FROM LATTICE MODELS TO NONLINEAR VOLATILITY AND FREE-BOUNDARY PROBLEMS. Antonio Paras (1 2) and Marco Avellaneda

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1 DYNAMIC HEDGING WITH TRANSACTION COSTS: FROM LATTICE MODELS TO NONLINEAR VOLATILITY AND FREE-BOUNDARY PROBLEMS Antonio Paras (1 ) Marco Avellaneda (1 3) Abstract. We study the dynamic hedging of portfolios of options other derivative securities in the presence of transaction costs. Following Bensaid, Lesne, Pages & Scheinkman (199), we examine hedging strategies which are risk-averse have the least initial cost, in the framework of a multiperiod binomial model. This paper considers the asymptotic limit of the model as the number of trading periods becomes large. This limit is characterized in terms of nonlinear diusion equations. If A = p k=( dt) < 1 (k is the roundrip transaction cost, is the volatility dt is the lag between trading dates), the optimal cost approaches the solution of a nonlinear Black-Scholes-type equation in which the volatility is dynamically adjusted upward to p p 1+A or downward to 1 ; A according to the local convexity of the solution. For A = 1, the upward adjustment is similar but the downward adjustment assigns zero nominal volatility tothe underlying asset for long-gamma positions. In the latter case, the optimal cost function is the solution of a free-boundary problem. We alsocharacterize the associated hedging strategies. We shown that if A < 1, it is optimal to replicate the nal payo via \nonlinear Delta hedging". On the other h, if A = 1, the optimal strategies are path-dependent, non-unique typically super-replicate the nal payo. (1) Courant Institute of Mathematical Sciences, New York University, N.Y.,N.Y.,1001. () These results were obtained as part of the rst author's doctoral dissertation. (3) Partially supported by the Institute for Advanced Study, Princeton, N.J.,

2 1. Introduction Statement of the Main Results The problem of accounting for transaction costs in the dynamic hedging of derivative securities has attracted considerable attention from theoreticians practitioners. S- trategies which account for transaction costs pricing models that incorporate the costs into the premium have considerable practical interest, especially for trading derivatives in markets with moderate or low liquidity. The problem can be formulated in terms of an agent that buys sells options on the stock of a company. At some point in time, he or she decides to hedge the \book", or options portfolio, against future price uctuations. The agent would like to determine the least costly strategy, taking into account the projected transaction costs due to dynamic hedging. The initial cost of such strategy can be interpreted as the minimal capital reserve needed to protect the portfolio against future market moves. 1 We shall make the assumption that the agent is totally risk-averse: only strategies which ensure non-negative cash-ows after closing all positions are deemed admissible. This assumption is important in the framework of this paper but by no means necessary. Several theorists have considered hedging strategies which allow for losses but maximize a utility function assigned to the agent (Constantinides (1979,1986), Hodges Neuberger (1989), Davis, Panas Zariphopolou (1993)). In principle, a utility-based approach may present greater exibility but has the disadvantage of producing utility-dependent results. For this other reasons, total risk-aversion plays a central role in the assessment of transaction costs in derivative strategies. Risk-averse hedging in the presence of transaction costs was rst considered by Lel (1985) for the log-normal model later by Boyle Vorst (199) for the binomial lattice. Both papers consider only the problem of hedging a single option. A non-trivial generalization of the Lel model applicable to option portfolios exotic options was proposed later by Hoggard, Whalley Wilmott (1993), based on dynamic replication of the payo. However, Boyle Vorst Hoggard et al. both observed that replication may not always be feasible since it can lead to innite option prices when transaction costs exceed a critical value the agent is long Gamma (as in the case of a long call position). Avellaneda Paras (1994) examined this issue, which is related to the mathematical ill-posedness of the replication equation. They proposed an explanation for the illposedness, which points to a fundamental dierence between long short options positions in markets with large bid-oer spreads. The risk-averse seller of an option is obliged to dynamically hedge his or her exposure in the cash market, regardless of transaction 1 Throughout this paper, we consider only dynamic hedging with shares of stock a money-market account. Of course, in practice, traders also hedge their books with other derivative securities to (among other things) diminish the transaction costs. We assume that the agent has already taken a denite position in the derivatives market seeks to hedge the \residual" exposure with a position in the cash market. The utility framework contains risk-aversion as a special case in which losses are assigned \innitely negative utility".

3 costs. On the other h, the buyer of the option risks only the initial premium: hedging is done primarily to oset the time-decay in the option's value. It is intuitively clear ( can be proved mathematically) that if transaction costs in the cash market are suciently large, Delta-hedging to oset time-decay is impossible. This key observation applies also to option portfolios: if transaction costs are high volatility is low, Delta-hedging a position which is long-gamma is counterproductive. In such situations, a temporary holding strategy is preferable, since it does not carry immediate market risk as long as capital reserves are suciently high. Based on this observation, Avellaneda Paras proposed a new scheme for pricing hedging option portfolios which is based on solving an obstacle problem for Lel's volatility-adjusted PDE. The corresponding dynamic hedges are non-markovian (path-dependent) dominate { or super-replicate { the nal payo. The aforementioned strategies are based on replication or super-replication. This raises the question of characterizing strategies which minimize the initial cost of hedging. How are the two concepts related? The least-initial-cost formulation was rst considered by Bensaid, Lesne, Pages Scheinkman (199) (BLPS) in the framework of a nite-period binomial tree, using a a dynamic programming algorithm (see Section, equation.10). Due to dependency on the previous stock holdings, the BLPS optimal hedging strategies are path-dependent, with some notable exceptions. For instance, in the case of a short option settled in shares, Bensaid et al. showed that it is always optimal to replicate the payo. This result has an interesting consequence: the BLPS least-initial-cost strategy for a short option corresponds, in the asymptotic limit of many trading periods, to a Black-Scholes pricing formula with volatility adjusted upwards to reect transaction costs, analogous to the Boyle Vorst (199) formula. 3 The least-initial-cost pricing hedging of complex options portfolios is the main subject of this paper. Our approach consists in characterizing the solutions of the BLPS algorithm in the limit of innitely many trading periods for general contingent claims. We show that the algorithm admits a simple interpretation in this asymptotic limit. In fact, the optimal initial costs satisfy nonlinear partial dierential equations analogous to those proposed by Hoggard et al. by Avellaneda Paras, with minor changes in the values of the adjusted volatilities that reect the dierence between the normal binomial statistics. As a consequence of this result, we determine precisely in which instances the BLPS algorithm gives rise to replicating strategies when the optimal strategies are path-dependent dominating. We also show that whenever path-dependency holds for BLPS, the hedging strategies are essentially analogous to those proposed by Avellaneda Paras (1994) for the log-normal model. Thus, path-dependency occurs when (i) transaction costs exceed a critical level (ii) the agent is long-gamma for some level of the spot price. 1.1 Overview of the results Following Bensaid, Lesne, Pages Scheinkman (199), we formulate the problem in 3 This is also the analogue of the Lel (1985) for the binomial tree. 3

