Dynamic Programming Algorithms for the Ask and Bid Prices

Dynamic Programming Algorithms for the Askand Bid Prices of American Options under Small

Proportional Transaction Costs

Krzysztof Tokarz and Tomasz ZastawniakDepartment of Mathematics, University of Hull

Cottingham Road, Kingston upon Hull, HU6 7RX, United Kingdom

[email protected], [email protected]

August 24, 2004

Abstract

Dynamic programming algorithms are developed for computing the askand bid prices of American contingent claims in a binary tree setting in thepresence of small proportional transaction costs, extending the recursiveconstruction of the Snell envelope. Associated with the pricing algorithmsare iterative procedures for computing optimal hedging strategies for thewriter as well as for the buyer of an American option. The bid and askprices of an American option are represented in terms of the expectationof the option payoff evaluated at an optimal stopping time with respectto an optimal martingale probability measure. As a by-product a simi-lar dynamic programming algorithm is obtained for pricing and hedgingEuropean contingent claims in the same setting.

Key words: American options, bid-ask spread, transaction costs, dynamicprogramming, Snell envelope.

1 Introduction

The pricing and hedging of European options and conditions for the lack of arbi-trage under proportional transactions costs have received considerable attentionin recent years. In a discrete time setting questions of this kind were stud-ied, for example, by Merton [Mer90], Dermody and Rockafellar [DR91], Boyleand Vorst [BV92], Bensaid, Lesne, Pages and Scheinkman [BLPS92], Edirs-inghe, Naik and Uppal [ENU93], Jouini and Kallal [JK95], Kusuoka [Kus95],Naik [Nai95], Shirakawa and Konno [SK95], Koehl, Pham and Touzi [KPT96],[KPT99], [KPT01], Stettner [Ste97], [Ste00], Perrakis and Lefoll [PL97],Rutkowski [Rut98], Touzi [Tou99], Jouini [Jou00], Ortu [Ort01], Palmer [Pal01a],[Pal01b], Kocinski [Koc04], and others.

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As compared to European options, rather less is known about the pricing andhedging of American type derivative securities under proportional transactioncosts. In discrete time Mercurio and Vorst [MV97] used a local risk minimisationcriterion to obtain option price bounds. Kocinski [Koc99] proved the existenceof a strictly replicating strategy for the seller of an American contingent claimin the binary tree model and gave sufficient conditions for the existence of anoptimal replicating strategy. In [Koc01] he obtained a lower bound for theseller’s price in a general discrete time model, and expressed the seller’s price ofan attainable American option as the cost of a replicating strategy under smalltransaction costs. Perrakis and Lefoll [PL00], [PL04] presented a procedurefor computing the ask and bid prices and optimal super-hedging strategies forAmerican calls on stocks paying known dividends and for American puts inthe binary tree model, subject to certain restrictions on the model parameters.Chalasani and Jha [CJ01] obtained general representations involving so-calledrandomised stopping times for the ask and bid prices of American contingentclaims.

The study of pricing and hedging of contingent claims in discrete time mod-els with transaction costs appears particularly significant because of a numberof results according to which the optimal super-hedge in continuous time is thetrivial buy-to-hold strategy, which is hardly acceptable in practice; see Soner,Shreve and Cvitanic [SSC95], Cvitanic, Pham and Touzi [CPT99], and Leventaland Skorohod [LS97] for European options, and Levental and Skorohod [LS97]and Jakubenas, Levental and Ryznar [JLR03] for American options. Otherways around this difficulty, not pursued here, include the expected utility max-imisation approach as, for example, in Hodges and Neuberger [HN89], Davis,Panas, Zariphopoulou [DPZ93], Davis, Zariphopoulou [DZ95], Constantinidesand Zariphopoulou [CZ01], and Constantinides and Perrakis [CP02], [CP04],or imperfect hedging with rebalancing the portfolio at discrete times only, asin Leyland [Lel85], Hoggard, Whalley and Wilmott [HWW94], Kabanov andSafarian [KS97], and others.

In the present paper dynamic programming algorithms are developed for thepricing and hedging of American contingent claims in a binary tree setting in thepresence of small proportional costs on transactions in the underlying security(or, equivalently, small bid-ask spreads). The pricing algorithms, which turnout somewhat different for the ask and bid option prices, can be viewed asextensions of the recursive construction of the Snell envelope in the friction freecase. Moreover, representations of the ask price (seller’s price, also known as theupper hedging price) and the bid price (buyer’s price, or lower hedging price)of an American contingent claim Y as, respectively,

πa(Y ) = maxτ

maxP

E(Yτ ), πb(y) = maxτ

minP

E(Yτ ) (1.1)

over stopping times τ and (suitably defined as in Jouini and Kallal [JK95]) mar-tingale probabilities P, with E being the expectation under P, are established.

As a by-product we also construct a dynamic programming type algorithmfor pricing and hedging European options in the same setting, and reprove, by

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a new method based on the algorithm, the known representations

πa(Y ) = maxP

E(X), πb(y) = minP

E(X)

for the ask and bid prices of a European contingent claim X in our setting.The representations (1.1) of the ask and bid prices, while resembling those

in Chalasani and Jha [CJ01], differ significantly in that standard stopping timesrather than randomised ones are used. This is contrary to a suggestion in [CJ01]that randomised stopping times are required to represent American option pricesunder proportional transaction costs.

Representations (1.1) are also very similar to those for the ask and bidAmerican option prices in incomplete models (with no transactions costs or anyother kind of friction) to be found in Harrison and Kreps [HK79] in continuoustime and in Follmer and Schied [FS02] in discrete time. However, in contrastto a friction free but possibly incomplete market, where

maxτ

minP

E(Yτ ) = minP

maxτ

E(Yτ ) (1.2)

(see Theorem 6.41 in [FS02]), if transaction costs are present, then the minimaxproperty (1.2) may be violated. The bid price of an American option is thengiven by the left-hand side of (1.2), but not necessarily by the right-hand side.

If there are no transaction costs, then our algorithms for computing theask and bid prices of an American option with payoff process Y reduce to thestandard construction of the Snell envelope Z of Y by backward recursion. Inthe presence of transaction costs Algorithm 4.1 for the ask price still resemblesthe Snell envelope construction, but an interesting novel feature appears: Inthe presence of transaction costs it becomes necessary to keep track of twoquantities Za

t and Zbt at each node of the tree at any time t prior to option

expiry, instead of the single quantity Zt = maxYt, Et(Zt+1) defining the Snellenvelope. Algorithm 4.2 for the bid price is slightly more complicated and theanalogy with the Snell envelope a little harder to see. Once again, there aretwo quantities Ua

t and U bt to keep track of, which can be seen as analogues of

the value of continuation Et(Zt+1) in the standard Snell envelope construction.(Here Et denotes the conditional risk neutral expectation at time t.)

The paper is organised as follows: In Section 2 we specify the model withproportional transaction costs, discuss the no-arbitrage conditions, recall somebasic definitions, notation, and facts, and state the small transaction costs as-sumption. Algorithm 3.1 for European options is considered in Section 3, notjust as a simple special case, but also because the results are needed later inLemma 4.7. Section 4 contains the main results of the paper, namely Algo-rithms 4.1 and 4.2 for computing the ask and bid prices of an American contin-gent claim, along with Theorems 4.4 and 4.8, which establish the correctness ofthe algorithms and provide various representations for the ask and bid prices.Finally, following some concluding remarks in Section 5, we provide a couple ofauxiliary technical propositions in the Appendix.

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2 Small Proportional Transaction Costs

2.1 Model Specifications

We adopt the binary tree model with trading times t = 0, . . . , T for some fixedpositive integer T . The corresponding probability space Ω consists of sequencesω1ω2 · · ·ωT with ω1, . . . , ωT ∈ u, d, where u and d stand for up anddown. We take F to be the σ-field consisting of all subsets of Ω, and Q to be aprobability measure on F such that Q ω > 0 for each ω ∈ Ω.

