A BARRIER OPTION OF AMERICAN TYPE

Transcription

1 A BARRIER OPTION OF AMERICAN TYPE IOANNIS KARATZAS Departments of Mathematics and Statistics, 619 Math. Bldg. Columbia University New York, NY 127 HUI WANG Department of Statistics 618 Mathematics Building Columbia University New York, NY 127 June 2, 27 Abstract We obtain closed form expressions for the prices and optimal hedging strategies of American put options in the presence of an up and out barrier, both with and without constraints on the short selling of stock. The constrained case leads to a stochastic optimization problem of mixed optimal stopping/singular control type. This is reduced to a variational inequality which is then solved explicitly in two qualitatively separate cases, according to a certain compatibility condition among the market coefficients and the constraint. Key Words: American option, barrier option, singular stochastic control, optimal stopping, variational inequality, hedging, elastic boundary condition, constrained portfolios. AMS 1991 Subject Classifications: Primary 93E2, 9A9, 6H3; Secondary 6G4, 6G44, 9A16. Supported in part by the National Science Foundation under Grant NSF-DMS

2 1 INTRODUCTION We solve in closed form the pricing and hedging problems for the up and out Barrier Put Option of American type, with payoff 1.1) Y t) = q St)) + 1 t<τh, t <. Here h > is the barrier and q, h) the strike price of the option, whereas 1.2) τ h = inf t / St) h is the time when the option becomes knocked out. The stock price per share S ) is assumed to satisfy the standard model 1.3) dst) = St) rdt + σdw t), S) = x, h) of Merton 1973) and Black & Scholes 1973), with r > the prevalent interest rate of the risk free asset bank account), σ > the volatility of the stock, and W ) a Brownian Motion under the risk neutral equivalent martingale measure. This analysis is carried out in Section 2. It is shown there that the optimal hedging portfolioweight is always negative, selling the stock short, and decreases without a lower bound as the stock-price approaches the barrier level h. As a consequence, the resulting portfolio is rather undesirable from the point of view of practical implementation. To remedy this situation, we discuss in Section 3 the same problem but now under a shortselling constraint: the hedging portfolio-weight is not allowed to fall below α, for some given constant α >. Using the theory developed by Karatzas & Kou 1998) for American contingent claims under constraints, we solve this problem also in closed form: first for α > 2r/σ 2 Section 4) and then for < α 2r/σ 2 Section 5). In the latter case the short-selling constraint is severe cf. Remark 3.4 for discussion) and the price of the option is given by the α-enlargement 1.4) φ α x) = sup e αν φxe ν ) = ν q q x ) ; x α q α x α α ; q < x < of the reward function φx) = q x) + for the American put option, introduced by Broadie, Cvitanić & Soner 1998). In the case α > 2r/σ 2 the short selling constraint is less severe; the value of the option dominates the quantity of 1.4), and is given by the solution to a stochastic optimization problem that involves both optimal stopping and singular control of the monotone follower type, as in Karatzas & Shreve 1984). We cast this problem in terms of a Variational Inequality, featuring the function φ α ) of 1.4), which can then be solved explicitly. The details are carried out in Theorem 4.2 proved in Appendix A) and in Proposition 4.3, respectively. The relevance of singular stochastic control to the pricing of European type barrier options was first brought out in Wystup 1997). An analysis of down-and-out barrier call-options, of both European and American type, is carried out in Section 8.9 of Merton 1973). 2

3 2 AMERICAN PUT OPTION OF BARRIER TYPE Let us consider the standard model of Merton 1973), Black & Scholes 1973) for a financial market: 2.1) dbt) = Bt)rdt, B) = 1 dst) = St) bt)dt + σdw t), S) = x >. This model consists of a money market with constant interest rate r > and price Bt) = e rt the so-called numéraire ), and of one stock with price-per-share St), constant volatility σ >, and appreciation rate bt) at time t. The driving process W = W t); t < is standard Brownian motion on a probability space Ω, F, P); we shall denote by F = Ft) t< the filtration generated by this process, namely Ft) = σw s); s t). It will be assumed that the appreciation rate process b ) is bounded, and progressively measurable with respect to F. In such a context, we are interested in the valuation problem for an American put option of the up and out barrier type, with payoff 2.2) Y t) = q St)) + 1 t<τh, t <. To place this in context, imagine a contract, signed at time t =, which confers to its holder the right but not the obligation, whence the term option ) to sell to the issuer one share of the stock, at the contractually specified price q > and at any time of the holder s choice, provided that a contractually specified) barrier h q, ) has not yet been reached or exceeded. In other words, the contract is knocked out for i.e., becomes worthless to) its holder at the first time 2.3) τ h = inft / St) h the stock price S ) reaches or exceeds the barrier level h. Clearly, if the holder of the contract exercises his option at time t, then effectively he receives from the issuer a payment of size Y t) as in 2.2). Such a contract is of potential value to a holder who believes that the stock price will fall below q, and to an issuer who believes otherwise but does not want to have to worry about hedging if the stock price should become too high i.e., reach or exceed the barrier h). How much should then the issuer charge his counterpart at t =, for signing this contract? In other words, what is the price, at time t =, of the American contingent claim Y ) in 2.2), 2.3)? From the standard theory on American contingent claims e.g. Karatzas 1996), 1.4) we know that this so-called hedging price is defined as the smallest initial captial ξ > that allows the issuer to cover his obligation successfully i.e., without risk), no matter when the holder should decide to exercise his option: 2.4) Hx) = infξ > / π, C) with X ξ,π,c τ) Y τ) τ S). Here S is the class of all F-stopping times; π ) is a portfolio process F-progressively measurable, with T π2 t) dt < a.s., for any < T < ); and C ) is a cumulative consumption process measurable and F-adapted, with values in, ) and increasing, right continuous paths with C) = a.s.). Finally, X ) X ξ,π,c ) is the wealth process corresponding to initial capital ξ, portfolio π ) and cumulative consumption C ), namely dxt) = πt) r dt + σ dw t) + Xt) πt)) r dt dct), X) = ξ, 3