4 terms of the binomial probability model. In this framework, there are N + 1 trading dates, the last one being the expiration date. The lag between successive trading dates is xed. The impact of imperfect liquidity is modeled through a bid-oer spread in the cash market, assuming that the agent will buy stock at the oer sell at the bid. The bid/oer prices for one share of stock are dened as S bid n = S n 1 ; k Sn ask = S n 1+ k where S n is the average between the bid oer k represents the (percentage) roundtrip transaction cost for buying/selling stock, i.e., 4 k = Sask n ; Sn bid : S n For simplicity, we assume that k is constant. The change in the average stock price over a single period is modeled by a two-state rom variable S n p 1-p S S n n U D with probabilities p 1 ; p for upward downward moves. We also assume that the dollar return for lending/borrowing over one trading period is R (constant), that U, D R satisfy the \pure" no-arbitrage condition D <1+R < U : Akey feature of this model is given in Denition 1.1. The stock is said to be risk-feasible if there is a positive probability of posting a prot by following either one of the following strategies: (i) Borrow money to purchase a share ofstock hold the stock for one trading period sell the stock pay back the loan. (ii) Short-sell a share ofstock deposit the proceeds in a money market account for one period close the account unwind the short position after the period. 4 Thus, an agent who buys immediately sells one share of stock assumes a loss of k S n dollars. 4

5 Mathematically, conditions (i) (ii) are satised if only if 1+k= 1 ; k= D < 1+R < U 1 ; k= 1+k= : The risk-feasibility of an asset depends on how much its price is expected to oscillate over a single trading period in relation to the round-trip transaction costs. As we shall see, the BLPS optimal strategies are very dierent according to whether risk-feasibility holds or not. If the stock is risk-feasible, replicating strategies are always optimal for arbitrary contingent claims. Otherwise, replication may not be optimal. To illustrate the sub-optimality of replicating strategies, we consider a simple example. Example 1. Assume there is a single trading period. An agent wishes to hedge a short position in a hypothetical claim contingent onthevalue of a non-risk-feasible stock. Assume that this claim pays $1 in the \up" state $0 in the \down" state. Consider rst the case in which the agent has no endowment in shares. Opening subsequently closing any position in shares after one period will result in an overall loss under both states of the world. Hence, it follows that the initial hedging cost of any replicating portfolio using shares an interest-bearing money-market account must exceed the present value of $1 (the maximum liability). On the other h, simply holding the present value of $1 (i.e., $(1 + R) ;1 ) in cash constitutes a (cheaper) dominating riskless strategy. Notice, however, that the situation is dierent if either (i) the agent has an endowment inshares or (ii) if the contingent claim is settled in shares rather than in cash. In such cases, a replicating strategy may becheaper than $(1 + R) ;1 because transaction costs are lower for the agent. As a matter of fact, the reader can verify that it is optimal to replicate when the agent's endowment in shares exceeds 1=[S 0 (1 ; k=)(u ; D)] shares in Case (i) 1=[S 0 (U ;D)] in Case (ii) (See Section 3). This suggests that, aside from the risk-feasibility of the underlying cash instrument, the agent's endowment the form of payment may determine whether the it is optimal to replicate or not. For a binomial model with a large number N of trading periods per year hence with small duration between successive trading dates dt = T=N 1 risk-feasibility is equivalent to A k p dt < 1 (1.1) where is the annualized stock volatility. We shall refer to A as the Lel number. Table 1.1 shows that the Lel number takes moderate values (between 0 10) for stard values of volatility, time-lag transaction costs ( volatility of 0%, time-lags ranging from one week to four times a day round-trip costs of up to 6%). From this table, we draw the heuristic conclusion that if the time-to-expiration is suciently long, then (i) the minimum lag between trades can be taken to be small enough to warrant a 5

6 k/ dt one week 1.9*10 - half week *10 one day *10 half day *10 quarter day * Table 1.1. Lel number A for dierent values of the minimum lag between trading dates dt (from one week to 1=4 day) with round-trip transaction costs between % 6% volatility of0%. continuum description of the system, (ii) at the same time, A can be considered to be nite. The interesting asymptotic regime of parameters for the asymptotic analysis of the BLPS algorithm is, in fact, dt 1 A = O(1). 5 Under these assumptions, the asymptotic analysis of the BLPS algorithm in the limit N %1yields the following results: (i) If the underlying asset is risk-feasible, the minimal initial cost of risk-averse hedging a European-style contingent claim with payo value F (S) expiration date T approaches uniformly to the solution of the nonlinear diusion ; + 1 S P + = 0 (1.) with P (S T )= F (S) 5 The regime dt 1, A 1, which might be considered relevant whendt is small, gives rise to trivial buy/hold strategies. The BLPS cost function for this regime is recovered by rst letting dt %1 then taking the limit A 1 in the asymptotic equations (1.4) derived hereafter (cf. Avellaneda Paras (1994)). 6

7 where P = 1+A P : (1.3) Here, r represents the annualized interest rate. Notice that, since the Lel number is less than unity, the nonlinear parabolic equation (1.) is well-posed. The optimal hedging strategy corresponds to maintaining a hedge-ratio shares of stock at each date. (ii) If the underlying asset is not risk-feasible, the minimum initial cost for hedging a European-style contingent claim with payo value F (S) expiring at time T can be approximated as N %1by the solution of the \obstacle problem": P (S t) e ;r(t ;t) F (e r(t + 1 S ( ; P 5 0 for all + 1 S ( ; P = 0 if P (S t) >e ;r(t ;t) F (e r(t ;t) S), with nal condition P (S T )= F (S) : (1.4) This obstacle problem reduces to a linear PDE analogous to Lel's equation if F SS is positive for all S. Otherwise, the problem splits the (S t)- plane into two regions: one where the PDE is satised, another, \the contact set", in which the value function coincides with the \obstacle" e ;r(t ;t) F (e r(t ;t) S) the nominal volatility is zero. The corresponding dynamic hedging strategies alternate between delta-hedging holding periods without hedge adjustments. The times at which the agent switches from deltahedging to static hedging are determined by the position of the spot price with respect to the contact set by the Delta of the agent's portfolio. In simple words, the asymptotic analysis of the BLPS algorithm shows that the impact on the total hedging cost arising from bid-oer spreads can be taken into account by adjusting the volatility upward to p 1+A when the agent is short \nonlinear Gamma" P=@S > 0 ) adjusting the volatility downward to p 1 ; A or to zero when the 7