A node ωt = ω1ω2 · · ·ωt of the tree at time t = 0, . . . , T , with ω1, . . . , ωt ∈u, d, will be identified with the event η ∈ Ω | η1 = ω1, . . . , ηt = ωt. Inparticular, ω0 will be identified with Ω. The family of all nodes ωt at time twill be denoted by Ωt. We take a filtration ∅, Ω = F0 ⊂ F1 ⊂ · · · ⊂ FT = F ,where Ft is the σ-field generated by the family Ωt for each t = 0, . . . , T . Weshall often identify Ft-measurable random variables on Ω with random variableson Ωt.

The market model will consist of a risky and a risk-free security, a stock anda bond. Trading in stock is subject to proportional transaction costs. A sharecan be bought for the ask price Sa

t or sold for the bid price Sbt at any time

t = 0, . . . , T , both price processes Sa and Sb being adapted to the filtration.For any t = 0, . . . , T − 1 and any node ωt ∈ Ωt the corresponding single-stepsubtree of stock prices can be depicted as

Sat+1(ωtu)

Sbt+1(ωtu)

Sat (ωt)

Sbt (ωt)

Sat+1(ωtd)

Sbt+1(ωtd)

Throughout this paper we shall work under the following assumption of smalltransaction costs, which simply means that the bid-ask spread intervals at eachnode do not overlap in any single-step tree fragment as above.

Assumption (small transaction costs) For each t = 0, . . . , T − 1 and eachωt ∈ Ωt

Sbt+1(ωtd) ≤ Sa

t+1(ωtd) < Sbt (ωt) ≤ Sa

t (ωt) < Sbt+1(ωtu) ≤ Sa

t+1(ωtu). (2.3)

Without loss of generality we shall assume the bond to be a risk-free securitywith zero interest rate, the bond price being 1 for all t = 0, . . . , T . Equivalently,all prices can be regarded as discounted prices.

Jouini and Kallal [JK95] have studied general conditions for the lack ofarbitrage in a model with proportional transaction costs. It follows directly fromtheir result, Theorem 2.1 below, that the small transaction costs assumptionadmits no arbitrage opportunities in the market model.

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2.2 Self-Financing Strategies, Arbitrage Opportunities,Martingale Measures

By a self-financing strategy we shall understand a pair (α, β) of predictableprocesses αt, βt for t = 0, . . . , T describing the positions in cask and stock suchthat β0 = 0 and

αt+1 − αt ≤ − (βt+1 − βt)+

Sat + (βt+1 − βt)

−Sb

t (2.4)

for each t = 0, . . . , T − 1. The set of all self-financing strategies will be denotedby Φ(Sa, Sb).

The time t = 0, . . . , T liquidation value of a strategy (α, β) ∈ Φ(Sa, Sb) willbe defined as

ϑt(α, β) = αt + β+t Sbt − β−t Sa

t .

We can also refer to ϑT (α, β) as the terminal value of the strategy.By an arbitrage opportunity we understand a strategy (α, β) ∈ Φ(Sa, Sb)

with α0 ≤ 0 and terminal value ϑT (α, β) ≥ 0 such that Q ϑT (α, β) > 0 > 0.Following Jouini and Kallal [JK95], we say that a probability measure P

equivalent to Q is a martingale measure if there is a martingale S under P suchthat Sb

t ≤ St ≤ Sat for each t = 0, . . . , T . The set of such martingale measures P

will be denoted by P.The following result, obtained by Jouini and Kallal [JK95], who used a

slightly different notion of arbitrage, referred to as ‘free lunch’ in their work, isalso valid under the above definition of an arbitrage opportunity in the presentsetting, see Tokarz [Tok04].

Theorem 2.1 (Jouini and Kallal [JK95]) There is no arbitrage opportunityif and only if P is non-empty.

In particular, it follows immediately that the small transaction costs assump-tion (2.3) admits no arbitrage opportunity.

We conclude this section with a simple property of self-financing strategies,which will prove useful later on. By T we denote the set of all stopping times τsuch that 0 ≤ τ ≤ T .

Lemma 2.2 If (α, β) ∈ Φ(Sa, Sb), P ∈ P, S is a martingale under P such thatSb ≤ S ≤ Sa, and E denotes the expectation under P, then

(a) α + βS is a supermartingale under P;

(b) E(ϑτ (α, β)) ≤ α0 for each stopping time τ ∈ T .

Proof (a) The self-financing condition (2.4) together with Sbt ≤ St ≤ Sa

t give

αt+1 − αt ≤ − (βt+1 − βt)+

Sat + (βt+1 − βt)

−Sb

t ≤ − (βt+1 − βt) St.

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As a result, for each t = 0, . . . , T − 1

E(αt+1 + βt+1St+1|Ft) = αt+1 + βt+1St ≤ αt + βtSt.

(b) Since Sbτ ≤ Sτ ≤ Sa

τ ,

ϑτ (α, β) = ατ + β+τ Sbτ − β−τ Sa

τ ≤ ατ + βτSτ .

Because α + βS is a supermartingale under P and β0 = 0 it follows that

E(ϑτ (α, β)) ≤ E(ατ + βτSτ ) ≤ α0,

as required.

3 European Options

We begin with the special case of European contingent claims, for which repre-sentations of the ask and bid prices in terms of optimal super- and sub-hedgingstrategies as well as in terms of expectations of the payoff under optimal mar-tingale probability measures (but not in terms of a dynamic programming algo-rithm) have already been studied by Jouini and Kallal [JK95]. The constructionsand results will then be extended to obtain new representations and algorithmsfor American option ask and bid prices in the presence of small proportionaltransaction costs. The results of the present section will also be needed later inthe proof of Lemma 4.7.

A European contingent claim can be characterised by a random variable X,the payoff at expiry time T . The time 0 ask and bid prices of such an optioncan be defined, respectively, as

πa(X) = minα0 | (α, β) ∈ Φ(Sa, Sb), ϑT (α, β) ≥ X, (3.5)

πb(X) = max−α0 | (α, β) ∈ Φ(Sa, Sb), −ϑT (α, β) ≤ X. (3.6)

The minimum and maximum are attained because the corresponding sets areclosed and, respectively, bounded below and above in the discrete setting. Thus,the ask price πa(X) is the lowest price that the option writer should demand tobe able to hedge the position without any risk of loss. The bid price πb(X) isthe highest amount that a buyer can raise to pay for the option such that hisor her position can be hedged without running any risk of loss. Clearly,

πb(X) = −πa(−X). (3.7)

Jouini and Kallal [JK95] obtained the following representation for the bid-ask spread of European option prices:

[πb(X), πa(X)] = E(X) |P ∈ P,

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where E is the expectation under P and where A denotes the closure of a setA ⊂ R. In particular, it follows that

πa(X) = maxP∈P

E(X), (3.8)

πb(X) = minP∈P

E(X). (3.9)

Here the maximum and minimum are attained because P is a compact set inthe discrete setting under the small transaction costs assumption (2.3), andP 7→ E(X) is a continuous function.

3.1 Algorithm

Neither the representations (3.5), (3.6) nor (3.8), (3.9) are particularly effectivefor computing the ask or bid prices of a European option. They involve the solu-tion of complex multidimensional optimisation problems. (Note that (3.8), (3.9)are not even linear optimisation problems because P is not convex, in general.)Here we put forward a dynamic programming type recursive algorithm to com-pute these prices quickly and efficiently. This will then be extended to the caseof American options.

The following assertion, which will be used in the proof of the correctness ofthe algorithm, follows directly from the results in Jouini and Kallal [JK95]. Westate and prove it here nevertheless so as to keep this paper self-contained.

Lemma 3.1 For any European option X

πb(X) ≤ minP∈P

E(X) ≤ maxP∈P

E(X) ≤ πa(X).

