Hedging problems for Asian options with transactions costs
NNoname manuscript No. (will be inserted by the editor)
Hedging problems for Asian options withtransactions costs
Serguei Pergamenchtchikov · AlenaShishkova
Received: date / Accepted: date
Abstract
In this paper, we consider the problem of hedging Asian optionsin financial markets with transaction costs. For this we use the asymptotichedging approach. The main task of asymptotic hedging in financial marketswith transaction costs is to prove the probability convergence of the terminalvalue of the investment portfolio to the payment function when the numberof portfolio revisions tends to be n to infinity. In practice, this means that theinvestor, using such a strategy, is able to compensate payments for all financialtransactions, even if their number increases unlimitedly. Keywords
Hedging strategy · Wiener process · Asian option · Stochasticdifferential equations · Brownian bridge
For a trader or an investor the main task is not only the saving but alsothe multiplication of its capital. Many risks can be avoided with the helpof one popular and very effective technique hedging. The option is hedgedto protect its value from the risk of price movement of the underlying assetin an unfavorable direction. To solve the hedging problem stochastic calculusmethods are used which became a powerful tool used in practice in the financialworld. Stochastic calculus is a well-developed branch of modern mathematics
Serguei PergamenchtchikovLaboratoire de Math´ematiques Raphael Salem, UMR 6085 CNRS- Universit´e de RouenNormandie, France and ILSSP&QF, National Research Tomsk State UniversityTel.: 02 32 95 52 22Fax: 02 32 95 52 86E-mail: [email protected] ShishkovaILSSP&QF, National Research Tomsk State UniversityE-mail: [email protected] a r X i v : . [ q -f i n . M F ] J a n Serguei Pergamenchtchikov, Alena Shishkova with a correct approach to analyzing complex phenomena occurring on worldstock markets.In the modern theory and practice of options the paper written by Blackand Scholes [1] has an important role. In this work the authors used economicknowledge in combination with PDE arguments which are similar to derivingthe heat equation from the first physical principles.In our paper we use a probabilistic approach and our main tool is repre-sentation theorem for the Wiener process. This theorem was formulated byJ.M.C. Clark in [2] also it can be obtained from the representation theoremstated in the paper [3] by K. Ito.It should be noted that the task of options pricing and the constructionof a hedging strategy is well studied for American and European options, forsuch derivatives there is a so-called delta strategy. But this technique is notenough developed for Asian options. Exotic options became more in demandin the late 1980s and early 1990s and their trade became more active in theover-the-counter market. Soon in the commodity and currency markets, Asianoptions were becoming popular.Mathematically, the value of an Asian option is reduced to calculating theconditional mathematical expectation of a payment function. Many authorshave studieded in their work the Asian pricing problem. H. Geman and M. Yor(1993) were among the first to consider derivatives are based on the averageprices of underlying assets [18]. Using the Bessel processes authors found thevalue of the Asian option. Moreover, applying simple probabilistic methodsthey obtained the following results about these options: calculated momentsof all orders of the arithmetic average of the geometric Brownian motion;obtained simple, closed form expression of the Asian option price when theoption is in the money. The exact pricing of fixed-strike Asian options is adifficult task, since the distribution of the average arithmetic of asset prices isunknown when its prices are distributed lognormally.L. C. G. Rogers and Z. Shi (1995) in their work [25] to compute the priceof an Asian option used two different ways. Firstly, exploiting a scaling prop-erty, they reduced the problem to the problem of solving a parabolic PDE intwo variables. Secondly, authors provided a lower bound which is so accuratethat it is essentially the true price. J. Vecer (2001) observed that the Asianoption is a special case of the option on a traded account and extended work[26] to the arithmetic average Asian option [27]. Using probabilistic techniqueshe established that the price of the Asian option is characterized by a simpleone-dimensional partial differential equation which could be applied to bothcontinuous and discreteaverage Asian option. J.Vecer and M.Xu (2004) studiedpricing Asian options in a semimartingale model [28]. They showed that the in-herently path dependent problem of pricing Asian options can be transformedinto a problem without path dependency in the payoff function. Authors alsoshowed that the price satisfies a simpler integro-differential equation in thecase the stock price is driven by a process with independent increments, Levyprocess being a special case. Pricing Asian options under Levy processes alsoconsidered in [29,30,31]. edging problems for Asian options with transactions costs 3