8 Short Call Position. V($) 0. A = 1.1 V($) 0. A = S($) S($) V($) 0. A = 0.71 V($) 0. A = S($) S($) Figure 1.1. Value of a short call position with strike price K =1with 6 months to expiration. The parameter values are k =1%, = 0% dt =0:01 (A =0:5) 0:005 (A = 0:71) 0:003(A =0:94) 0:0 (A =1:1). The curves corresponding to the PDEs the BLPS algorithm (thick line) are visually indistinguishable. The dotted lines represent the intrinsic value of the call. agent is long \nonlinear Gamma" (@ P=@S 5 0). 6 When A = 1, an adjustment to zero volatility should be made in the latter case. In addition to providing a characterization of least-initial-cost dynamic hedging, the asymptotic PDEs provide an ecient computational procedure for solving approximately the BLPS algorithm. In fact, BLPS involves two state-variables price stock 6 Since P represents costs the agent's cost (liabilities), the prot-loss is ;P. Hence our denition of Gamma. 8

9 Long Call Position. V($) A = 1.1 V($) A = S($) S($) V($) A = 0.71 V($) A = S($) S($) Figure 1.. Same as Figure 1.1 for a long call position. Again, the agreement between BLPS the nonlinear PDE s is remarkable. Notice that the long call position is priced at intrinsic value (as if the volatility were zero) for A =1:1. holdings (see equation (.11) in Section ), while the PDEs involve only the spot price. More importantly, the BLPS scheme has exponential complexity in N when the underlying asset is not risk-feasible, unlike the nite-dierence PDE solvers which areo(n ). The error which arises from the use of a nite-dierence scheme for the PDEs (1.) or (1.4) to approximate the BLPS algorithm is essentially proportional to p dt, hence small. The quality of the approximation is exhibited in Figures 1.1, 1., for various contingent claims under realistic values of the model parameters for equity markets. To conclude this Introduction, we address an important issue regarding risk-feasibility 9

10 Short Buttery Position. V($) 0.1 A = 1.1 V($) 0.1 A = S($) S($) V($) 0.1 A = 0.71 V($) 0.1 A = S($) S($) Figure 1.3. Comparison between the value functions obtained from the dierential equations the Bensaid-Lesne-Pages-Scheinkman algorithm for a short 0:9;1:0;1:1 buttery spread with six months to expiration. The smooth curves are the solutions of the PDEs (1.) or (1.4) the oscillating curves are solutions of the BLPS algorithm. The dotted lines represents the intrinsic value of the spread in present dollars. The parameter values are as in Figure 1.1. risk-aversion. As shown above, the initial cost the strategy to follow are determined by the Lel number A or equivalently, by the minimum lag between adjustments, dt. The question of how to choose the Lel number in practice poses itself naturally. Clearly, the BLPS model does not address this issue, requiring an endogenous specication of dt. In practice, a choice of the mesh-size cannot be made without taking into account the agent's aversion to hedge slippage between trading dates. This remark also applies to 10

11 Short Digital Position. V($) 0.1 A = 1.1 V($) 0.1 A = S($) S($) V($) 0.1 A = 0.71 V($) 0.1 A = S($) S($) Figure 1.4. Same as Figure 1.3 for a digital option with 6 months to expiration. The payo is $1 if the stock isworth more than $1 at expiry $0 otherwise. the asymptotic PDEs. This dependency on the time-lag is a conceptual disadvantage of the model vis a vis the utility-dependent approaches such asdavis, Panas Zariphopolou (1993) which determine dt endogenously. It is clear however that small Lel numbers correspond to large intervals between adjustments, with greater slippage risk, large Lel numbers correspond to smaller lags between adjustments thus to less risk but more transaction costs. A practical way to specify dt would be to view the slippage risk as being determined by the magnitudes of the nonlinear Delta (@P=@S) nonlinear Gamma P=@S ). Thus, a choice of dt can be made using either a \worst-case scenario" time-step, or a time-step that varies according to the P to expiration. In practice, specication of dt can be done by analyzing prot/loss his- 11

12 tograms obtained by Monte Carlo simulation (Avellaneda Paras (1994)). The overall result would be to have avariable Lel number A a PDE that would combine the features of (1.)-(1.3) (1.4). 7 The remaining sections of this of the paper contain a detailed mathematical analysis of the passage from the binomial lattice model to the asymptotic PDEs. In Section, we review the BLPS algorithm. In Section 3, we study two special conditions under which the optimal hedging strategy is to replicate the payo thus BLPS reduces to a backwardinduction algorithm on the binomial lattice. This analysis pertains to the properties of the BLPS algorithm at the discrete level. In Section 4, we study the asymptotic behavior of the algorithm in the cases when replicating strategies are optimal. The technique used in the proofs is to derive approximate solutions of the backward-induction algorithm using the solutions of the PDE (1.). This requires some regularity properties of the solutions as well as a \comparison principle", adapted to the nonlinear induction relation, to evaluate rigorously the approximation error. Section 5 studies the case A = 1, when optimal dynamic hedging is path-dependent. To characterize this regime, we construct super-replicating strategies using the solution of the obstacle problem following Avellaneda Paras (1994). This leads to an upper bound on the BLPS solution in terms of (1.4). A lower bound in terms of P (S t) is obtained by comparing the BLPS optimal cost with the values of certain \barrier options" that knock out on the boundary of the contact set of the obstacle problem. The main asymptotic results are contained in Theorems 4.4, in Sections 4 5. For the reader's convenience, the regularity properties of equations (1.) (1.4) used in the proofs are presented in an Appendix. 7 The use of variable-dt strategies was suggested in Whalley Wilmott (1993) for small values of A for which equation (1.3) remains well-posed. More recently, Grannan Swindle (1995) investigated strategies for hedging stard options using Lel's framework with variable time-steps. Other recent papers which investigate option replication with transaction costs the time-lag issue include Hodges Clelow (1993,1994), Flesaker Houghston (1994), Albanese Tompaidis (1995) Cvitanic Karatzas (1995). 1