Proof The middle inequality is obvious.Take a strategy (α, β) ∈ Φ(Sa, Sb) such that πa(X) = α0 and X ≤ ϑT (α, β).

By Lemma 2.2, for each P ∈ P

E(X) ≤ E(ϑT (α, β)) ≤ α0 = πa(X),

which proves the last inequality.Now take (α, β) ∈ Φ(Sa, Sb) such that πb(X) = −α0 and −ϑT (α, β) ≤ X.

Thus, again by Lemma 2.2, for each P ∈ P

E(X) ≥ −E(ϑT (α, β)) ≥ −α0 = πb(X).

This proves the first inequality.

Next, let us introduce some notation, which will also be used in subsequentsections on American options. For any u, v, w ∈ a, b, any t = 0, . . . , T − 1 andωt ∈ Ωt, and any Ft+1-measurable R2-valued random variables G = (Ga, Gb)and H = (Ha,Hb) we put

Euvwt (G;H|ωt) = puvw

t (ωt)Gv(ωtu) + (1− puvw

t (ωt))Hw(ωtd),

puvwt (ωt) =

Sut (ωt)− Sw

t+1(ωtd)

Svt+1(ωtu)− Sw

t+1(ωtd).

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We shall write Euvwt (G;H) to denote the Ft-measurable random variable ωt 7→

Euvwt (G;H|ωt).

This notation is slightly more complicated than necessary in the case ofEuropean options, for which we shall always take G = H. However it willbecome necessary to allow G 6= H when discussing the bid price of an Americanoption later on.

Algorithm 3.1 (ask price of European option) Given a European optionwith payoff X and expiry time T , an R2-valued process Z = (Za, Zb) is con-structed by backward induction as follows:

1. PutZa

T = ZbT = X;

2. For each t = 1, . . . , T and each u ∈ a, b take

Zut−1 = max

v,w∈a,bEuvw

t−1 (Zt;Zt). (3.10)

Claim 3.1 The ask price of the European option is given by

πa(X) = maxZa0 , Z

b0.

The claim will be verified in what follows, see Theorem 3.4.

Remark 3.1 (bid price of European option) Due to (3.7) Algorithm 3.1can also be used to compute the bid price πb(X) of a European option.

The proof of correctness of Algorithm 3.1 will involve a number objects thatneed to be constructed. We begin by constructing processes S, Z such that:

1. For some u ∈ a, bS0 = Su

0 ,

Z0 = Zu0 ,

andZu0 = maxZa

0 , Zb0. (3.11)

2. For each t = 0, . . . , T − 1 and each ωt ∈ Ωt there are v, w ∈ a, b suchthat

St+1(ωtu) = Svt+1(ωtu), St+1(ωtd) = Sw

t+1(ωtd),Zt+1(ωtu) = Zv

t+1(ωtu), Zt+1(ωtd) = Zwt+1(ωtd),

andZu

t (ωt) = Euvwt (Zt+1;Zt+1|ωt) (3.12)

for each u ∈ a, b. Such v, w exist by Proposition 6.1.

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Remark 3.2 The processes S, Z may not be unique. The lack of uniquenessmay arise whenever there is more than one pair v, w ∈ a, b satisfying (3.12),or there is more then one u ∈ a, b such that (3.11) holds. In such cases onecan choose any S, Z satisfying the conditions above.

Let P be the probability measure turning S into a martingale. It is welldefined and equivalent to Q because, by the small transaction costs assump-tion (2.3), St+1(ωtu) > St(ωt) > St+1(ωtd) for each t = 0, . . . , T − 1 and eachωt ∈ Ωt. We denote by E the expectation under P.

Lemma 3.2 (a) Z is a martingale under P;

(b) E(X) = Z0.

Proof (a) Take any t = 0, . . . , T − 1 and any ωt ∈ Ωt. By the construction ofS, Z there are u, v, w ∈ a, b such that

St(ωt) = Sut (ωt), St+1(ωtu) = Sv

t+1(ωtu), St+1(ωtd) = Swt+1(ωtd),

Zt(ωt) = Zut (ωt), Zt+1(ωtu) = Zv


and (3.12) holds. Thus

Zt(ωt) = Zut (ωt) = Euvw

t (Zt+1;Zt+1|ωt) = E(Zt+1|ωt).

(b) Since ZaT = Zb

T = X, it follows that ZT = X. Because Z is a martingaleunder P we obtain E(X) = E(ZT ) = Z0.

Next, let α, β be predictable processes such that β0 = 0 and for each t =0, . . . , T

αt + βtSt = Zt. (3.13)

Since S and Z are martingales under P, it follows that for each t = 0, . . . , T − 1

αt+1 + βt+1St = αt+1 + βt+1E(St+1|Ft)

= E(αt+1 + βt+1St+1|Ft) = E(Zt+1|Ft) = Zt. (3.14)

Observe that (α, β) is the self-financing strategy hedging the European op-tion X in the standard (friction free) model with stock price process S. Namelyαt + βtSt = αt+1 + βt+1St for all t = 0, . . . , T − 1, and αT + βT ST = X. Nev-ertheless, it does not immediately follow that (α, β) is also a hedging strategyfor X in the model with transaction costs. This will be proved in Lemma 3.3.

Lemma 3.3 (a) (α, β) ∈ Φ(Sa, Sb).

(b) ϑT (α, β) = X.

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Proof (a) By (3.13) and (3.14) the self-financing condition (2.4) can be writtenas

(βt+1 − βt)+(Sa

t − St) + (βt+1 − βt)−(St − Sb

t ) ≤ 0

or, equivalently, as

(βt+1 − βt)(Srt − St) ≤ 0 ∀r ∈ a, b. (3.15)

We shall prove that (α, β) satisfies (3.15) for each t = 0, . . . , T − 1.First we shall verify (3.15) for t = 0. Consider the tree fragment

ω0u

ω0

ω0d

For brevity, we shall omit ω0 in the expressions below. By the construction ofS, Z there are u, v, w ∈ a, b such that at the respective nodes

S1(u) = Sv1 (u)

Z1(u) = Zv1 (u)

S0 = Su0

Z0 = Zu0

S1(d) = Sw1 (d)

Z1(d) = Zw1 (d)

andZu0 = Euvw

0 (Z1;Z1).

Observe that

β1 =Z1(u)− Z1(d)

S1(u)− S1(d)=

Zv1 (u)− Zw

1 (d)Sv1 (u)− Sw

1 (d).

By the construction of Z0 we know that Z0 = Zu0 ≥ Zr

0 for each r ∈ a, b, andby the construction of Zr

0 in Algorithm 3.1, Zr0 ≥ Ervw

0 (Z1;Z1). This gives

Euvw0 (Z1;Z1) ≥ Ervw

0 (Z1;Z1),

which can be transformed into

β1(Sr0 − Su

0 ) ≤ 0

for each r ∈ a, b. Because β0 = 0 this implies (3.15) for t = 0, as required.Next, we need to demonstrate that (α, β) satisfies (3.15) for each t = 1, . . . ,

T − 1. Let us take any node ωt−1 ∈ Ωt−1 and consider the tree fragment

ωt−1uu

ωt−1u

ωt−1

ωt−1ud

ωt−1d

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We shall verify (3.15) at node ωt−1u. For brevity, ωt−1 will be omitted in theexpressions below. By the construction of S, Z there are u, v, w, g, h ∈ a, bsuch that at the respective nodes

St+1(uu) = Sgt+1(uu)

Zt+1(uu) = Zgt+1(uu)

St(u) = Svt (u)

Zt(u) = Zvt (u)

St−1 = Sut−1

Zt−1 = Zut−1

St+1(ud) = Sht+1(ud)

Zt+1(ud) = Zht+1(ud)

St(d) = Swt (d)

Zt(d) = Zwt (d)

andZu

t−1 = Euvwt−1 (Zt;Zt), Zv

t (u) = Evght (Zt+1;Zt+1|u).

