 1/48, which approximately correspods to a weekly valuatio. By = 1/48 ad a maturity of T = 8 it correspods to a tree with 384 steps. Table 3 has the same structure as Table 1 ad shows the ifluece of the coversio ratio γ o G() ad CCB(). The volatility of the firm s value is kept to be costat, i.e. σ V =.2. The chage of coversio ratio γ does ot affect the price of the straight coupo bod ad it oly chages the value of G(). The icrease of γ from 1.5 to 2. makes the game optio compoet more valuable, thus i total the callable ad covertible bod more valuable 7. The case by κ =.4, c = 2 ad γ = 2 is ot a good cotract desig. As with CCB() = 78.27, ad S() = 37.38, the iitial price of the bod is almost equal to the iitial coversio value, which meas that the coversio may take place very quickly after the iceptio of the cotract, because a slight icrease of the firm s value will make coversio the optimal choice of the 6 I Example 5.2, the value of the callable ad covertible bod icreases i volatility, but oe caot argue it geerally, as it depeds also o other factors e.g. default barrier ad maturity. 7 Agai we caot take it as a geeral result, as it depeds also o the parameters m ad. 15

18 γ = 1.5 γ = 2 κ c SB() G() CCB() S() SB() G() CCB() S() Table 3: Ifluece of the coversio ratio o the o-arbitrage price of the callable ad covertible bod (384 steps) bodholder. Usually it is ot the itetio of the issuer to issue a bod which will be coverted or called immediately after the iceptio of the cotract. T = 8 T = 6 κ c SB() G() CCB() S() SB() G() CCB() S() Table 4: Ifluece of the maturity o the o-arbitrage price of the callable ad covertible bod (384 steps) Table 4 is also structured i the same way as Tables 3 ad 1. It demostrates the ifluece of the maturity T o G() ad CCB(). The volatility of the firm s value ad coversio ratio are σ V =.2 ad γ = 1.5. Comparig the case T = 8 with T = 6, we observe that the straight bod is more valuable with shorter maturity, because the default probability is lower ad by positive iterest rate the pricipal is more valuable if it is paid earlier. The game optio compoet G() is less valuable i the case of shorter maturity. It is due to two effects: first, shorter maturity meas less coversio chaces for the bodholder, ad secodly, a icrease of the value of the straight bod reduces the value of the equity thus the coversio value. The reductio of G() may i tur icrease the value of equity, here the fial result is that reductio i maturity icreases the value of the callable ad covertible bod CCB(). The value of the game optio compoet ca be restricted whe the call level is reduced. This effect is cofirmed by the results i Table 5. The reductio of the call level is achieved by makig the call level to be time depedet H(t) = e ω(t t)h, ω. (31) The value of H(t) icreases i time ad reaches H at maturity T. By ω =, the call level reaches its maximum ad is a costat H. The impact of the call level o the o-arbitrage price of game optio compoet is stroger i the case of higher coupo rate c ad lower volatility of the firm s value σ V. 16

19 κ c ω =,σ V =.2 ω =.4,σ V =.2 ω =,σ V =.4 ω =.4,σ V = Table 5: Ifluece of the call level o the o-arbitrage price of the game optio compoet (384 steps) 6 Stochastic Iterest Rate 6.1 Recursio schema I this sectio we solve the optimizatio problems stated i Equatios (22) ad (25) by allowig stochastic iterest rate. Similar as i Sectio 5, the cotiuous time problem is approximated with a discrete time oe ad the o-arbitrage value is derived by a recursive formula. We discretize the forward price of the firm s value process modeled i Sectio 2.2. Accordigly, the call level ad coupos are adjusted to the forward value. The recursio is carried out o the T -forward adjusted values, see Figure 5, where F V (t,t) is the forward price of the firm s value just before payout ad CCB F (t ) is the T -forward value of the callable ad covertible bod at time t. At the termial date T, F V (T,T) = V T thus CCB F (T) = CCB(T). ν t is the default barrier. The coupo c t will oly be paid out if the firm s value is above certai level, i.e. V t > η t. The o-arbitrage price of the callable ad covertible bod equals B(,T)CCB F (). For =,1,...,N 1, ad CCB F (t ) = { { H ct mi max max B(t,T), γf V (t,t) m γ { γfv (t,t) m γ, E P T [CCB F (t 1 ) F t ] F V (t,t) CCB(T) = { γvt max V T m γ, L c t N }, c } } t if F V (t,t) > ν t B(t,T) } if F V (t,t) ν t if V T > (L c tn ) if V T (L c tn ) (32) (33) Figure 5: Mi-max recursio callable ad covertible bod, T -forward value 6.2 Some coditioal expectatios The recursio formula, Equatio (32) cotais both F V (t,t) ad B(t,T) as variables. I order to circumvet a two-dimesioal tree, we solve CCB F (t,t) as coditioal expectatio give F V (t,t). To achieve the aalytical closed-form solutio, we first explore the relatioship betwee F V (t, T) ad B(t, T). 17