13 . The Bensaid-Lesne-Pages-Scheinkman Algorithm Consider a binomial tree with N periods, starting at the initial date t 0 ending at the expiration date t N.Atrading strategy is dened as a sequence of portfolios ( n B n ), n =0... N, to be held during each period. Here, n represents the number of shares of the underlying security held long or short B n the balance of a money-market account. Adjustments of the portfolio are done at the beginning of each trading period: at time t n an amount n ; n;1 of shares is traded the balance of the operation (transaction cost included) is added to the money-market account. A self-nanced strategy is one for which the balance in the money-market account after adjusting the share holdings is exactly equal to B n.we will only consider self-nanced strategies. At time t n, but before readjusting the position, the portfolio is composed of n;1 shares B n;1 (1 + R) in cash. The cost of trading n ; n;1 shares is where ( n ; n;1 )S n + k j n ; n;1 js n ( n ; n;1 )S n (y) = 8 >< Therefore, a self-nanced strategy satises >: (1 + k ) y y = 0 (1 ; k ) y y<0: B n = B n;1 (1 + R) ; ( n ; n;1 )S n n =1 ::: N: (.1) Applying this relation recursively, we obtain B n = B 0 (1 + R) n ; nx j=1 ( j ; j;1 )S j (1 + R) n;j (.) so that at time t N B N = B 0 (1 + R) N ; NX j=1 ( j ; j;1 )S j (1 + R) N;j : (.3) This formula shows that a self-nanced strategy is uniquely determined by the initial money-market balance B 0 the (projected) sequence of stock holdings f 0... N g. 8 A European-style derivative security is a contingent claim with payo 8 It is assumed throughout the paper that the sequence f n g dening a trading strategy is nonanticipating with respect to the ow of information, i.e., n = n (S 0 S 1 ::: S n ). 13

14 N = N (S N) B N = B N (S N) in shares cash, respectively, attimet N. A dominating (or risk-averse) strategy of a short position in this claim is such that N = N B N = B N (.4) for all nal states. Note that if a dominating strategy satises N > N for some nal state, the excess of shares can be sold the prot credited to the money-market account. Therefore, we can restrict our attention to risk-averse hedging strategies for which If a hedging strategy satises the more stringent conditions for all nal states, we say that it is a replicating strategy. N = N B N = B N : (.5) N = N B N = B N (.6) Given an adapted sequence of stock holdings f 0... N = Ng, it follows from (.3) that B 0 = max fs j g N j=1 8 < : B N(1 + R) ;N + NX j=1 ( j ; j;1 )S j (1 + R) ;j 9 = (.7) is the minimum initial cash reserve required to constitute a dominating strategy of the contingent claim ( N B N ).9 Assume that at time t 0 an agent is short the contingent claim has an inventory of ;1 shares (long or short). The minimum investment required to implement f 0... N = Ng as a risk-averse hedging strategy is given by ;1 S 0 + ( 0 ; ;1 )S 0 + B 0 which represents the combined sum of the value of the initial endowment in shares, the cost of trading 0 ; ;1 shares, the minimum initial cash reserves necessary to ensure that the strategy f j g N j=0 dominates the nal payo. An optimal hedging strategy is one that, given the initial endowment ;1, minimizes the sum of the last two terms, ( 0 ; ;1 )S 0 + B 0. By denition, this is the eective cost of the strategy. 9 Here, the maximum is taken over all forward paths f S j g N j=0 followed by the stock price. 14

15 According to this denition, the minimum eective cost over all possible risk-averse hedging strategies (minimum eective cost, for short) is V 0 ( ;1 S 0 )= min f ( 0 ; ;1 )S 0 + B 0 g f j g N j=0 = min max f j g N j=0 fs j g N j=1 8 < : B N(1 + R) ;N + NX j=0 ( j ; j;1 )S j (1 + R) ;j 9 = : (.8).1 Dynamic Programming Equation. A similar argumentshows that if an agent who has an endowmentof n;1 shares at time t n sells the derivative security with payo ( N B N ) plans to implement the strategy f j g N j=n, the minimum cash reserve she requires to generate a dominating strategy is bb n = max fs j g N j=n+1 8 < : B N(1 + R) ;(N;n) + NX j=n+1 ( j ; j;1 )S j (1 + R) ;(j;n) 9 = : (.9) Hence, the minimum eective cost at time t n is V n ( n;1 S n )= = min f j g N j=n max fs j g N j=n+1 min f Bn b + ( n ; n;1 )S n g f j g N j=n 8 < : B N(1 + R) ;(N;n) + NX j=n ( j ; j;1 )S j (1 + R) ;(j;n) 9 = (.10) for n =0... N. Proposition.1 The function V n ( S) satises the equation V n ( n;1 S n )= min n ( n ; n;1 )S n R maxfv n+1( n S n U) V n+1 ( n S n D)g (.11) 15

16 with nal condition 10 V N ( N;1 S N )=B N(S N ) + ( N(S N ) ; N;1 )S N : (.1) Proof: The minimum cash reserve in eq. (.9) can be rewritten in the form bb n = max S n+1 =fs n U S n Dg ( bbn+1 1+R R ( n+1 ; n )S n+1 ) = 1 n o 1+R max bbn+1 + ( n+1 ; n )S n+1 fu Dg : Therefore, we have min f j g N j=n+1 bb n = 1 min f j g N 1+R maxf Bn+1 b + ( n+1 ; n )S n+1 g U D j=n+1 1 = 1+R max U D min f j g N j=n+1 f b Bn+1 + ( n+1 ; n )S n+1 g = 1 1+R maxfv n+1( n S n U) V n+1 ( n S n D)g: Hence, substituting this into (.10), we conclude that V n ( n;1 S n )= = min n min f ( n ; n;1 )S n + Bn b g f j g N j=n ( n ; n;1 )S n R maxfv n+1( n S n D) V n+1 ( n S n D)g : (.13) We claim that the opposite inequality also holds. In fact, recall that V n+1 ( n S n+1 ) represents the minimum eective cost at time t n+1 given that the endowment is n the spot price is S n+1. Because of this, the maximum of the two amounts 10 The algorithms for solving the BLPS equation can have complexity O(e N ) with > 0 for some contingent claims. This is due to increasing complexity with the time-to-maturity of the quantity maxfv n+1 ( n S n U) V n+1 ( n S n D)g as a function of n. This function must be stored in memory to deduce V n from V n+1. 16