Observe that

βt =Zt(u)− Zt(d)

St(u)− St(d)=

Zvt (u)− Zw

t (d)Sv

t (u)− Swt (d)

,

βt+1(u) =Zt+1(uu)− Zt+1(ud)

St+1(uu)− St+1(ud)=

Zgt+1(uu)− Zh

t+1(ud)

Sgt+1(uu)− Sh

t+1(ud).

By the construction of Zut−1 in Algorithm 3.1, for any r ∈ a, b

Euvwt−1 (Zt;Zt) ≥ Eurw

t−1 (Zt;Zt),


βt (Srt (u)− Sv

t (u)) ≥ Zrt (u)− Zv

t (u).

Furthermore, for any r ∈ a, b

Zrt (u) ≥ Ergh

t (Zt+1;Zt+1|u)

by the construction of Zrt (u) in Algorithm 3.1. As a result,

Zrt (u)− Zv

t (u) ≥ Erght (Zt+1;Zt+1|u)− Evgh

t (Zt+1;Zt+1|u)

= βt+1(u) (Srt (u)− Sv

t (u)).

It follows that(βt+1(u)− βt)(S

rt (u)− Sv

t (u)) ≤ 0

for any r ∈ a, b, that is, (3.15) holds at node ωt−1u, as required. The argumentto verify (3.15) at node ωt−1d is similar.

The above proves (3.15) and therefore (2.4) at all nodes for all times t =0, . . . , T − 1.

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(b) Since αT + βT ST = ZT = X, the condition ϑT (α, β) = X can be written as

β+T (ST − SbT ) + β−T (Sa

T − ST ) = 0,

or, equivalently, asβT (Sr

T − ST ) ≥ 0 ∀r ∈ a, b. (3.16)

Take any ωT−1 ∈ ΩT−1 and consider the tree fragment

ωT−1u

ωT−1

ωT−1d

We shall verify (3.16) at node ωT−1u. For brevity, ωT−1 will be omitted in theexpressions below. By the construction of S, Z there are u, v, w ∈ a, b suchthat at the respective nodes

ST (u) = SvT (u)

ZT (u) = ZvT (u)

ST−1 = SuT−1

ZT−1 = ZuT−1

ST (d) = SwT (d)

ZT (d) = ZwT (d)

andZu

T−1 = EuvwT−1(ZT ;ZT ).

Observe that

βT =ZT (u)− ZT (d)

ST (u)− ST (d)=

ZvT (u)− Zw

T (d)Sv

T (u)− SwT (d)

.

By the construction of ZuT−1 in Algorithm 3.1, for any r ∈ a, b

EuvwT−1(ZT ;ZT ) ≥ Eurw

T−1(ZT ;ZT ),


βT (SrT (u)− Sv

T (u)) ≥ 0

using the fact that ZrT (u) = Zv

T (u) = X(u). This implies (3.16) and thereforeϑT (α, β) = X at node ωT−1u, as required. The argument to verify ϑT (α, β) = Xat node ωT−1d is similar. This, then, covers every node at time T .

Theorem 3.4 In a model with proportional transaction costs subject to thesmall transaction costs assumption (2.3) the ask price πa(X) of a Europeanoption X can be represented as

πa(X) = α0 = Z0 = maxZa0 , Z

b0 = E(X) = max

P∈PE(X).

In particular, this implies the correctness of Algorithm 3.1 for computing πa(X)asserted in Claim 3.1.

12

Proof Observe that πa(X) ≤ α0 by the definition (3.5) of πa(X) and byLemma 3.3. Furthermore, α0 = Z0 by the construction of α0, Z0 = E(X) byLemma 3.2, E(X) ≤ maxP∈P E(X) because P ∈ P, and maxP∈P E(X) ≤ πa(X)by Lemma 3.1. Finally, Z0 = maxZa

0 , Zb0 by the construction of Z, completing

the proof.

4 American Options

An American contingent claim can be characterised by an adapted process Y ,where Yt is the payoff at time t = 0, . . . , T .

The time 0 ask and bid prices of an American option with payoff process Yare defined as

πa(Y ) = minα0 | (α, β) ∈ Φ(Sa, Sb), ∀ τ ∈ T : ϑτ (α, β) ≥ Yτ, (4.17)

πb(Y ) = max−α0 | (α, β) ∈ Φ(Sa, Sb), ∃ τ ∈ T : −ϑτ (α, β) ≤ Yτ, (4.18)

where T is the set of all stopping times τ such that 0 ≤ τ ≤ T . The minimumand maximum are attained because the corresponding sets are closed and, re-spectively, bounded below and above in the discrete setting. Similarly as forEuropean options, the ask price πa(Y ) is the lowest price the writer of the op-tion should demand, and the bid price πb(Y ) is the highest amount an optionbuyer could raise to pay for the option so as to be able to hedge their respectivepositions without any risk of loss.

We shall obtain dynamic programming type algorithms for computing theAmerican option ask and bid prices πa(Y ) and πb(Y ) in a market with smalltransaction costs, that is, subject to assumption (2.3). The algorithms can beviewed as extensions of the Snell envelope construction from the case withoutfriction. We shall also construct hedging strategies for both the writer andthe buyer of an American option under small transaction costs. Moreover,representations for the American option ask and bid prices in terms of optimalmartingale measures and stopping times will be established.

We begin with the following assertion, extending Lemma 3.1 to Americanoptions.

Lemma 4.1 For any American option Y

πb(Y ) ≤ maxτ∈T

minP∈P

E(Yτ ) ≤ maxτ∈T

maxP∈P

E(Yτ ) ≤ πa(Y ).

Proof The middle inequality is obvious.Take a strategy (α, β) ∈ Φ(Sa, Sb) such that πa(Y ) = α0 and Yτ ≤ ϑτ (α, β)

for all τ ∈ T . By Lemma 2.2, for each τ ∈ T and each P ∈ P

E(Yτ ) ≤ E(ϑτ (α, β)) ≤ α0 = πa(Y ),

which proves the last inequality.

13

Now let us take a strategy (α, β) ∈ Φ(Sa, Sb) such that πb(Y ) = −α0 and−ϑτ (α, β) ≤ Yτ for some stopping time τ ∈ T . Then, again by Lemma 2.2,

E(Yτ ) ≥ −E(ϑτ (α, β)) ≥ −α0 = πb(Y )

for each P ∈ P. This proves the first inequality.

4.1 Ask Price Algorithm

We propose the following algorithm for computing the ask price πa(Y ) of anAmerican option Y under the small transaction costs assumption (2.3).

Algorithm 4.1 (ask price of American option) Given an American optionwith payoff process Y and expiry time T , construct an R2-valued process Z =(Za, Zb) by backward induction as follows:

1. PutZa

T = ZbT = YT ;


Zut−1 = maxYt−1, max

v,w∈a,bEuvw

t−1 (Zt;Zt).

Claim 4.1 The ask price of the American option is given by

πa(Y ) = maxZa0 , Z

b0.

The claim will be verified in Theorem 4.4.

For each u ∈ a, b and t = 0, . . . , T we put

V ut =

max

v,w∈a,bEuvw

t (Zt+1;Zt+1) if t < T,

YT if t = T,(4.19)

so that Zut = maxYt, V

ut , and we construct processes S, V , Z such that:

1. For some u ∈ a, bS0 = Su

0 ,

V0 = V u0 ,

Z0 = Zu0 ,

andV u0 = maxV a

0 , V b0 . (4.20)

14

2. For each t = 0, . . . , T − 1 and each ωt ∈ Ωt there are v, w ∈ a, b suchthat


t+1(ωtd),Vt+1(ωtu) = V v

t+1(ωtu), Vt+1(ωtd) = V wt+1(ωtd),

Zt+1(ωtu) = Zvt+1(ωtu), Zt+1(ωtd) = Zw

t+1(ωtd),

andV u

t (ωt) = Euvwt (Zt+1;Zt+1|ωt) (4.21)

for each u ∈ a, b. Such v, w exist by Proposition 6.1.