20 Accordig to the assumptios o the firm s value made i Sectio 2.2, uder P T the auxiliary forward price of the firm s value FV κ (t, T) ad the T -forward price of the default free zero B(t, s) coupo bod F B (t, s, T) :=, t s < T are both martigales, ad satisfy B(t, T) with df κ V (t, T) = F κ V (t, T) σ F (t, T)dW T t. df B (t, s, T) = F B (t, s, T) σ B (t, s, T)dZ T t σ 2 F(t, T) = σ 2 B(t, s, T) = t t σ 2 V 2ρσ V b(u, T) b 2 (u, T)du (b(u,s) b(u, T)) 2 du ad b(t, s) = σ r b (1 e b(s t) ). Wt T ad Zt T are two correlated stadard Browia motio with costat coefficiet of correlatio equals ρ. Hece FV κ(t, T) ad F B(t, t) B(t, t,t) = B(t, T) = 1 are bivariate ormally distributed ad B(t, T) have the followig variaces, expectatios ad covariaces 8 t σ1 2 := V P T [lfv κ(t, T)] = (σv 2 2ρσ V b(s,t) b 2 (s,t))ds σ 2 2 := V P T [lf B (t, t,t)] = 1 2b 3(1 e 2bt )b(t, T) 2 µ 1 := E P T [lf κ V (t, T)] = lf κ V (,T) 1 2 σ2 1 ad µ 2 := E P T [lb(t, T)] = E[ lf B (t, t,t)] = l B(,T) B(,t) 1 2 σ2 2 γ := Cov P T (lf κ V (t, T),lB(t, T)) = Cov P T (lfv κ (t, T),lF B (t, T)) t ( ) = ρσ V (b(u,t) b(u, t)) (b(u, t)b(u, T) b(u, t) 2 ) du. Give these relatioships the expectatio ad variace of l B(t, T) coditioal o the forward price of the firm s value ca be derived with the followig formulas [ ] µ 3 := E lb(t, T) lfv κ (t, T) = w = µ 2 γ σ1 2 (l w µ 1 ), (34) [ ] σ3 2 := V lb(t, T) lfv κ (t, T) = w = σ2 2 γ2 σ1 2. (35) Therefore, coditioal o lfv κ (t, T) = w the radom variable l(b(t, T)) equals ( ) lb(t, T) lfv κ (t, T) = w = µ 3 σ 3 x 8 For details see Mekveld ad Vorst (2). 18