17 ( n ; n;1 )S n R V n+1( n S n U) ( n ; n;1 )S n R V n+1( n S n D) is sucient to cover the value of a riskless hedge at time t n S n+1 = S n U or S n D. Consequently, wehave regardless of whether V n ( n;1 S n ) 5 min n ( n ; n;1 )S n R maxfv n+1( n S n U) V n+1 ( n S n D)g : (.14) Combining (.13) (.14), we obtain the nonlinear dynamic programming equation (.10). The nal condition in (.11) follows from the denition of eective cost. Q.E.D. Remark.. The argument generalizes to derivative securites which deliver a sequence of payos at dierent dates, contingent on the value of the stock ateach date. 11 Any such derivative security can be specied by its sequence of payos ( n B n), where n = n(s n ) B n = B n(s n ): (.15) The eective cost function the optimal hedging strategy for this security can be found by solving the modied dynamic programming equation V n ( n;1 S n )= min n B n + ( n + n ; n;1 )S n R maxfv n+1( n S n U) V n+1 ( n S n D)g (.16) with nal condition (.1). Also, the same equation applies to derivative securities in which the nal date t N is a rom stopping time, like barrier options, since the proof of Proposition.1 does not use the fact that N is deterministic. 11 Examples of such\contingent claims" include portfolios of European options with dierent maturities equity-linked debt instruments with coupon payments contingent on the price of a stock or a stock index. 17

18 Remark.3In the formalism presented here, forward contracts bonds correspond to contingent claims such that N B N are constant. It is easy to verify that for these claims with no optionality the unique optimal strategy is then n = N for all n. The minimum eective costattimet n is given by V n ( n;1 S n ) = ( N ; n;1 ) S n + = N S n + 1 (1 + R) N;n B N 1 (1 + R) N;n B N + k j N ; n;1 j S n ; n;1 S n : (.17) Here, the term in brackets corresponds to the well-known cost-of-carry formula (without transaction costs). The other two terms represent the transaction cost of the initial share purchase/sale minus the value of the agent's endowment. More generally, assume that the nal payo has the form N = N(S N )+ (0) BN = B N(S N )+B (0) where (0) B (0) are constants. It is then easy to verify that if [f n gn n=0 B 0] is an optimal strategy for hedging the nal payo ( N (S N ) B N (S N )) given initial endowment ;1, then [f n + (0) g N n=0 B 0 + B (0) (1 + R) ;N ]isanoptimal strategy for hedging the payo ( N B N )given an initial endowment ;1 + (0).. A stability property of the BLPS algorithm. Replication of the nal payo can lead to mathematical instabilities if transaction costs are large (Whalley Wilmott (1993), Boyle Vorst (199), Avellaneda Paras (1994) ). Even though there are nitely many trading periods, replication may be more expensive than an (already expensive) buy--hold strategy. The BLPS algorithm does not have this pathology. Proposition.4: (1) N B(1) N satisfy Suppose that two European-style derivative securities with payos are such that their respective eective costs at expiration () N B() N V (1) N ( S N) 5 V () N ( S N) (.18) for all all S N. Then V (1) n ( S n) 5 V () n ( S n) (.19) for all 0 5 n 5 N S n. 18

19 Proof: Using the inequality (.18) equation (.11), we nd that for n = N ; 1, we have V (1) N;1 ( S N;1) 5 V () N;1 ( S N;1) : This argument can be applied recursively to obtain (.19) for all n. Q.E.D. 19

20 3. Replication vs. Super-replication: \Fine Structure " of the BLPS Algorithm In this section, we show that the risk-feasibility of the underlying asset is sucient to guarantee that replicating strategies are optimal. The dynamic programming equation (.11) (or (.16)) then reduces to a backward-induction relation for the pairs ( n B n ) (see Boyle Vorst (199)). Another condition under which replication is optimal is the k-convexity of the nal payo. This assumption can be seen as a discrete version of convexity of the value the nal payo. Prices state-variables corresponding to dierent nodes of the binomial tree are represented using double-index notation e.g. S j n represents the price at time t n at the node (n j). 3.1 Risk-Feasibility. Proposition 3.1. If the underlying asset is risk-feasible, i.e., if 1+k= 1 ; k= D < 1+R < U (3.1) 1 ; k= 1+k= then (i) the optimal sequences of stock holdings f n g of cash balances fb n g are independent of the initial endowment ;1 (ii) the optimal hedging strategy is replicating (iii) given the pairs ( j n+1 Bj n+1 solution of the equation ) (j+1 n+1 Bj+1 n+1 ), the hedge ratio j n is the unique ( j+1 n+1 ; j n )Sj+1 n+1 + Bj+1 n+1 = (j n+1 ; j n )Sj n+1 + Bj n+1 (3.) the money-market account balance B j n is given by B j n = 1 1+R ( (j+1 n+1 ; j n)s j+1 n+1 + Bj+1 n+1 ) = 1 1+R ( (j n+1 ; j n )Sj n+1 + Bj n+1 ) : (3.3) 0

21 Proof. Assume rst that the starting date is t N;1, that the starting state is j ( i:e: S = S j N;1 ) that the initial endowment is N;. We abbreviate the notation, setting S N;1 = S j N;1 S U N = S N;1 U = S j+1 N S D N = S N;1 D = S j N ( U N BU N)=( j+1 N Bj+1 N ) ( D N BD N )=( j N Bj N ) : According to Equation (.11), the optimal hedge ratio N;1 minimization problem is found by solving the min max f U ( N;1 ) D ( N;1 )g (3.4) N;1 where the functions U D are dened by U () ( ; N; )S N; R f (U N ; )S N;1U + B U N g (3.5) D () = ( ; N; )S N; R f (D N ; )S N;1D + BN D g: (3.6) We claim that U () D () are, respectively, strictly decreasing strictly increasing functions of. To see this, we dierentiate formally Equations (3.5) (3.6). Accordingly, 1

22 d U () 1 d = k S N;1 ; 1 1 k S N;1 U 1+R 5 = 1+ k S N;1 ; 1 1 ; k S N;1 U 1+R 1+ k S N;1 1 ; (1 ; k=)u (1 + k=)(1 + R) < 0 where risk-feasibility was used to obtain the last inequality. A similar calculation shows that d D () 1 d = k S N;1 ; 1 1 k S N;1 D 1+R = = 1 ; k S N;1 ; 1 1+ k S N;1 D 1+R 1 ; k S N;1 1 ; (1 + k=)d (1 + R)(1 ; k=) > 0 : Since U is decreasing D increasing, we have U () for < N;1 maxf U () D ()g = D () for > N;1 where N;1 is the point where the two graphs intersect. Clearly, N;1 is also the value of N;1 at which the minimum of maxf U () D ()g is achieved. Hence, the optimal hedge-ratio satises U ( N;1) = D ( n;1)