4 or equivalently 2.5) e rt X ξ,π,c t) = ξ where,t 2.6) W t) = W t) + e rs dcs) + σ t t bs) r σ e rs πs) dw s), t <, ds, t <. From this same general theory, we also know that the hedging price Hx) of 2.4) can be computed as the optimal expected reward in a problem of optimal stopping: 2.7) Hx) = Gx) = sup τ S E e rτ q Sτ)) + 1 τ<τh = sup τ S E e rτ q Sτ)) + 1 max t τ St)<h. We have denoted by E the expectation under the probability measure P on Ω, F) with dp t ) 2.8) bs) r dp = exp dw s) 1 t ) bs) r 2 ds Ft) σ 2 σ for every t, ). Under this measure P, the process W ) of 2.6) is standard Brownian motion by the Girsanov theorem, and we may rewrite 2.1) in the form of 1.3). 2.1 REMARK: We refer the reader to the discussion on pp in Karatzas & Shreve 1991) for the Girsanov theorem on an infinite time horizon, and to 1.7, 2.6 in Karatzas & Shreve 1998) for the measure theoretic subtleties associated with questions of hedging on an infinite time horizon. How are we then to solve the optimal stopping problem of 2.7)? By analogy with Example in Karatzas 1996), we cast this problem as a variational inequality. For an alternative approach, that does not rely on the so called principle of smooth fit, the reader may wish to consult and apply the results of Salminen 1985). 2.2 VARIATIONAL INEQUALITY: Find a number b, q) and a convex, decreasing function g ) in the space C, )) Cb 1, ) \ h) C2 b, ) \ b, h), such that: 2.9) 2.1) 2.11) 2.12) 2.13) σ 2 2 x2 g x) + rxg x) rgx) = ; b < x < h σ 2 2 x2 g x) + rxg x) rgx) < ; < x < b gx) > q x) + ; x > b gx) = q x) + ; x b gx) = ; h x <. 2.3 THEOREM: If the pair b, g )) solves the Variational Inequality 2.2, then g ) coincides with the optimal expected reward of the stopping problem in 2.7), and the stopping time 2.14) τ b = inft / St) b 4

5 is optimal for this problem: 2.15) gx) = Gx) = sup τ S E e rτ q Sτ)) + 1 τ<τh = E e rτ b q Sτ b )) + 1 τb <τ h. Proof: All claims are obvious for x h, so let us concentrate on a fixed x, h). On the interval, h), the functions g ), g ) are bounded, continuous; and the function g ) is bounded, continuous on, h) \ b. An application of Itô s rule cf. Problem 3.7.3, p. 219 in Karatzas & Shreve 1991)) to the process e rt gst)), t < yields then, in conjunction with 2.8) 2.1): 2.16) gx) e rτ τ h) gsτ τ h )) + σ τ τh e rt ξg ξ) ) ξ=st) dw t) τ τh σ = e rt 2 2 ξ2 g ξ) + rξg ξ) rgξ)) dt, P a.s. ξ=st) for every τ S. Now the P expectation of the stochastic integral in 2.16) is equal to zero; indeed, from the convexity and decrease of g ) we have ξg ξ) g) gξ) g) for < ξ <, and thus E e 2rt St)g St))) 2 dt g 2 )/2r <. We obtain gx) E e rτ τh) gsτ τ h )) = E e rτ gsτ))1 τ<τh + E e rτ 2.17) h gsτ h ))1 τh τ, τ h < = E e rτ gsτ))1 τ<τh E e rτ q Sτ)) + 1 τ<τh from 2.11) 2.13); in other words, gx) Gx). On the other hand, thanks to 2.9) and 2.12), all the inequalities in 2.16), 2.17) hold as equalities for the choice τ τ b of 2.14). The claims of 2.15) now follow readily. It remains to construct the solution of the Variational Inequality PROPOSITION: Let b be the unique solution of the equation 2.18) 1 + β ) b = β + q ) b β h in the interval, q), where β = 1 + 2r/σ 2), and define q ) x ; x b 2.19) gx) = x q b h/x) β 1 b ; b < x < h h/b) β 1. ; h x < Then the pair b, g )) solves the Variational Inequality 2.2. Proof: The general solution of the equation 2.9) is given by gx) = Ax γ + + Bx γ for suitable constants A, B, where γ + = 2r/σ 2, γ = 1 are the roots of σ 2 γγ + 1)/2 rγ r =. The requirements gh ) =, gb+) = q b continuity of g ) at the points x = h and x = b) 5