Remark 4.1 The processes S, V , Z may not be unique. The lack of uniquenessmay arise whenever there is more than one pair v, w ∈ a, b satisfying (4.21),or there is more then one u ∈ a, b such that (4.20) holds. In such cases onecan choose any S, V , Z satisfying the conditions above.

Let P be the probability measure turning S into a martingale. It is welldefined and equivalent to Q because, by the small transaction costs assump-tion (2.3), St+1(ωtu) > St(ωt) > St+1(ωtd) for each t = 0, . . . , T − 1 and eachωt ∈ Ωt. We denote by E the expectation under P. We also define a stoppingtime τ ∈ T by

τ = mint | Zt = Yt.

Lemma 4.2 (a) Vt = E(Zt+1|Ft) for each t = 0, . . . , T − 1;

(b) Z is the Snell envelope of Y under P;

(c) E(Yτ ) = Z0.

Proof (a) By the construction of S, V , Z, for any t = 0, . . . , T − 1 and anyωt ∈ Ωt there are u, v, w ∈ a, b such that



Vt(ωt) = V ut (ωt), Vt+1(ωtu) = V v

t+1(ωtu), Vt+1(ωtd) = V wt+1(ωtd),

Zt(ωt) = Zut (ωt), Zt+1(ωtu) = Zv


and (4.21) holds. Thus

Vt(ωt) = V ut (ωt) = Euvw

t (Zt+1;Zt+1|ωt) = E(Zt+1|ωt).

(b) Because ZaT = Zb

T = YT , we have

ZT = YT .

Since Zut = maxYt, V

ut for each u, it follows from (a) that for each t =

0, . . . , T − 1Zt = maxYt, Vt = maxYt, E(Zt+1|Ft).

(c) Because Z is the Snell envelope of Y , we know that the stopped process Zτ∧t

is a martingale under P. Since Zτ = Yτ , it follows that E(Yτ ) = E(Zτ ) = Z0.

15

We define α, β to be predictable processes such that β0 = 0 and for eacht = 0, . . . , T

αt + βtSt = Zt. (4.22)

By Lemma 4.2 and since S is a martingale under P, it follows that for eacht = 0, . . . , T − 1


= E(αt+1 + βt+1St+1|Ft) = E(Zt+1|Ft) = Vt. (4.23)

Clearly, (α, β) is a hedging strategy for the American option Y in thestandard (friction free) model with stock price process S. That is to say,αt + βtSt ≥ αt+1 + βt+1St for each t = 0, . . . , T − 1, and αt + βtSt ≥ Yt

for each t = 0, . . . , T . Nevertheless, it does not immediately follow that (α, β)is also a hedging strategy for Y in the model with transaction costs. This willbe shown in Lemma 4.3.

Lemma 4.3 (a) (α, β) ∈ Φ(Sa, Sb).

(b) ϑt(α, β) ≥ Yt for each t = 0, . . . , T .


(βt+1 − βt)+(Sa

t − St) + (βt+1 − βt)−(St − Sb

t ) ≤ Zt − Vt,


(βt+1 − βt)(Srt − St) ≤ Zt − Vt ∀r ∈ a, b. (4.24)

Thus, we need to demonstrate that (α, β) satisfies (4.24) for each t = 0, . . . , T−1.First we shall verify (4.24) for t = 0. Consider the tree fragment:

ω0u

ω0

ω0d

For brevity, we shall omit ω0 in the expressions below. By the construction ofS, V , Z there are u, v, w ∈ a, b such that at the respective nodes

S1(u) = Sv1 (u)

V1(u) = V v1 (u)

Z1(u) = Zv1 (u)

S0 = Su0

V0 = V u0

Z0 = Zu0

S1(d) = Sw1 (d)

V1(d) = V w1 (d)

Z1(d) = Zw1 (d)

16

andV u0 = Euvw

0 (Z1;Z1).

Observe that

β1 =Z1(u)− Z1(d)

S1(u)− S1(d)=

Zv1 (u)− Zw

1 (d)Sv1 (u)− Sw

1 (d).

By the construction of V0 and by (4.19) we know that V0 = V u0 ≥ V r

0 ≥Ervw0 (Z1;Z1) for each r ∈ a, b. This gives

Euvw0 (Z1;Z1) ≥ Ervw

0 (Z1;Z1),


β1(Sr0 − Su

0 ) ≤ 0

for each r ∈ a, b. Because β0 = 0 and Z0 ≥ V0, this implies (4.24) for t = 0,as required.

Next we shall verify (4.24) for any t = 1, . . . , T − 1. Let us take any nodeωt−1 ∈ Ωt−1 and consider the tree fragment

ωt−1uu

ωt−1u

ωt−1

ωt−1ud

ωt−1d

We are going to verify (4.24) at node ωt−1u. For brevity, we shall omit ωt−1 inthe expressions to follow. By the construction of S, V , Z there are u, v, w, g, h ∈a, b such that at the respective nodes


Vt+1(uu) = V gt+1(uu)

Zt+1(uu) = Zgt+1(uu)

St(u) = Svt (u)

Vt(u) = V vt (u)

Zt(u) = Zvt (u)

St−1 = Sut−1

Vt−1 = V ut−1

Zt−1 = Zut−1


Vt+1(ud) = V ht+1(ud)

Zt+1(ud) = Zht+1(ud)

St(d) = Swt (d)

Vt(d) = V wt (d)

Zt(d) = Zwt (d)

andV u

t−1 = Euvwt−1 (Zt;Zt), V v

t (u) = Evght (Zt+1;Zt+1|u).

17

Observe that

βt =Zt(u)− Zt(d)

St(u)− St(d)=

Zvt (u)− Zw

t (d)Sv

t (u)− Swt (d)

,

βt+1(u) =Zt+1(uu)− Zt+1(ud)

St+1(uu)− St+1(ud)=

Zgt+1(uu)− Zh

t+1(ud)

Sgt+1(uu)− Sh

t+1(ud).

By (4.19), for any r ∈ a, b


t−1 (Zt;Zt),

which after a few transformations can be written as

βt (Srt (u)− Sv


t (u).

Next, again by (4.19), for any r ∈ a, b

V rt (u) ≥ Ergh

t (Zt+1;Zt+1|u).

It follows that

V rt (u)− V v

t (u) ≥ Erght (Zt+1;Zt+1|u)− Evgh

t (Zt+1;Zt+1|u)

= βt+1(u) (Srt (u)− Sv

t (u)) .

As a result, since Zrt (u) ≥ V r

t (u),

(βt+1(u)− βt) (Srt (u)− Sv

t (u)) ≤ (V rt (u)− V v

t (u))− (Zrt (u)− Zv

t (u))

≤ Zvt (u)− V v

t (u)

for any r ∈ a, b, which implies (4.24) at node ωt−1u. The argument to ver-ify (4.24) at node ωt−1d is similar.

This, therefore, proves (4.24), and consequently (2.4), at all nodes for alltimes t = 0, . . . , T − 1.

(b) Using (4.22), we can write the inequality ϑt(α, β) ≥ Yt as

β+t (St − Sbt ) + β−t (Sa

t − St) ≤ Zt − Yt,


βt(Srt − St) ≥ Yt − Zt ∀r ∈ a, b. (4.25)

We shall verify (4.25) for each t = 0, . . . , T .For t = 0 this is trivially satisfied because β0 = 0 and Z0 ≥ Y0. For any

t = 1, . . . , T and any ωt−1 ∈ Ωt−1 consider the tree fragment

ωt−1u

ωt−1

ωt−1d

18

We shall verify (4.25) at node ωt−1u. For brevity, ωt−1 will be omitted in theexpressions below. By the construction of S, V , Z there are u, v, w ∈ a, b suchthat at the respective nodes

St(u) = Svt (u)

Vt(u) = V vt (u)

Zt(u) = Zvt (u)

St−1 = Sut−1

Vt−1 = V ut−1

Zt−1 = Zut−1

St(d) = Swt (d)

Vt(d) = V wt (d)

Zt(d) = Zwt (d)

andV u

t−1 = Euvwt−1 (Zt;Zt).