23 which is equivalent to ( U N ; N;1)S N;1 U + B U N = ( D N ; N;1)S N;1 D + B D N : (3.7) Notice that Equation (3.7) does not involve the initial endowment N;. In particular, N;1 is completely determined by the nal cash-ows (U N BU N )(D N BD N ) for the two connecting nodes at time N. Furthermore, according to Equation (.9), we have B N;1 = 1 1+R f (U N ; N;1)S N;1 U + B U Ng = 1 1+R f (D N ; N;1)S N;1 D + B D Ng: (3.8) Substituting the right-h sides of (3.8) into the self-nancing equation (.1), we see that the strategy ( N;1 B N;1 ) replicates the nal payo. Equations (3.7) (3.8) show that N;1 B N;1 are both functions of the spot price S N;1, independent of the initial endowment N;. The procedure outlined here for time t N;1 can be repeated at earlier times t N;, t N;3, etc. Thus, Proposition follows by induction on n. 3. k-convexity. Another condition which ensures that ( n B n) is path-independent replicating is k-convexity. Denition 3.. The payo ( N (S N ) B N (S N )) is said to be k-convex if the following conditions are satised: (i) (ii) j N 5 j+1 N for 0 5 j 5 N ; 1 (3.9) 1 ; k S j N (j+1 N ; j n ) 5 Bj N ; Bj+1 5 N 1+ k S N ( j+1 N ; j ) N : (3.10) The denition is motivated by the following important special case. Proposition 3.3. Let F (S) be a convex function set N = F 0 (S N ) B N = F (S N ) ; S N F 0 (S N ): (3.11) Then ( N B N ) is k-convex for any k = 0. 3

24 Proof. According to (3.11) j+1 SZN N ; j = N F 0 (S j+1 ) N ; F 0 (S j )= N F 00 (S)dS j+1 S j N j S Z N B j N ; Bj+1 = N S j+1 N (F (S) ; SF 0 (S)) 0 ds = ; In particular, since F 00 (S) = 0, (3.9) holds. Moreover, S Z j N S j+1 N SF 00 (S)ds = S j+1 ZN S j N SF 00 (S)ds: B j N ; Bj+1 N 5 Sj+1 N S j+1 ZN F 00 (S)ds = S j+1 5 N S j N 1+ k (j+1 N ; j N ) S j+1 N (j+1 N ; j N ) for all k = 0. Similarly, B j N ; Bj+1 N = Sj N S j+1 ZN F 00 (S)ds S j N = S j N (j+1 N ; j N ) = 1 ; k S j N (j+1 N ; j N ) for all k = 0. Q.E.D. Example 3.4 Stock options with settlement in shares have k-convex payos. In fact, the payo for a call option is 1 S = K (S) = 0 S<K ;K S = K B(S) = 0 S<K: Thus, the equations in (3.11) are satised with 4

25 F (S) =(S) S + B(S) = Max(S ; K 0) which isconvex in S. Cash-settled options do not have k-convex payos. Clearly, since j = 0 for all j, condition (3.10) would require N B N to be constant, which is not the case for cash-settled options. 1 Proposition 3.5. Suppose that ( N B N ) is k-convex. Then (i) the optimal sequences f ng fb ng are path-independent, independent ofthe initial endowment ;1, constitute a replicating strategy (ii) ( n B n) is k-convex for all n (iii) j n+1 5 j n 5 j+1 n+1 (iv) the portfolios ( j n Bj n ) satisfy the backward-induction equations j n (1 + k=) Sj+1 = n+1 j+1 n+1 ; (1 ; k=) Sj n+1 j n+1 (1 + k=)s j+1 n+1 ; (1 ; k=)sj n+1 B j n = 1 1+ k ( j+1 n+1 1+R ; j n )S j+1 n+1 j+1 + B n+1 + B j+1 n+1 ; Bj n+1 (3.1) = 1 1 ; k ( j n+1 1+R ; j n )S j n+1 + Bj n+1 : (3.13) Proof. Assume that at time t N;1 the initial endowment is N;. The optimal hedge ratio N;1 is the solution to the minimization problem min max U D f U () D ()g with U D as in (3.5), (3.6). Since the transcation cost function is convex, U D are also convex so is () = max f U () D ()g : Therefore, a necessary sucient condition for the minimum of () to be achieved at = is 1 The argument shows that the only cash-settled contingent claims with k-convex payos are bonds. The contingent claim of Example 1. can be viewed as a one-period cash-settled option. 5

26 d() d() lim 5 0 lim = 0: (3.14) " d # d We claim that this minimum is unique, that it is given by the unique solution of the problem 8 >< >: U ( N;1 )= D( N;1 ) D N 5 N;1 5 U N : (3.15) To show this, we shall prove rst prove that if the payo is k-convex then (3.15) has a unique solution. Notice that (3.15) is equivalent to 8 >< >: ; 1+ k ( U N ; N;1 )S N;1U + B U N = ; 1 ; k ( D N ; N;1 )S N;1D + B D N D N < N;1 < U N : Elementary linear algebra shows that a solution to (3.16) exists if only if 1 ; k ( UN ; DN ) S N;1 D 5 B DN ; BUN 5 1+ k ( U N ; D N ) S N;1 U: (3.16) But this is precisely condition (3.10) of the denition of k-convexity, so the claim follows. It remains to show that the solution to (3.16) achieves the minimum of. For this purpose, we verify that the rst-order optimality condition (3.15) holds. We have d U ( N;1 ) d N;1 = 5 = 1 k S N;1 ; 1 1+ k S N;1 U 1+R 1+ k S N;1 ; 1 1+ k S N;1 U 1+R 1+ k S N;1 1 ; U 1+R < 0 (3.17) 6

27 d D ( N;1 ) d N;1 = = = 1 k S N;1 ; 1 1 ; k S N;1 D 1+R 1 ; k S N;1 ; 1 1 ; k S N;1 D 1+R 1 ; k S N;1 1 ; D 1+R > 0 (3.18) where we used the no-arbitrage condition D<1 + R<U. Using (3.17)-(3.18) the fact that ( N;1 )= U( N;1 )= D( N;1 ), we conclude that d() lim " N;1 d = d U( N;1 ) d N;1 d() lim # N;1 d = d D( N;1 ) d N;1 < 0 > 0 as desired. We have shown that N;1 is the optimal hedge-ratio. Since the problem (3:16) does not involve the endowment N;, the optimal hedge-ratio is independent of the agent's holdings. The values of N;1 B N;1 can be computed from (3.16). They are N;1 = (1 + k=)u N S N;1U ; (1 ; k=) D N S N;1D + B U N ; BD N (1 + k=)s N;1 U ; (1 ; k=)s N;1 D (3.19) B N;1 = 1 1+ k ( U N 1+R ; N;1 )S N;1U + B U N = 1 1 ; k ( D N ; 1+R N;1)S N;1 D + BN D 7 : (3.0)