6 and g b+) = 1 continuity of g ) at the point x = b; principle of smooth fit) lead then to the equation 2.18) for b, and to the formulae A = q/b) 1 b β h β) 1, B = Ah β for the two constants. These, in turn, yield the expression of 2.19). In order to see that b, q) is determined uniquely by the equation 2.18), note that F u) = u ) ) β u β 1 + β, u < h q is convex, with F ) = β 1 >, F q) = q/h) β 1 <. Thus F ) has exactly one root in the interval, q). Furthermore, notice from 2.19) that ) x ; < x < b 2.2) xg x) = x q b γ+ h/x) β +1 b ; b < x < h h/b) β 1. ; h < x < In the interval, b), we have σ2 2 x2 g x) + rxg x) rgx) = rq < ; in the interval b, h) we have g ) <, since 1 + γ + h/x) β > 1 + γ + = β >. In other words, g ) is strictly decreasing on, h). Furthermore, by 2.19) and 2.2), we have σ 2 2 x2 g x) = rgx) xg x) >, b < x < h and thus g ) is strictly convex on b, h). It is now clear that the pair b, g )) solves the Variational Inequality 2.2. The theory of 1.4 in Karatzas 1996) also provides the optimal hedging portfolio process ˆπ ) and the cumulative consumption or cash flow ) process Ĉ ), in the form xg x) ˆπt) = x=st) ; t < τ h 2.21) ; τ h t < t τh ) σ 2 Ĉt) = 2 x2 g x) + rxg x) rgx) 1,b) x) du x=su) 2.22) = rq t τh 1,b) Su)) du, t <. The corresponding value process is also given explicitly, by 2.23) X gx),ˆπ,ĉt) ˆXt) gst)) ; t < τh = ; τ h t <. In particular, 2.24) ˆpt) = ) ˆπt)/ ˆXt) = xg x) gx) x=st) ; t < τ h ; τ h t < are the optimal hedging portfolio weights, or proportions. Finally, the stopping time τ b of 2.14) is the optimal exercise time for this American put option of up and out barrier type, by its holder. 6

7 From 2.19), 2.2) and 2.24), we notice that the leverage ratio 2.25) xg x) gx) = 1 β 1 x/h) β, b < x < h decreases without bound i.e., down to ), as x h. In other words, as the stock price approaches the barrier h, the optimal hedging portfolio sells short vast amounts of stock, measured in units of the option s value. Such a portfolio is rather undesirable, from the point of view of practical implementation. In order to get around this problem, we shall impose throughout the remainder of this paper restrictions on the short selling of stock, in the form of constraints on the portfolio weights pt) = πt)/xt). 3 CONSTRAINED SHORT SELLING Suppose now that we decide to constrain the short selling of stock, by requiring that the portfolio weight πt)/xt) ; if Xt) > 3.1) pt) = ; if Xt) = i.e., the proportion of the wealth Xt) that is invested in stock) should always exceed a given lower bound α. In other words, we impose the leverage constraint 3.2) πt) αx ξ,π,c t), t < for some given real constant α >. We saw at the end of the last section that such a requirement is in fact desirable from a practical point of view, when one is trying to hedge the American put option of barrier type. How does this requirement affect the hedging price of the American barrier option of 2.2)-2.3)? In other words, what is the analogue 3.3) H α x) = infξ > / π, C) s.t. 3.2) holds, and X ξ,π,c τ) Y τ) τ S) of 2.4) under this new, additional constraint? In general terms, the answer to this question is provided by the theory developed in Karatzas & Kou 1998); it is shown there that H α x) can be computed, in principle, in terms of a double stochastic optimization problem 3.4) H α x) = G α x) = sup sup E λ e τ r+δλt))) dt q Sτ)) + 1 max t τ St)<h λ ) D τ S of mixed stochastic control/optimal stopping type. In the notation of that paper, K = α, ) is the interval where the portfolio weight process of 3.1) is constrained to take values; and 3.5) δν) = αν ; for ν sup pν) = p K ; for ν < 7

8 is the support function of the set K =, α. We shall denote by K = ν/δν) < =, ) the effective domain of δ ), and by D the space of all bounded, F-progressively measurable processes λ ) :, ) Ω K =, ). Finally, for every λ ) D, E λ denotes expectation with respect to the probability measure P λ with 3.6) dp λ dp for t <. = Z λ t) = exp Ft) t bu) r + λu) dw u) 1 σ REMARK: For any λ ) D and with Λ ) = λu) du, the process t ) bu) r + λu) 2 du σ t 3.7) W λ t) = bs) r + λs) W t) + ds = W t) + σ 1 Λt), σ t < is Brownian motion under the probability measure P λ of 3.6). Furthermore, 1.3) shows that we have 3.8) dst) = St) r λt)) dt + σ dw λ t) = St) r dt + σ dw λ t) dλt), S) = x; and we may write 3.8) in the equivalent form 3.9) ds λ t) = S λ t) r dt + σ dw λ t), S λ ) = x for the process 3.1) S λ ) = S )e Λ ), Comparing 3.9) with 1.3), we see that the process S λ ) of 3.1) has the same law under P λ, as the law of the original stock price process S ) under P. This suggests that the quantity 3.11) G α x) = sup sup λ ) D τ S E λ e ) + rτ+αλτ)) q e Λτ) S λ τ) 1max t τe Λt) Sλt))<h of 3.4), might be written equivalently as 3.11) G α x) = sup sup λ ) D τ S E e ) + rτ+αλτ)) q e Λτ) Sτ) 1max t τe St))<h. Λt) We shall verify this later, in the proofs for Theorems 4.2 and 5.1 below. The processes λ ) of D play the role of Lagrange multipliers, that enforce the short-selling constraint of 3.2). It turns out that the supremum over λ ) D in 3.11) is not attained; the supremum is attained, however, if one enlarges the class D in the manner of Wystup 1997). More precisely, consider the space L + of F-adapted processes Λ :, ) Ω, ) with continuous, increasing paths and Λ) =. Then, as we shall see in Appendix B and in Lemma 5.2, we can take the supremum in 3.11) over this larger class without increasing its value: 8