Observe that

βt =Zt(u)− Zt(d)

St(u)− St(d)=

Zvt (u)− Zw

t (d)Sv

t (u)− Swt (d)

.

By (4.19), for any r ∈ a, b


t−1 (Zt;Zt),


βt (Srt (u)− Sv


t (u) ≥ Yt(u)− Zvt (u) ≥ 0.

This implies (4.25) at node ωt−1u. The argument at node ωt−1d is similar.We have therefore verified (4.25), and consequently the hedging condition

ϑt(α, β) ≥ Yt at each node for every t = 0, . . . , T .

We are now in a position to summarise our results obtained so far in thefollowing theorem, providing various representations of the ask price πa(Y ) ofan American option Y .

Theorem 4.4 In a model with proportional transaction costs subject to thesmall transaction costs assumption (2.3) the ask price πa(Y ) of an Americanoption Y can be represented as

πa(Y ) = α0 = Z0 = maxZa0 , Z

b0 = E(Yτ ) = max

τ∈TmaxP∈P

E(Yτ ).

In particular, this implies the correctness of Algorithm 4.1 for computing πa(Y )asserted in Claim 4.1.

Proof By the definition (4.17) of πa(Y ) and Lemma 4.3 we know that πa(Y ) ≤α0. Next, α0 = Z0 by the construction of α0, Z0 = E(Yτ ) according to

19

Lemma 4.2, and E(Yτ ) ≤ maxτ∈T maxP∈P E(Yτ ) since τ ∈ T and P∈P. More-over, maxτ∈T maxP∈P E(Yτ ) ≤ πa(Y ) by Lemma 4.1. Finally, Z0 = maxZa

0 , Zb0

by the construction of Z. This completes the proof.

Algorithm 4.1 for the ask price of an American option is similar to thestandard iterative construction of the Snell envelope. However, in this formit does not readily extend to the bid price case. We shall now present analternative and somewhat more complicated version of the algorithm, whichwill be extended to the bid price case in the next section.

Algorithm 4.1a (ask price of American option, reformulated) Denoteby Y = (Y a, Y b) the R2-valued process with Y a = Y b = Y , where Y is thepayoff process of an American option with expiry time T , and construct anR2-valued process V = (V a, V b) by backward induction as follows:

1. PutV a

T = V bT = YT ;


V ut−1 = max

M,N∈Yt,Vtmax

v,w∈a,bEuvw

t−1 (M;N).

Claim 4.1a The ask price of the American option is given by

πa(Y ) = maxY0, maxV a0 , V b

0 .

Proof Observe that V ut is equal to that given by (4.19) and that Zu

t =max Yt, V

ut for each t = 0, . . . , T and each u ∈ a, b. As a result, this

version of the algorithm can also be used to compute the ask price as πa(Y ) =maxZa

0 , Zb0 = maxY0, maxV a

0 , V b0 .

4.2 Bid Price Algorithm

Algorithm 4.1a can be modified to apply to the bid price case as follows.

Algorithm 4.2 (bid price of American option) Denote by Y = (Y a, Y b)the R2-valued process with Y a = Y b = Y , where Y is the payoff process ofan American option with expiry time T , and construct an R2-valued processU = (Ua, U b) by backward induction:

1. We putUa

T = U bT = YT ;

2. For each t = 1, . . . , T and each u ∈ a, b we take

Uut−1 = max

M,N∈Yt,Utmin

v,w∈a,bEuvw

t−1 (M;N). (4.26)

20

Claim 4.2 The bid price of the American option is given by

πb(Y ) = maxY0, minUa0 , U b

0.

The claim will be proved in Theorem 4.8.

Let us construct R-valued processes S, U and an R2-valued process U suchthat:

1. For some M ∈ Y0,U0 and u ∈ a, b

S0 = Su0 ,

U0 = Mu,U0 =M,

andMu = minMa,M b = maxY0, minUa

0 , U b0. (4.27)

2. For each t = 0, . . . , T −1 and each ωt ∈ Ωt there are K,L ∈ Yt+1,Ut+1and v, w ∈ a, b such that


t+1(ωtd),Ut+1(ωtu) = Kv(ωtu), Ut+1(ωtd) = Lw(ωtd),Ut+1(ωtu) = K(ωtu), Ut+1(ωtu) = L(ωtu),

andUu

t (ωt) = minc,d∈a,b

Eucdt (K;L|ωt) = Euvw

t (K;L|ωt) (4.28)

for each u ∈ a, b. The existence of such K,L and v, w follows, respec-tively, from Propositions 6.2 and 6.1.

Remark 4.2 The processes S, U , U may not be unique. The lack of uniquenessmay arise whenever there is more than one pair K,L ∈ Yt+1,Ut+1 or morethan one pair v, w ∈ a, b such that (4.28) holds for each u ∈ a, b, or thereis more then one M ∈ Y0,U0 or more than one u ∈ a, b such that (4.27)holds. In such cases we can choose any S, U , U satisfying the conditions above.

Let P be the probability measure turning S into a martingale. It is welldefined and equivalent to Q because, by the small transaction costs assump-tion (2.3), St+1(ωtu) > St(ωt) > St+1(ωtd) for each t = 0, . . . , T − 1 and eachωt ∈ Ωt. Since Sb

t ≤ St ≤ Sbt for each t = 0, . . . , T , we have P ∈ P.

We also define a stopping time τ ∈ T by

τ = mint | Ut = Yt.

Observe that the definition of τ involves the R2-valued processes U and Y. Thisis in contrast to the stopping time τ , which was introduced in Section 4.1 inconnection with the ask price of an American option and defined in terms ofR-valued processes, namely Z and Y .

21

Lemma 4.5 (a) The stopped process Uτ∧t is a martingale under P;

(b) E(Yτ ) = U0.

Proof (a) By the construction of S, U , U, for any t = 0, . . . , T − 1 and anyωt ∈ Ωt there are u, v, w ∈ a, b, K,L ∈ Yt+1,Ut+1 andM ∈ Yt,Ut suchthat



Ut(ωt) = Mu(ωt), Ut+1(ωtu) = Kv(ωtu), Ut+1(ωtd) = Lw(ωtd),Ut(ωt) =M(ωt), Ut+1(ωtu) = K(ωtu), Ut+1(ωtd) = L(ωtd),

and (4.28) holds. If t < τ on ωt, then M(ωt) = Ut(ωt), so that Mu(ωt) =Uu

t (ωt), and we obtain

Ut(ωt) = Uut (ωt) = Euvw

t (K;L|ωt) = E(Ut+1|ωt).

This means that on t < τ

Uτ∧t = E(Uτ∧(t+1)|Ft).

On the other hand, on t ≥ τ we have Uτ∧t = Uτ∧(t+1), and since Uτ∧t isFt-measurable,

Uτ∧t = E(Uτ∧t|Ft) = E(Uτ∧(t+1)|Ft),

completing the proof that Uτ∧t is a martingale under P.

(b) By the definition of τ we have Uτ = Yτ , which implies Uτ∧T = Uτ = Yτ ,so that E(Yτ ) = E(Uτ∧T ) = U0 by (a).