28 We nowshow that the optimal portfolio at time t N;1,( j N;1 B j N;1 )N;1 j=0,isak-convex payo. For this, we consider the diagram. UU N U N;1. N; (S N; =S) DU N. D N;1 DD N The inequality in (3.16) implies that therefore UD N 5 U N;1 5 UU N DD N 5 D N;1 5 DU N (3.1) D N;1 5 U N;1 : (3.) Using these inequalities, the formulas for Bn D<(1 + R) <U,we nd that in (3.0) the no-arbitrage condition BN;1 D ; BN;1 U = 1 1+R SUD UD N + 1 ; k U N;1 ; 1+ k D N;1 = 1 1+R SUD k UD N ; k U N; k ( U N;1 ; D N;1) R < 1+ k SUD( U N;1 ; D N;1 ) 1+ k SU( U N;1 ; D N;1) (3.3) 8

29 B D N;1 ; BU N;1 = 1 1+R SUDfkUD N ; kd N;1 +(1; k=)(u N;1 ; D N;1 )g = 1 1+R (1 ; k=)sud(u N;1 ; D N;1) > (1 ; k=)sd( U N;1 ; D N;1 ): (3.4) Inequalities (3.), (3.3) (3.4) show that ( N;1 B N;1 )isak-convex payo. We have therefore proved Proposition 3.5 for n = N ; 1. The general proof now follows by induction on n. 3.3 The backward-induction equations. To better describe the minimum eective cost the optimal hedging strategy, we introduce the state-variables 13 P n n S n + B n Q n n S n : These variables can be interpreted, respectively, as the value of the portfolio the value of its equity portion at time t n. 14 In the case of risk-feasible underlying assets, explicit backward-induction equations for the pairs ( P n Q n ) can be derived by solving equation (3.7). We shall avoid the use of superscripts by using the notation 8 >< >: P U n+1 = U n+1 S nu + B U n+1 Q U n+1 =U n+1 S nu 8 >< >: P D n+1 = D n+1 S nd + B D n+1 Q D n+1 =D n+1 S nd : (3.5) 13 In the rest of this section, we drop the superscript use f n g fb n g to represent the optimal portfolio. 14 In this interpretation, the stock inventory is valued at the mid-price between the bid the oer. 9

30 Proposition 3.6 If the underlying asset is risk-feasible, the pairs ( P n Q n ) satisfy 8 >< ; P n = 1 1+R U Pn+1 U + D Pn+1 D + (;1) a k (1+R) ; U Q U n+1 +(;1) b D Q D n+1 >: Q n = 1 ; P U n+1 ; P D n+1 + (;1) a k ; Q U n+1 +(;1) c Q D n+1 (3.6) where D U a b c, dened hereafter, are functions of ( P n+1 Q n+1 ) (or, equivalently, of ( n+1 B n+1 )). Specically, we have Case 1: 1 ; D n+1 U n+1 k S n D( U n+1 ; D n+1) 5 Bn+1 D ; Bn+1 U 5 1+ k S n U( U n+1 ; D n+1) : (3.7) In this case, = 1+ k U ; 1 ; k D U = D = (1 + R) ; (1 ; k=)d (1 + k=)u ; (1 + R) a =+1 b= ;1 c =+1: (3.8) Case : U n+1 < D n+1 1+ k S n D( D n+1 ; U n+1) 5 Bn+1 U ; Bn+1 D 5 1 ; k S n U( D n+1 ; U n+1): (3.9) 30

31 In this case, = 1 ; k U ; 1+ k D U = (1 + R) ; (1 + k=)d D = (1 ; k=)u ; (1 + R) a = ;1 b= ;1 c=+1: (3.30) Case 3: Either D n+1 U n+1 1 ; k S n D( U n+1 ; D n+1 ) >BD n+1 ; BU n+1 or U n+1 < D n+1 Bn+1 U ; BD n+1 > 1 ; k S n U( D n+1 ; U n+1 ): (3.31) = 1 ; k (U ; D) U = (1 + R) ; (1 ; k=)d D = (1 ; k=)u ; (1 + R) a = ;1 b=+1c = ;1 : (3.3) Case 4: Either D n+1 U n+1 31

32 Bn+1 D ; Bn+1 U > 1+ k S n U( U n+1 ; D n+1) or U n+1 < D n+1 1+ k S n D( D n+1 ; U n+1 ) >BU n+1 ; BD n+1 : (3.33) In this case, = 1+ k (U ; D) U = D = (1 + R) ; (1 + k=)d (1 + k=)u ; (1 + R) a =+1 b=+1c = ;1 : (3.34) Proof: The proof follows by solving for N;1 arbitrary values of n. in Equation (3.7) generalizing to Proposition 3.7. If the nal payo is k-convex then the pairs (P n Q n ) satisfy the linear backward-induction relation 8 >< >: ; P n = 1 1+R U Pn+1 U + D Pn+1 D + k (1+R) Q n = 1 ; P U n+1 ; P D n+1 + k ; Q U n+1 + Q D n+1 ; U Q U n+1 ; D Q D n+1 (3.35) with = 1+ k U ; 1 ; k D 3

33 U = (1 + R) ; (1 ; k=)d D = (1 + k=)u ; (1 + R) : Proof: The proof is immediate, since the portfolios ( n B n ) satisfy the conditions in (3.7) for all n. Remark 3.8.The eective cost of hedging a short position in a European option settled in shares can be calculated from (3.35) for arbitrary values of k. On the other h, a long option position corresponds to a linear backward-induction equation dened by Case of Proposition 3.6 provided that the underlying asset is risk-feasible. In both cases the corresponding values coincide with those obtained by Boyle Vorst (199). Remark 3.9. In general, D U satisfy D + U = 1. For k > 0, D U are positive (in all four cases) only if the underlying asset is risk-feasible. If this condition holds, D U can be interpreted as \risk-neutral" probabilities for no-arbitrage pricing with transaction costs. Otherwise, since at least one of the inequalities in (3.1) does not hold, we have D U < 0 in Case (cf. (3.30)). Thus, the scheme (3.6) becomes numerically unstable replicating strategies may result in negative or innite prices as N %1. If k =0,wehave U = 1+R ; D U ; D D = U ; 1 ; R U ; D the backward-induction equations reduces to the classical result of Cox, Ross Rubinstein(198) 8 >< >: P n = 1 1+R (U P U n+1 + D P D n+1) Q n = P U n+1 ;P D n+1 U;D : (3.36) 33