9 3.11) G α x) = sup sup Λ ) L + τ S We have denoted by 3.12) P Λ ) = e Λ ) S ) E e rτ+αλτ)) q P Λ τ)) + 1 max t τ P Λ t)<h. the Λ ) reduced stock price process, which satisfies the equation 3.13) dp Λ t) = P Λ t) r dt + σ dw t) dλt), P Λ ) = x. It should be noted that the resulting double stochastic optimization problem in 3.11) involves both optimal stopping, and singular stochastic control of the monotone follower type. 3.2 REMARK : For a given Λ ) L +, the second maximization in 3.11) amounts to solving an optimal stopping problem analogous to that of 2.7), but with new, reduced numéraire B Λ t) = e rt αλt) and stock-price process P Λ ) as in 3.12), in place of of B ) and S ), respectively. Subjecting the reduced process P Λ ) to the barrier h >, is the same as raising the barrier for the original stock-price process S ) to the new height he Λ ), which is both time-varying and random. The purpose of the first maximization in 3.11), is to select a Λ ) L + that leads to a raised-barrier he Λ ) which is just right for computing the quantity H α x) in 3.3), 3.4). 3.3 REMARK : We can identify a process Λ ) L + that attains the supremum over Λ ) L + in 3.11), as follows: )) Su) ) Λ t) = max log, t <. u t h This is the smallest of all processes Λ ) L + with the property P Λ t) h, t < ; in particular, 3.15) h P Λ t)) dλ t) =. In other words, the process Λ ) of 3.14) reduces pushes to the left, in the stochastic equation 3.13)) the stock price S ), only just enough to prevent the resulting process P Λ ) of 3.12) from ever crossing the barrier h. The paths of Λ ) are continuous, but singular with respect to Lebesgue measure. As we shall see in Theorems 4.2 and 5.1 below, this process Λ ) attains the supremum in 3.11), namely 3.11) G αx) = sup E e rτ+αλ τ)) q P Λ τ)) + 1 max t τ P τ S Λ t)<h. Notice that this last problem is one of Optimal Stopping for the process P Λ t) of 3.12) 3.15). In the next section we shall solve this optimal stopping problem by associating with it a Variational Inequality, similar to that of 2.9) 2.13). 3.4 REMARK : The expression of 2.25) for the unconstrained-optimal leverage ratio, suggests 9

10 α = 2r/σ 2 as a critical level for the constant α > in 3.2), in the sense that for α α the unconstrained-optimal portfolio ˆπ ) of 2.21) violates the constraint 3.2) for every t, τ h ), or equivalently that xg x) gx) < α holds for every stock-price level x, h). In other words, a constraint of the type 3.2) with α 2r/σ 2 is severe, whereas α > 2r/σ 2 can be construed as mild. These two cases will be discussed separately in Sections 5 and 4, respectively), as the solution of the stochastic optimization problem of 3.4) exhibits qualitatively distinct features in each one of them. 4 A VARIATIONAL INEQUALITY By analogy with Theorem 2.3 and 2.21) 2.24), let us postulate the existence of a function g α ) and of a process Λ α ) in L +, such that we have ˆπ α t) = e αλαt) xg αx)) x=pλα t), ˆXα t) ˆX gαx),ˆπα,ĉα t) = e αλαt) g P Λα t)) for the optimal hedging portfolio/consumption process pair ˆπ α ), Ĉα )) and their corresponding value-process ˆX α ), under the constraint 3.2) on the short selling of stock. In order for this constraint to hold, the function g α ) must satisfy 4.1) αg α x) + xg αx), on, h). On the other hand, the function g α ) must dominate on the interval, h) the α-enlargement φ α ) of the reward function φx) = q x) +, namely 4.2) g α x) φ α x), x, h), where 4.3) φ α x) = sup = ν K e δν) φxe ν ) q = sup e αν q xe ν) + ν q x ; x α ) x α ; < x < is the function of Broadie, Critanić & Soner 1997); see also Karatzas & Kou 1998) and Wystup 1997) for other applications of this enlargement. We are thus led to a variational inequality similar to that of Section 2. ) 4.1 VARIATIONAL INEQUALITY : Find a number b α, and a convex, decreasing func- 1