Next, we define predictable processes α, β such that β0 = 0 and for eacht = 0, , . . . , T

αt + βtSt = −Uτ∧t. (4.29)

Since S and the stopped process Uτ∧t are martingales under P, it follows thatfor each t = 0, . . . , T − 1


= E(αt+1 + βt+1St+1|Ft) = −E(Uτ∧(t+1)|Ft) = −Uτ∧t. (4.30)

Also observe that βt = 0 and αt = −Uτ on τ < t.

Lemma 4.6 (a) (α, β) ∈ Φ(Sa, Sb),

(b) ϑτ (α, β) = −Yτ .


(βt+1 − βt)+(Sa

t − St) + (βt+1 − βt)−(St − Sb

t ) ≤ 0.

22

This, in turn, is equivalent to

(βt+1 − βt)(Srt − St) ≤ 0 ∀r ∈ a, b. (4.31)

We shall demonstrate that (α, β) satisfies (4.31) for each t = 0, . . . , T − 1.First, suppose that t = 0. If τ = 0, then β0 = β1 = 0 and (4.31) at t = 0

follows trivially.If t = 0 and τ > 0, then we consider the tree fragment

ω0u

ω0

ω0d

In the expressions below ω0 will be omitted for brevity. By the constructionof S, U , U there are K,L ∈ Y1,U1, M ∈ Y0,U0 and u, v, w ∈ a, b suchthat at the respective nodes

S1(u) = Sv1 (u)

U1(u) = Kv(u)U1(u) = K(u)

S0 = Su0

U0 = Mu

U0 =M

S1(d) = Sw1 (d)

U1(d) = Lw(d)U1(d) = L(d)

andUu0 = min

c,d∈a,bEucd0 (K;L) = Euvw

0 (K;L). (4.32)

Observe that

β1 = − U1(u)− U1(d)

S1(u)− S1(d)= −Kv(u)− Lw(d)

Sv1 (u)− Sw

1 (d).

Since τ > 0, we know that M = U0, so Uu0 = Mu ≤ Mr = Ur

0 for eachr ∈ a, b. By (4.32) and Proposition 6.2

Ur0 = min

c,d∈a,bErcd0 (K;L) ≤ Ervw

0 (K;L).

As a result, for each r ∈ a, b

Euvw0 (K;L) ≤ Ervw

0 (K;L),


β1(Sr0 − Su

0 ) ≤ 0.

Since β0 = 0, this gives (4.31) for t = 0 and τ > 0.

23

Next suppose that t = 1, . . . , T − 1 and τ < t. Then βt = βt+1 = 0 and(4.31) is self evident.

If t = 1, . . . , T − 1 and τ ≥ t, then consider the tree fragment

ωt−1uu

ωt−1u

ωt−1

ωt−1ud

ωt−1d

We shall verify (4.31) at ωt−1u. For brevity ωt−1 will be omitted in the ex-pressions below. By the construction of S, U , U there are I,J ∈ Yt+1,Ut+1,K,L ∈ Yt,Ut, M ∈ Yt−1,Ut−1 and u, v, w, g, h ∈ a, b such that at therespective nodes


Ut+1(uu) = Ig(uu)Ut+1(uu) = I(uu)

St(u) = Svt (u)

Ut(u) = Kv(u)Ut(u) = K(u)

St−1 = Sut−1

Ut−1 = Mu

Ut−1 =M


Ut+1(ud) = Jh(ud)Ut+1(ud) = J(ud)

St(d) = Swt (d)

Ut(d) = Lw(d)Ut(d) = L(d)

and

Uut−1 = min

c,dEucd

t−1(K;L) = Euvwt−1 (K;L), (4.33)

Uvt (u) = min

c,dEvcd

t (I;J|u) = Evght (I;J|u). (4.34)

Observe that

βt = − Ut(u)− Ut(d)

St(u)− St(d)= −Kv(u)− Lw(d)

Svt (u)− Sw

t (d).

From (4.33) we deduce that for each r ∈ a, b

Euvwt−1 (K;L) ≤ Eurw

t−1 (K;L),


− βt (Srt (u)− Sv

t (u)) ≤ Kr(u)−Kv(u). (4.35)

24

Now if τ = t at ωt−1u, then βt+1(u) = 0 and K(u) = Yt(u), that is, Kr(u) =Kv(u) = Yt(u), so that (4.35) implies (4.31) at node ωt−1u. If, on the otherhand, τ > t at ωt−1u, then

βt+1(u) = − Ut+1(uu)− Ut+1(ud)

St+1(uu)− St+1(ud)= − Ig(uu)− Jh(ud)

Sgt+1(uu)− Sh

t+1(ud).

From (4.34), by Proposition 6.2, we know that for each r ∈ a, b

Urt (u) = min

c,dErcd

t (I;J|u) ≤ Erght (I;J|u).

As a result, by (4.33)

Urt (u)− Uv

t (u) ≤ Erght (I;J|u)− Evgh

t (I;J|u),


Urt (u)− Uv

t (u) ≤ −βt+1(u) (Srt (u)− Sv

t (u)) . (4.36)

But τ > t at ωt−1u also means that K(u) = Ut(u), that is, Kr(u) = Urt (u) and

Kv(u) = Uvt (u), so (4.35) and (4.36) imply (4.31) at node ωt−1u. In a similar

manner, one can verify (4.31) at each node ωt−1d for t = 1, . . . , T − 1 in thecase when τ ≥ t, completing the proof of (a).

(b)We want to show that ατ +β+τ Sbτ−β−τ Sa

τ = −Yτ . Because ατ +βτ Sτ = −Uτ

and Uτ = Yτ this equality can be written as

β+τ(Sτ − Sb

τ

)+ β−τ

(Sa

τ − Sτ

)= 0,

which is equivalent to

βτ (Srτ − Sτ ) ≥ 0 ∀r ∈ a, b. (4.37)

We shall show that (α, β) satisfies (4.37).If τ = 0, then (4.37) follows immediately since β0 = 0.Suppose that τ > 0. If τ = t at a node ωt−1u, then we shall consider the

tree fragmentωt−1u

ωt−1

ωt−1d

and we shall verify (4.37) at ωt−1u. For brevity, ωt−1 will be omitted in theexpressions below. By the construction of S, U , U there are K,L ∈ Yt,Ut,

25

M ∈ Yt−1,Ut−1 and u, v, w ∈ a, b such that at the respective nodes

St(u) = Svt (u)

Ut(u) = Kv(u)Ut(u) = K(u)

St−1 = Sut−1

Ut−1 = Mu

Ut−1 =M

St(d) = Swt (d)

Ut(d) = Lw(d)Ut(d) = L(d)

andUu

t−1 = minc,d

Eucdt−1(K;L) = Euvw

t−1 (K;L).

Observe that

βt = − Ut(u)− Ut(d)

St(u)− St(d)= −Kv

t (u)− Lwt (d)

Svt (u)− Sw

t (d).

It follows that for all r ∈ a, b

Euvwt−1 (K;L) ≤ Eurw

t−1 (K;L),


− βt (Srt (u)− Sv

t (u)) ≤ Kr(u)−Kv(u). (4.38)

Because τ = t at ωt−1u, we know that K(u) = Yt(u), so that Kr(u) = Kv(u) =Yt(u), and (4.38) implies (4.37) at ωt−1u, as required. The argument to verify(4.37) if τ = t at a node ωt−1d is similar, completing the proof.

Lemma 4.7maxτ∈T

minP∈P

E(Yτ ) ≤ U0.

Proof We take any τ ∈ T , consider a European option with payoff X = −Yτ

expiring at time T , and construct processes Za, Zb as in Algorithm 3.1. Observethat for each u ∈ a, b

−Zuτ = Yτ .