21 where x is a stadard ormal radom variable. Thus the followig coditioal expectatio ca be derived after some elemetary itegratio [ 1 ] ( E lfv κ (t, T) = w = exp µ 3 1 ) B(t, T) 2 σ2 3 (36) [ ] ( p ) E B(t, T) q lf κ V (t, T) = w = h ( p ) x e 2 e µ 3σ q 2 3 x dx 2π = p e µ 3 σ N(h σ3 ) q N(h) (37) with h = (l(p/q) µ 3 ))/σ 3 for some p, q R. Here, N( ) deotes the cumulative distributio fuctio of a stadard ormal distributio. 6.3 Implemetatio with biomial tree For the implemetatio of the recursio schema displayed i Figure 5 we apply the method developed by Mekveld ad Vorst (1998) which is a combiatio of a aalytical approach ad a oe-dimesioal biomial tree approach. A simple recombiig biomial tree for the forward price F V (t, T) := V t /B(t, T) of the firm s value ca be costructed with the trick that the iterval [,T] is ot divided ito periods of equal legth, but ito periods of equal volatility. Recursio is the carried out alogside the T -forward risk adjusted tree. The iterval [, T] is divided ito periods = t < t 1 <... < t N = T of equal volatility σ N F := 1 N T (σ 2 V 2ρσ V b(s,t) b 2 (s,t))ds. The stochastic evolutio of the forward price of the firm s value is the modeled by with F() = V ()/B(,T) ad F V (, j) = F()u j d jˆκ, j =,...,, = 1,...,N u = e σn F, d = e σf N, κ ˆ = e κ, = t t 1, where F V (, j) deotes the forward price of the firm s value after payout, at time t after j up-movemets. F() is the iitial forward price of the firm s value. The expressios show that u ad d are time ad state idepedet. ˆκ is time depedet as the time steps are o loger of equal legth. The (time depedet) coupo paymet is give by c() = c. The forward martigale measure P T exists because d < 1 < u ad the trasitio probability is give by p T := 1 d u d. Thus the coditioal expectatio i the recursio schema ca be calculated as EV (, j) := p T CCB F ( 1,j) (1 p T ) CCB F ( 1,j 1) The forward price of the firm s value at time t after j up movemets ad just before payout is F V (,j) := F V (, j) 1 ˆκ. 19

22 At each ode (, j) we calculate the expected value of the mi-max strategy uder the measure P T coditioal o the available iformatio F V (, j). The calculatio is tedious but ca be solved aalytically. We make first some simplificatios of the otatios which are oly used for the calculatio of CCB F (, j). H(, j) ad c(, j) are writte as H ad c, ad CV := γf V (,j) m γ EV := EV (, j) which are coversio ad simple recursio value. Accordig to the recursio formula Equatio (32), { { H c } { CCB F (, j) = mi max B(t,T),CV c } }, max CV, EV B(t,T) { [ ] H c [ ] } = mi B(t,T) CV c CV, EV B(t,T) CV CV [ ] c = CV EV B(t,T) CV [ ] H B(t,T) EV 1 { Hc B(t,T) >CV }1 {EV c B(t,T) >CV }. (38) Equatio (38) ca be further calculated i two cases. (i) CV EV [ ] c CCB F (, j) = EV B(t,T) H B(t,T) EV (39) because i this case the secod term of Equatio (38) is certaily positive ad CV icludes also the case (ii) CV > EV where H c B(t,T) > CV. [ c CCB F (, j) = CV B(t,T) [ H B(t,T) EV [ H MIN := mi EV, H c CV, ] (CV EV ) H B(t,T) > ] 1 {B(t,T)>MIN} (4) ] c. CV EV Accordig to the coditioal expectatios give i Equatios (36) ad (37), the aalytical solutio of Equatios (39) ad (4) ca be derived as coditioal expectatios give FV κ (, j) = F V (, j)e κt = w. (i) CV EV [ CCB F (, j) = EV c exp µ 3 σ2 ] [ 3 H exp 2 µ 3 σ2 ] 3 N(h 1 σ 3 ) EV N(h 1 ) 2 where h 1 := l H EV µ 3 σ 3. 2

23 (ii) CV > EV where [ CCB F (, j) = CV c exp [ H exp µ 3 σ2 3 2 µ 3 σ2 3 2 h 2 := l c CV EV µ 3 σ 3 h 3 := lmin µ 3 σ 3. Ad µ 3 ad σ 3 have bee defied i Equatios (34) ad (35). ] N(h 2 σ 3 ) (CV EV )N(h 2 ) ] N(h 3 σ 3 ) EV N(h 3 ) I the followig umerical example we compute the o-arbitrage price of a callable ad covertible bod with stochastic iterest rates. Example 6.1. The iitial term structure is flat, choose T = 8, σ V =.2, K = 4, ω = 13, σ r =.2, b =.1, V () = 1, L = 1, K = 4, m = 1, = 8, H = 12, γ = 1.5, r =.6. 9 The recursios are carried out alogside a tree with 384 steps. The o-arbitrage prices of a straight bod, a callable ad covertible bod ad the game optio compoet i America-style with ad without stochastic iterest rates are preseted i Table 6. No stads for o iterest rate risk, -.5 ad.5 give the correlatio coefficiet of the iterest rate ad firm s value. The values are derived for differet payout ad coupo combiatios. G() CCB() SB() κ c No.5.5 No.5.5 No Table 6: No-arbitrage prices of the o-covertible bod, callable ad covertible bod ad game optio compoet i America-style with stochastic iterest rate (384 steps) Icreasig correlatio betwee the iterest rate ad the firm s value causes icreasig volatility of the forward price of the firm s value. The default probability rises with icreasig volatility, which results i a reductio of the value of the straight bod SB(). But o the other side, the value of the game optio compoet G() icreases i volatility. Therefore i geeral the total effect is ucertai, i our cocrete example the total value declies with icreasig correlatio. Moreover, the ifluece of the iterest rate risk is relatively small which is recogized by the value of the covertible bod, the results listed i the colums uder CCB(). 9 The default barrier is η t = KB(t, T)e κt, ad the coupos are to be paid if the firm s value is above η t = ωb(t, T)e κt. 21