11 tion g α ) in the space C, h)) C 1 b, h)) C2 b, h) \ b α), such that 4.4) 4.5) 4.6) 4.7) 4.8) 4.9) 4.1) σ 2 2 x2 g αx) + rxg αx) rg α x) = ; b α < x < h σ 2 2 x2 g αx) + rxg αx) rg α x) < ; < x < b α g α x) > φ α x) ; b α < x < h g α x) = φ α x) ; x b α αg α x) + xg αx) > ; < x < h αg α h ) + hg αh ) = g α x) = ; h x <. The condition 4.9) suggests that h > is an elastic barrier for the process P Λ ) of 3.12) 3.15); see section 6.4 in Karatzas & Shreve 1991) for a study of elastic boundary conditions for Brownian motion. 4.2 THEOREM: If the pair b α, g α )) solves the Variational Inequality 4.1, then g α ) coincides with the optimal expected reward in the stopping problem of 3.11), and the stopping time 4.11) τ b α = inf t /PΛ t) b α attains the supremum in 3.11). More specifically, G α x) = g α x) = E 4.12) with the notation of 3.14) and 4.13) τ Λ h = sup τ S e rτ bα αλ τbα ) q P Λ τb α ) ) + 1τ bα <τh E e rτ αλ τ) q P Λ τ)) + 1 τ<τ h = sup sup Λ ) L + τ S E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h = inf t /P Λ t) > h, τ h τ Λ h. Furthermore, we can replace in the expressions of 4.12) the function q ) + = φ ) by its α- enlargement φ α ) of 4.3). The proof of this result is deferred to Appendix A. We present now the solution of the Variational Inequality 4.1, in the case α > 2r/σ 2 of a mild constraint ). The case < α 2r/σ 2 of a severe constraint ) is treated in the next section. 4.3 PROPOSITION: Suppose that α > 2r/σ 2. With the notation β = 1 + 2r/σ 2 ) of Proposition 2.4, denote by b α the unique solution of the equation ) bα 4.14) 1 + β = β + 1 β ) ) β bα q 1 + α h 11

12 ) in the interval,, and define 4.15) g α x) = x q bα q x ; x b α ) h/x) β 1+ β b α h/b α ) β 1+ β ; b α < x < h ; h x < Then the pair b α, g α )) solves the Variational Inequality Proof : The general solution of the equation 4.4) is given by g α x) = A α x γ + + B α x γ for suitable real constants A α, B α, and γ ± as in the proof of Proposition 2.1. The requirements g α b α +) = q b α, g αb α +) = 1 continuity of the function g α ) and of its derivative at x = b α ) and 4.9), lead to the equation 4.14) for b α and to the expressions 4.16) A α = ) q bα b α h β h b α ) β 1 β These, in turn, yield the expression of 4.15) for g α ). ), B α = A α 1 β ) h β. 1 + α To show that the equation 4.14) has a unique solution in the interval form F α b α ) =, where F α ) is the strictly convex function F α u) = β + 1 β ) u ) β u 1 β 1 + α h) q recall here that 1 + α > β > 1). We have F α ) = β 1 > as well as ) F α = β β ) α 1 + α 1 + α 1 + α q h ),, write it in the = F u) β u ) β, u < 1 + α h ) β αβ 1 + α < β β Consequently, there is a unique root b α of the equation F α u) in the interval satisfies F b α ) = β b ) β h >, so we have also bα < b. The derivatives of the function g α ) in 4.15) are given as 1 ; < x < b α g αx) = B α A α γ + x β ; b α < x < h ; x > h and g αx) = Aα βγ + x 1+β) ; b α < x < h ; < x < b α or x > h. 1 + α, ) αβ 1 + α =. ). This root Thus, in order to verify the decrease and convexity of g α ), it is enough to check that B α < < A α ; but this follows readily from the expressions of 4.16) under our assumptions. To verify 4.5) and 4.8), note that we have αg α x) + xg αx) = x) x > 1 + α)b α >, on, b α ) 12

13 as well as σ 2 2 x2 g αx) + rxg αx) rg α x) = rx rq x) = rq <, on, b α ) αg α x) + xg αx) = α A α x γ + + B α x ) ) + x A α γ + x β + B α = A α xα γ + )x β h β ) >, on b α, h). The equality 4.7) is clear from 4.3) and b α <, so it remains to show 4.6), namely 4.17) 4.18) x α g α x) > g α x) > q x ; b α < x 1 + α q ) α ; 1 + α 1 + α 1 + α < x < h. The inequality 4.17) follows easily, from the strict convexity of g α ) on the interval b α, h) and from g αb α +) = 1, g α b α +) = q b α. To see 4.18), notice that ) 4.19) the function x x α g α x) is strictly increasing on 1 + α, h since x α g α x)) = x α 1 αg α x) + xg αx)) > from 4.8); consequently, x α g α x) > ) α ) ) α g α q ) = q ) α. 1 + α 1 + α 1 + α 1 + α 1 + α 1 + α 4.4 REMARK : In terms of the process Λ ) L + of 3.14), the optimal hedging portfolio ˆπ α ) and the optimal cumulative consumption processes Ĉα ) are given by e αλ t) xg 4.2) ˆπ α t) = αx)) x=pλ t) ; t < τh ; τh t <, t τ h 4.21) Ĉ α t) = rq e αλ s) 1,bα ) P Λ s)) ds, respectively. The wealth process 4.22) ˆXα t) X gαx),ˆπα,ĉα t) = e αλ t) g α P Λ t)) ; t < τ h ; τ h t < is then the value process for the American put option of barrier type, under the short selling constraint of 3.2). Note that this constraint is indeed satisfied by the portfolio proportion process 4.23) ˆp α t) = ˆπα t)/ ˆX ) α t) = xg α x) g αx) ; t < τh x=pλ t), ; τh t < for which we clearly have ˆp α ) α. Finally, the stopping time τ = inf time of the option by its holder. t / Ĉ α t) > τb α of 4.11) is the optimal exercise 13