34 4. k-convex Payoffs or Risk-Feasible Underlying Assets: Asymptotics for N % +1 This section studies the backward-induction equations for N % +1. We recall rst the stard Cox-Ross-Rubinstein scaling of the binomial tree, in which the parameters U D R, N are related to the annualized volatility, the interest rate, the maturity of the contingent claim. We then show that the optimal dynamic portfolio can be approximated by the solution of the partial dierential equation (1.)-(1.3) its derivative with respect to the spot price. 4.1 Adjusting the model parameters. We denote by T the time-to-maturity of a European-style derivative security by dt = T N the time-interval between successive adjustments of the hedging portfolio. Following the classical Cox, Ross, Rubinstein (198) scaling for the binomial model, we set 8 >< >: U = e p dt + dt D = e ;p dt + dt (4.1) R = e rdt ; 1 = rdt dt 1: (4.) Here, r are the annualized volatility interest rate, respectively. The parameter represents the subjective drift (not adjusted for risk). 15 The Lel number A k p dt : (4.3) represents the \ percentage transaction costs per stard deviation for a single period". Lemma 4.1. For dt suciently small, the underlying asset is risk-feasible if only if A<1. 15 As in the classical CRR theory, the parameter will not appear in the asymptotic equations. 34

35 Proof. It follows from (3.1) that the underlying asset is risk-feasible if only if k U ; (1 + R) < min U +1+R 1+R ; D : (4.4) 1+R + D Using the scaling relations (4.1), we nd that, for dt 1, p U ; (1 + R) p U +(1+R) = dt + O(dt) pdt dt +O p 1+R ; D 1+R + D = dt + O(dt) pdt p dt +O : Thus, for dt suciently small, risk-feasibility isequivalent tok= < p dt =, or just A<1. Q.E.D. 4.. A Comparison Principle. Denition 4.. A hedging strategy the payo ( N B N ) if: (i) h i fe j g N j=n Bn e + (ii) the eective cost of hfe j g Nj=n e Bn i is a dominating strategy for ( N B N ) hfe j g Nj=n e Bn i of ( N B N ) by at most dollars, i.e., is said to be -optimal at time t n for diers from the optimal eective cost V ( S n ) V ( S n ) ; ebn + (e n ; )S n 5 : Let us represent formally the non-linear backward-induction equations (3.6) as Pn Q n = Pn+1 Q n+1 n =0... N ; 1 : (4.5) 35

36 Proposition 4.3. Consider a nal payo ( N B N ). Assume that either this payo is k-convex or that the underlying asset is risk-feasible. Suppose that there exist sequences ( e Pn e Qn ) (p n q n ) having the following properties: (i) epn epn+1 pn = + n =0... N ; 1 (4.6) eq n eq n+1 q n (ii) ep N = P N e QN = Q N (4.7) (iii) if the underlying asset is not risk-feasible, then for every n the intermediate \portfolios" Dene are k-convex for all n. n en eb n 1 ; (1 + R) ;(N;n) (1 + R) ;1 R Then, for each n 0 5 n 5 N ; 1 the strategy eqn =S n ep n ; Qn e max jp n 0 >n S n0(s)j + k jqn0(s)j hfe j g Nj=n e Bn i is n ; optimal. : (4.8) Proof. Let us introduce an auxiliary contingent claim with nal payo ( N B N ) with intermediate payos (\coupons") dn qn =S n = (4.9) b n p n ; q n for n =0 ::: N ; 1. The payos of this auxiliary security are represented in Diagram 4.1. Let e Vn ( S n ) represent its optimal eective cost given an endowment at time t n. >From the assumptions (i) through (iii), we conclude that (e n e Bn ) is the optimal riskaverse strategy for hedging this security (see Remark.), that ( e Pn e Qn ) represents the optimal portfolio in the variables (P Q). The minimum eective cost e Vn ( S n ) can be estimated by stripping the intermediate coupons from the nal payo pricing them separately. For n 0 = n, the coupon ( p 0 n q0 n ) can be purchased at p n 0 + k jq n 0j 5 max jp n" (S)j + k n">n S jq n"(s)j " n (4.10) dollars. Its value at time t n is therefore at most " n (1 + R) ;(n0 ;n). Since the eective cost of the auxiliary claim at time t n is at most equal to the eective cost of the original 36

37 . ( 3 3 B3 3 ) (d b. ) (d 0 b0 0 ). (d 1 b1 1. ). (d 0 1 b0 1 ) (d 1 b1 ) ( 3 B 3 ) ( 1 3 B1 3 ). (d 0 b0 ) ( 0 3 B0 3 ) Diagram 4.1. contingent claim plus the present value of the estimated costs of the intermediate portfolios, we conclude that X N;1 ev n ( S n ) 5 V n ( S n )+" n (1 + R) ;(j;n) = V n ( S n )+ n : (4.11) j=n Similarly, alower bound for Vn e ( S n ) is obtained by considering an auxiliary derivative security that delivers, in addition to the i intermediate coupons, " dollars at each trading date. The strategy hfe j g Nj=n Bn e + n is clearly a dominating strategy for such security. But, since the intermediate payments are all positive on account of (4.10), the strategy is also dominating for the European contingent claim with one single payo ( N B N )at time t N.Thus, we have V n ( S n ) 5 e Vn ( S n )+ n : (4.1) The proof of Proposition 4.3 is complete. 37

38 4.3 Asymptotic analysis. We consider rst payos of the form 8 >< >: N (S N )=F 0 (S N ) B N (S N )=F (S n ) ; S N F 0 (S N ) (4.13) where F (S) is a four-times continuously dierentiable function satisfying kf ksup S 8 < 4X : j=0 S j d j F (S) ds j 9 = < 1 : (4.14) The boundedness regularity assumption (4.14) is mathematically convenient but not essential. In Section 4.4 we extend the results derived for the payos (4.13) to option portfolios. The main result of this section is: Theorem 4.4. Let F (S) be a function satisfying the regularity condition (4.14). Assume that A < 1 or F(S) is convex. Let P (S t) be the solution of the nal-value problem 8 >< h1+a + sign P (S T )=F (S): ; rp =0 (4.15) Set 8 >< >: ep n (S n )=P (S n t n ) eq n (S n (S n t n ) (4.16) en eqn =S n = : eb n ep n ; Qn e (4.17) Then: (i) There exists a constant C = C(r A T) independent off dt, suchthat ; epn (S n ) e Qn (S n ) satises the backward-induction equation 38