24 7 Ucertai Volatility of Firm Value I practice it is ofte a difficult problem to calibrate a model to the available data. Here oe major drawback of the structural model approach is that it specifies a certai firm s value process. As the firm s value, however, is ot always observable, e.g. due to icomplete iformatio, determiig the volatility of this process is a o-trivial problem. Moreover, the iterest rate risk ad the ucertaity about the correlatio of the iterest rate ad firm value process are other cotributors to the ucertaity of the volatility. To relax the assumptio of costat volatility of the firm s value, oe ca specify volatility as a particular fuctio of the firm s value, or model volatility itself with a stochastic process. However, specificatio of a reasoable model for the volatility dyamics ad precise estimatio of the parameters would be a difficult task. We circumvet these problems by assumig that the volatility of the firm s value process lies betwee two extreme values. The volatility is o loger assumed to be costat or a fuctio of uderlyig ad time. It is istead assumed to lie betwee two extreme values σ mi ad σ max, which ca be viewed as a cofidece iterval for the future volatilities. This assumptio is less striget compared to the approaches where the volatility is modeled as a fuctio of the uderlyig or as a stochastic process. It eeds also less parameter iputs. We treat the callable ad covertible bod with ucertai volatility by applyig the model of Avellaeda et al. (1995) ad Lyos (1995) ad combiig it with the results of Kallse ad Küh (25) o game optio i icomplete market such that certai pricig bouds ca be derived. The bodholder selects the stoppig time which maximizes the expected payoff give the miimizig strategy of the issuer, ad the expectatio is take with the most pessimist estimate from the aspect of the bodholder. The optimal strategy of the bodholder ad his choice of the pricig measure determie the lower boud for the o-arbitrage price. Whereas the issuer chooses the stoppig time that miimizes the expected payoff give the maximizig strategy of the bodholder ad the expectatio is also the most pessimist oe but from the aspect of the issuer, thus the upper boud of the o-arbitrage price ca be derived. The volatility is selected dyamically from the two values σ mi ad σ max i a way that always the oe with the worse effect, thus the most pessimist pricig measure is chose. 7.1 Ucertai volatility model The ucertai volatility model is first proposed idepedetly by Avellaeda et al. (1995) ad Lyos (1995). It is a extesio of the Black-Scholes framework to deal with the biased estimate of the historical volatility or the smile effect of the implied volatility 1. Avellaeda et al. (1995) study the case of derivatives writte o a sigle uderlyig asset. The volatility of the asset is ot assumed to be a costat or a fuctio of the uderlyig or rather stochastic. Istead, it is oly assumed to lie betwee two extreme values σ mi ad σ max, which ca be viewed as a cofidece iterval for the future volatilities. This assumptio is less striget compared to other approaches ad it eeds also less parameter iputs. The derivatio of a o-arbitrage pricig boud is based o a super-hedgig strategy which is a worst case estimatio. At each (t, x) the volatility is selected dyamically from the two values σ mi ad σ max i a way that always the oe with the worse effect o the value of the derivative from aspect of seller or buyer is chose. For a give martigale measure Q, suppose the stock price evolves accordig to the followig 1 The volatility implied from the traded optios, plotted as a fuctio of the strike price, ofte exhibits a specific U-shape, which is referred to as the smile effect. 22