14 5 THE CASE < α 2r/σ 2 Now let us return to the case α 2r/σ 2 of severely constrained short selling of stock. In this setting the hedging problem admits a very simple solution, given by the function φ α ) of 4.3). 5.1 THEOREM: Suppose that < α 2r/σ 2. Then the hedging price of 3.4) is given as φα x) ; < x < h 5.1) G α x) =, ; h x < where φ α ) is the α-enlargement of the function φ ) = q ) +, as in 4.3). We shall devote much of the remainder of this section to the proof of Theorem 5.1 for x, h). Let us start by noticing that the function φ α ) of 4.3) is convex, decreasing and of class C, )) Cb 1, )) C2 b, ) \ ), with φ αx) = 1 ; < x < ) x ) ; x and In particular, we have φ αx) = 5.2) αφ α x) + xφ αx) = ) ; < x < α x 2+α) ; x > x1 + α) > ; < x < ; x. 5.3) σ 2 2 x2 φ αx) + rxφ αx) rφ α x) = ) rq ) ; < x < ασ 2 α 2r 1 rqx α ; x >. For fixed < x < h, and an arbitrary Λ ) L +, let us apply Itô s rule e.g. Karatzas & Shreve 1991), Problem 3.7.3, p.219) to the process e rt αλt) φ α P Λ t)), t <. By analogy with 6.1)-6.3) in the Appendix A, and using again the inequalities of 5.2) and 5.3) in conjuction with the dynamics of 3.13) for the process P Λ ), we obtain φ α x) E e rτ αλτ) φ α P Λ τ)) 1 τ<τ Λ h E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h, τ S. Therefore, 5.4) φ α x) sup sup E e rτ αλτ) φ α P Λ τ)) 1 τ<τ Λ Λ ) L + τ S h sup sup E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ Λ ) L + τ S h sup sup E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h. λ ) D τ S 14

15 5.2 LEMMA: The inequalities of 5.4) are in fact valid as equalities. Proof : Clearly, it suffices to verify 5.5) sup E e rt αλt ) q P Λ T )) + 1 T <τ Λ h λ ) D T φ α x), where the expression on the left hand side is the value of the European-type barrier option with fixed exercise time T >. From Wystup 1997), we know that this value equals 5.6) sup E e rt αλt ) q P Λ T )) + 1 T <τ Λ Λ ) L + h r T ) where we have set 5.7) Λ T t) = = E max u t log ) max Λ T T ), log ST ) + e rt αλ T T ) q P Λ T T )) 1T <τ h )) + Su) h ; t < T ; t = T In other words, Λ T ) pushes the stock-price just enough to ) avoid crossing the barrier h, and makes a possible) jump at t = T that pushes P Λ T T ) to whenever the stock-price ST ) at time t = T exceeds this threshold. Accordingly, we have denoted in 5.6) by L + r T ) the space of adapted processes Λ :, T Ω, ) whose paths are increasing and right continuous on, T ) with Λ) =. With this notation, we have clearly Λ T T ) P Λ T T ) 1 T <τ h T T T )) + logx log, 1 + α ) min x,, 1 + α 1, a.s. By the Bounded Convergence Theorem, the limit of the left-hand side in 5.6) as T, is equal to ) + exp α logx log q x )) = φ α x), 1 + α 1 + α., as claimed. Now fix x < h and, for arbitrary λ ) D, apply Itô s rule to the process e rt αλt) φ α St)), t <. By analogy with 6.5)-6.6) in Appendix A, and using the inequalities of 5.2) and 5.3), we obtain in conjunction with the dynamics of 3.8) for the stock price process S ): φ α x) E λ e rτ αλτ) φ α Sτ)) 1 τ<τh E λ e rτ αλτ) q Sτ)) + 1 τ<τh, τ S. Consequently, 5.8) φ α x) sup sup λ ) D τ S sup sup λ ) D τ S E λ e rτ αλτ) φ α Sτ)) 1 τ<τh E λ e rτ αλτ) q Sτ)) + 1 τ<τh =: G α x). 15