We claim that for each u ∈ a, b and for each t = 0, . . . , T

−Zut ≤ Uu

t

on τ > t. This claim can be proved by backward induction on t. Sinceτ > T is empty, the claim is trivially satisfied for t = T . Now suppose thatthe claim is valid for some t = 1, . . . , T . At each node ωt−1 ∈ Ωt−1 such thatτ > t− 1 at ωt−1 (that is, ωt−1 ⊂ τ > t− 1) there are up to four possibilities:

26

1. If τ > t at ωt−1u and at ωt−1d, then −Zvt ≤ Uv

t at ωt−1u and −Zwt ≤ Uw

t

at ωt−1d for each v, w, so by (3.10) and (4.26) for each u

−Zut−1 ≤ min

v,w∈a,bEuvw

t−1 (Ut;Ut) ≤ Uut−1;

2. If τ = t at ωt−1u and τ > t at ωt−1d, then −Zvt = Yt at ωt−1u and

−Zwt ≤ Uw

t at ωt−1d for each v, w, so by (3.10) and (4.26) for each u

−Zut−1 ≤ min

v,w∈a,bEuvw

t−1 (Yt;Ut) ≤ Uut−1;

3. If τ > t at ωt−1u and τ = t at ωt−1d, then −Zvt ≤ Uv

t at ωt−1u and−Zw

t = Yt at ωt−1d for each v, w, so by (3.10) and (4.26) for each u

−Zut−1 ≤ min

v,w∈a,bEuvw

t−1 (Ut;Yt) ≤ Uut−1;

4. Finally, if τ = t at ωt−1u and at ωt−1d, then −Zvt = Yt at ωt−1u and

−Zwt = Yt at ωt−1d for each v, w, so by (3.10) and (4.26) for each u

−Zut−1 ≤ min

v,w∈a,bEuvw

t−1 (Yt;Yt) ≤ Uut−1.

This verifies the claim. It follows that −Zu0 ≤ Uu

0 on τ > 0 for each u.Moreover, −Zu

0 = Y0 on τ = 0 for each u. As a result,

−maxZa0 , Z

b0 = min−Za

0 ,−Zb0 ≤ maxY0, minUa

0 , U b0 = U0.

By Theorem 3.4 we therefore obtain

minP∈P

E(Yτ ) = −maxP∈P

E(−Yτ ) = −maxZa0 , Z

b0 ≤ U0,

completing the proof because of the arbitrariness of τ .

Theorem 4.8 In a model with proportional transaction costs subject to thesmall transaction costs assumption (2.3) the bid price πb(Y ) of an Americanoption Y can be represented as

πb(Y ) = −α0 = U0 = maxY0, minUa0 , U b

0 = E(Yτ ) = maxτ∈T

minP∈P

E(Yτ ).

In particular, this implies the correctness of Algorithm 4.2 for computing πb(Y )asserted in Claim 4.2.

Proof By the definition (4.18) of πb(Y ) and Lemma 4.6 we know that πb(Y ) ≥−α0. Moreover, −α0 = U0 by the construction of α0, U0 ≥ maxτ∈T minP∈P E(Yτ )by Lemma 4.7, and maxτ∈T minP∈P E(Yτ ) ≥ πb(Y ) by Lemma 4.1. Finally,U0 = E(Yτ ) by Lemma 4.5, and U0 = maxY0, minUa

0 , U b0 by the construc-

tion of U0, completing the proof.

27

5 Concluding Remarks

We have extended the construction of the Snell envelope to compute the bid andask prices of American contingent claims under small proportional transactioncosts. In addition, we have provided iterative constructions of optimal hedgingstrategies for the seller as well as for the buyer of an American option. Asa special case, we have also considered European options in the same setting.The pricing algorithms are based on backward induction, involving the solutionof an optimisation problem at each tree node, and can be viewed as dynamicprogramming procedures. An interesting new feature of these algorithms is thatit is necessary to keep track of two quantities at each node, rather than a singleone as in the well known case with no friction. Otherwise, the algorithms havemany features in common with the recursive construction of the Snell envelope,to which they in fact reduce in the absence of transaction costs.

We have only considered contingent claims with cash delivery and assumedimplicitly that the option buyer or seller have no position in the underlying assetinitially. This would, of course, be no consequence in a friction free market.However, in the presence of transaction costs the algorithms would need to beadapted for an option buyer or seller with a long or short initial position in theunderlying or to allow for options with physical delivery or, generally, mixeddelivery with the payoff being a portfolio of cash and underlying stock.

A natural next step is to try to construct similar algorithms under relaxedassumptions of small proportional transaction costs. In the case of Europeanoptions this is addressed in Tokarz [Tok04] for arbitrary proportional transactioncosts, as long as the model remains free of arbitrage. The procedure is similarto the algorithms in the present paper, except that one has to keep track ofmore than two quantities at any node for which the underlying bid-ask spreadoverlaps with that at an adjacent node.

Other directions in which the algorithms could be extended in future workinvolve more general models, including incomplete ones, and other kinds offriction, such as, for example, different lending and borrowing rates, differentlypriced short and long positions, short selling restrictions, or fixed transactioncosts.

6 Appendix

Here we shall state and outline the proofs of two technical propositions concern-ing a single-step tree

Sa1 (u)

Sb1(u)

Sa0

Sb0

Sa1 (d)

Sb1(d)

28

with small proportional transaction costs, i.e. subject to assumption (2.3). Theresults also apply to any single-step fragment of a larger binary tree model,which is how they are used in the preceding sections. The full details of theproofs, which are elementary but somewhat tedious, can be found in [Tok04].

Consider any two R2-valued random variables A = (Aa, Ab) and B =(Ba, Bb) in the single-step model. Then the following propositions hold.

Proposition 6.1 For any c, d ∈ a, b the following conditions are equivalent:

(a) minv,w∈a,b

Eavw0 (A;B) = Eacd

0 (A;B);

(b) minv,w∈a,b

Ebvw0 (A;B) = Ebcd

0 (A;B).

The conditions remain equivalent if the minima are replaced by maxima.

Proof Outline The main steps of the proof are:

1. For any c, d, e, f, u ∈ a, b such that c 6= e and d 6= f show that theinequalities

Eucd0 (A;B) ≤ Eued

0 (A;B), (6.39)

Eucd0 (A;B) ≤ Eucf

0 (A;B) (6.40)

implyEucd0 (A;B) ≤ Euef

0 (A;B),

and deduce that (6.39), (6.40) are equivalent to

minv,w∈a,b

Euvw0 (A;B) = Eucd

0 (A;B).

2. Verify that for any c, d, e, f ∈ a, b the inequalities (6.39), (6.40) withu = a are equivalent to (6.39), (6.40) with u = b.

The equivalence of (a) and (b) follows directly from steps 1 and 2 above.

Proposition 6.2 For any C,D ∈ A,B the following conditions are equiva-lent:

(a) maxV,W∈A,B

minv,w∈a,b

Eavw0 (V;W) = min

v,w∈a,bEavw0 (C;D);

(b) maxV,W∈A,B

minv,w∈a,b

Ebvw0 (V;W) = min

v,w∈a,bEbvw0 (C;D).

Proof Outline The main steps of the proof are:

29

1. For any C,D,E,F ∈ A,B such that E 6= C and F 6= D and for anyu ∈ a, b show that the inequalities

minv,w∈a,b

Euvw0 (C;D) ≥ min

v,w∈a,bEuvw0 (E;D), (6.41)

minv,w∈a,b

Euvw0 (C;D) ≥ min

v,w∈a,bEuvw0 (C;F) (6.42)

implymin

v,w∈a,bEuvw0 (C;D) ≥ min

v,w∈a,bEuvw0 (E;F),

and deduce that (6.41), (6.42) are equivalent to

maxV,W∈A,B

minv,w∈a,b

Euvw0 (V;W) = min

v,w∈a,bEuvw0 (C;D).

2. Verify that for any C,D,E,F ∈ A,B the inequalities (6.41), (6.42) withu = a are equivalent to (6.41), (6.42) with u = b.

The equivalence of (a) and (b) follows directly from steps 1 and 2 above.

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