16 5.3 PROPOSITION: The inequalities of 5.8) can be replaced by equalities. Proof : It suffices to show the inequality φ α x) G α x); but this is proved in exactly the same way as in the Appendix A proof of 6.7)). This completes the proof of Theorem REMARK : By analogy with Remark 4.3, the optimal hedging pair ˆπ α ), Ĉ α )) is now given as e αλ t) xφ 5.9) ˆπ α t) = αx)) x=pλ t) ; t < τh ; τh t < and 5.1) 5.11) t τ h Ĉ α t) = rq e αλ u) 1, ) P Λ u)) ) ) α + 1 ασ2 1 2r P Λ u),h) P Λ u)) du, for t <, whereas the corresponding value process is 5.12) ˆXα t) X φαx),ˆπα,ĉα t) = e αλ t) φ α P Λ t)) ; t < τ h ; τ h t <. The associated portfolio-proportion process 5.13) ˆp α t) = ˆπα t)/ ˆX α t) = ) xφ α x) φ α x) x=pλ t) ; t < τh ; τh t < clearly satisfies the constraint ˆp α ) α. The optimal time for the holder to exercise his option, is / 5.14) ˆτ α = inf t Ĉα t) >. This coincides with the stopping time τ = inf / t P Λ t) /1 + α) if α 2r/σ 2 ; but for α < 2r/σ 2, we have ˆτ α in 5.13). Both these claims follow readily from 5.1). 5.5 REMARK: The limiting case α corresponds to prohibition of short selling in 3.2). In this limiting case, the cheapest way for the issuer to hedge the American barrier option of 2.2) is the trivial one of setting aside the strike price itself at t =, namely 5.15) lim α H α x) = lim α G α x) = as can be checked easily from 5.1), 4.3). limα φ α x) ; < x < h ; h x < = q ; < x < h ; h x <, 16

17 6 APPENDIX A We devote this section to the proof of Theorem 4.2. All the terms in 4.12) are equal to zero for x h, so we shall concentrate on the case < x < h. On the interval, h), the functions g α ), g α ) are bounded, continuous; and the function g α ) is bounded, continuous on, h)\b α. For any given Λ ) L + and τ S, apply Itô s e.g. Karatzas & Shreve 1991), Problem 3.7.3, p. 219) rule to the process e rt αλt) g α P Λ t)), t < τh Λ in conjunction with 3.13), to obtain 6.1) = g α x) e rτ τ h Λ) αλτ τ h Λ) τ τ Λ g α PΛ τ τh Λ )) h + σ τ τ Λ h e rt αλt) ξg αξ) + αg α ξ) ) ξ=pλ t) dλt) e rt αλt) ξg αξ) ) ξ=pλ t) dw t) τ τ Λ h σ e rt αλt) 2 2 ξ2 g αξ) + rξg αξ) rg α ξ)) dt, P a.s. ξ=pλ t) from 4.4), 4.5) and 4.8), 4.9). Note also that, thanks to 4.4), 4.9) and 3.15), the inequality of 6.1) holds as equality for 6.2) Λ ) Λ ), τ τ b α defined in 3.14) and 4.11), respectively. The same argument as in the proof of Theorem 2.3 shows that the stochastic integral in 6.1) has P expectation equal to zero. We deduce g α x) E e rτ τ h Λ) αλτ τ h Λ) g α PΛ τ τh Λ )) 6.3) E e rτ αλτ) g α P Λ τ))1 τ<τ Λ h E e rτ αλτ) φ α P Λ τ))1 τ<τ Λ h E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h, and note that the inequalities of 6.3) hold as equalities for the choices of 6.2). Therefore, g α x) = E e rτ bα αλ τbα ) q P Λ τb α ) ) + 1τ b α <τ h 6.4) = sup E e rτ αλ τ) q P Λ τ)) + 1 τ<τ h τ S = sup sup E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ Λ ) L + τ S h = sup sup E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h, λ ) D τ S where the last equality comes from Appendix B. Similar equalities hold if one replaces the function q ) + = φ ) in the equalities of 6.4), by its α enlargement φ α ) of 4.3). This proves all the equalities of 4.12) except the first, namely, G α x) = g α x). Proof of G α x) g α x): For any given λ ) D and τ S, and with Λ ) λu) du, 17

18 apply Itô s rule to the process e rt αλt) g α St)), t < τ h in conjunction with 3.8) and 2.3), to obtain 6.5) = τ τh g α x) e rτ τ h) αλτ τ h ) g α Sτ τ h )) + σ τ τh e rt αλt) ξg αξ) + αg α ξ) ) ξ=st) λt) dt e rt αλt) ξg αξ) ) ξ=st) dw λ t) τ τh σ e rt αλt) 2 2 ξ2 g αξ) + rξg αξ) rg α ξ)) dt, P λ a.s. ξ=st) thanks to 4.4) 4.9), by analogy with 6.1). Just as before cf. proof of Theorem 2.3), the P λ expectation of the stochastic integral in 6.5) is equal to zero, which gives 6.6) g α x) E λ e rτ τ h) αλτ τ h ) g α Sτ τ h )) E λ e rτ αλτ) g α Sτ)) 1 τ<τh E λ e rτ αλτ) q Sτ)) + 1 τ<τh by analogy with 6.3). Taking the supremum in 6.6) with respect to λ ) D, τ S and recalling 3.4), 3.5), we obtain the inequality g α x) G α x). Proof of G α x) g α x): In view of 3.4), 3.3) and 6.4), it suffices to show that 6.7) ξ E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h, λ ) D, τ S holds for all ξ > H α x). We can rewrite the right hand side of 6.7) as dp 6.8) Ẽ λ d P e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h λ Fτ) 1 = Ẽλ Zλ τ)) e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h, where Ẽλ denotes expectation with respect to the probability measure P λ that satisfies d 6.9) P t ) λ = dp Z λ t) = λs) exp dw s) 1 t ) λs) 2 ds σ 2 σ Ft) for all t <. 6.1 REMARK: Under the measure P λ, the process t 6.1) Wλ t) = λu) W t) σ du, t < is Brownian motion, and thus 6.11) dp Λ t) = P Λ t) r dt + σ d W λ t) 18

19 from 3.13). In other words, the process P Λ ) has the same law under P λ, as the law of process S ) under P. Furthermore, we have the dynamics ) ) 1 λt) ) ) d Zλ t) = Zλ t) d Wλ t), σ ) 1 Zλ ) 1 for the inverse of the process Z λ ) in 6.9). From the assumption ξ > H α x), we know that there exists a portfolio / consumption process pair π, C) such that, for the corresponding wealth process X λ ) X ξ, π, C ) as in 6.13) e rt Xλ t) = ξ e rs d Cs) + σ,t by analogy with 2.5)), we have both 6.14) α X λ t) + πt), t < and t e ru πu) d W λ u) 6.15) Xλ t) q P Λ t)) + 1 t<τ Λ h, t < by analogy with 3.2) and 2.4), 2.2)). Now in order to prove 6.7), it suffices to show that the process 6.16) Qλ t) = Zλ t)) 1 e rt αλt) Xλ t), t < is a P λ supermartingale; because then from the optional sampling theorem and 6.15), we have ξ = Q 1 λ ) Ẽλ Qλ τ) = Ẽλ Zλ τ)) e rτ αλτ) Xλ τ) 1 Ẽλ Zλ τ)) e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ = E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h, τ S holds for any λ ) D, which is 6.7). In order to prove that Q λ ) is a P λ supermartingale, observe from 6.12), 6.13) that d Q ) 1 λ t) = Zλ t) e rt αλt) λt) α X ) λ t) + πt) dt d Ct) + σ πt) λt) ) σ X λ t) d W λ t). Thanks to 6.14), we deduce that Q λ ) is a local supermartingale under P λ ; but Q λ ) is also nonnegative, and thus it is actually a supermartingale under P λ. The proof of Theorem 4.2 is complete. h 19

20 7 APPENDIX B We shall establish here the equality 7.1) sup sup λ ) D τ S = sup sup Λ ) L + τ S E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h which was used in the proof of Theorem 4.2 last equality in 6.4); recall the notation of 3.12) and 4.13)). Clearly, it is enough to show that, for any given Λ ) L + and any bounded stopping time τ S, we have 7.2) L = sup λ ) D E e rτ τ λu) du q e τ + Sτ)) λu) du 1 max u τ e u ) λs) ds Su) <h =: R. E e rτ αλτ) q P Λ τ)) + 1 τ<τ Λ h Proceeding as in Wystup 1997), we introduce a sequence Λ n ) of truncated left mollifications of Λ ), as follows: 7.3) Λ n t, ω) = where χu) = C exp every ω Ω, i) 1 1 2u+1) 2 1 min n, Λ t + u ) n, ω χu) du, t <, and the constant C > is chosen so that χu) du 1. For Λ n, ω) is absolutely continuous ii) Λ n, ω) Λ, ω) uniformly on compact intervals as n. Thus L lim inf n E lim inf n E e ) + rτ αλ nτ) q e Λnτ) Sτ) 1max u τe Su))<h Λnu) ) + e rτ αλ nτ) q e Λnτ) Sτ) 1max u τe Su))<h) Λnu) E e rτ αλτ) q e Λτ) Sτ) ) + 1max u τe Λu) Su))<h = R, by Fatou s Lemma, the Bounded Convergence Theorem, and the fact that we have 7.4) lim inf 1 n max u τe Λnu) Su))<h 1 max u τe Λu) Su))<h. This is certainly true on max u τ e Λu) Su) ) h ; on the other hand, on the event max u τ e Λu) Su) ) < h we have max u τ e Λ nu) Su) ) < h for all n large enough, and 7.4) follows. 2

21 Acknowledgement: We are indebted to Professor Steven Shreve for a talk at Columbia University on Uwe Wystup s dissertation, which was the inspiration for this work. We also want to thank Drs. Jakša Cvitanić, Mark Davis, Paavo Salminen and Qiang Zhang, for helpful discussions. References 1 Black, F. & Scholes, M. 1973) The pricing of options and corporate liabilities. J. Political Economy 81, Broadie, M., Cvitanić, J. & Soner, M. 1998) Optimal replication of contingent claims under portfolio constraints. Review of Financial Studies, Vol. 11, No. 1, Karatzas, I. 1996) Lectures on the Mathematics of Finance. CRM Monograph Series, Vol. 8, American Mathematical Society. 4 Karatzas, I. & Kou, S.G. 1998) Hedging American contingent claims with constrained portfolios. Finance and Stochastics 3, Karatzas, I. & Shreve, S. 1984) Connections between optimal stopping and singular stochastic control I: Monotone follower problems. SIAM J. Control & Optimization 22, Karatzas, I. & Shreve, S. 1991) Brownian Motion and Stochastic Calculus. Second Edition, Springer-Verlag, New York. 7 Karatzas, I. & Shreve, S. 1998) Methods of Mathematical Finance. Springer-Verlag, New York. 8 Merton, R.C. 1973) Theory of rational option pricing. Bell J. of Econom. & Management Sci. 4, Salminen, P. 1985) Optimal stopping of one-dimensional diffusions. Mathematische Nachrichten 124, Wystup, U. 1997) Valuation of exotic options under short-selling constraints as a singular stochastic control problem. Doctoral Dissertation, Carnegie Mellon University. 21