When explicit valuation formulas are not available, sharp lower and upperbounds intervals for option prices can be useful in improving the quality ofthe approximations adopted: some results in this direction are provided in thepapers [32,33,34]. In [35] authors considered geometric average Asian optionand showed that the lower and upper bounds can be expressed as a portfolio ofdelayed payment European call options. Pricing Asian options with stochasticvolatility considered in [36,37,38].A large number of works are connected with the numerical approach.Kemna and Vorst were among the first who solved the task [20]. In theirwork the pricing strategy includes Monte Carlo simulation with elements ofdispersion reduction and improves the pricing method. Furthermore, the au-thors showed that the price of an option with an average value will always belower than of a standard European option. Carverhill and Clewlow [21] used afast Fourier transform to calculate the density of the sum of random variables,as convolution of individual densities. Then the payoff function is numericallyintegrated against the density. In this direction other authors continued towork, applying to the calculations improved methods of numerical simulation[22,23,24]. Unfortunately, these methods do not provide information on thehedging portfolio.In the articles above, the authors focus on calculating the value of theoption, but do not consider in detail the hedging problem, use only generalexistence theorems. In works [41,40] authors consider the problem of hedgingwith the payoff f = (cid:18) S t + ... + S t n n − K (cid:19) + and use the moment recurrence technique, i.e. get recurrence equations. Wecan not use this technique, as we are considering the following payoff f = (cid:90) S u du − K + . General equations for the hedging strategy based on the martingal repre-sentation M t = E ( f |F t ) = M + t (cid:90) α s dW s , where α s = (cid:104) M, W (cid:105) s . This theory is well developed only for options whose payoffs depend only onthe price at the last moment in time f = f ( S T ). And further, to computea strategy, it is necessary to study only one random variable S T , which is ageometric Brownian motion, i.e. density is known. In the case of Asian options,the payoff is a functional of the whole path and it is required to study the Serguei Pergamenchtchikov, Alena Shishkova density of the integrals t (cid:82) S u du in order to calculate (cid:104) M, W (cid:105) s , that is, it isnecessary to average over an infinite-dimensional distribution.Yor and Dufrance [18,19] obtained the pricing in the explicit form, but theystudied not the density, they considered the functional f , and for it they got arepresentation in the form of infinite series on a special orthogonal basis, whichis not possible to study analiticaly the regularity properties. There is anothermethod to study this properties of the density one can use the Brownian bridge,which proposed by Kabanov Yu.M. and Pergamenshchikov S.M. (2016) [39].Using this method we construct the hedging strategy. This is made in [17].It is worth noting that the option pricing model in work [1] has an idealcharacter, i.e. it is assumed that it is friction-free market without costs. Thistheory is no longer true when we need to take into account transaction costs κ n J n = κ n − α J n because there is no unimprovable hedge. Therefore optionpricing and replication with nonzero trading costs are different from that inthe Black-Scholes setting.Models with proportional transaction costs were considered as early asthe 1970s. Magill and Constantinides [4] suggested in 1976 the consumption-investment model which is generalization of the Merton model of 1973 [5].However, the article written by H. Leland [6] in 1985 became more importantfor practical application. Leland’s strategy provides an easy way to effectivelyeliminate the risks associated with transaction costs. This method is basedon the idea that transaction costs can be offset by increasing the volatilityparameter in the Black-Scholes strategy, that is the delta strategy obtainedfrom a changed Black-Scholes equation with an appropriate modified volatilityensures an approximately complete replication as expected. The major goal inLeland’s algorithm is to explore the asymptotic behavior of the hedging error(difference between the terminal value of portfolio and the payoff function) asthe number of transaction goes to infinity.Leland suggested that if transaction costs are fixed, i.e. α = 0 then thevalue of the portfolio converges in probability to the payoff function as n → ∞ .He also suggested that this result will be true in the case of α = 1 /
2. Laterthis fact has proved by K. Lott in his thesis [7]. Later Yu. Kabanov and M.Safarian [8] extended Lotts work to any α ∈ (0 , / α = 0, i.e. constant transaction cost. The authors proved thatthe hedging error admits a non-zero limit. The obtained result was used byH.Ahn end others [9] for the hedging problem with transaction costs in generaldiffusion models.There are a lot of studies using Lelands algorithm and extend it to varioussetting. For example, S. Pergamenshchikov in [10] studied the convergencerate of approximation in the case of constant costs. He obtained technicallydifficult result since used nontrivial procedure. This result is important becauseit not only provides asymptotic information about the hedging error but alsogives a reasonable way to solve the hedging problem, namely, the investor canget a portfolio whose final value exceeds the desired profit by choosing theappropriate value of the modified volatility. edging problems for Asian options with transactions costs 5 The important result had been obtained by E. Lepinette [11] in the caseof time-depending volatility models. He used a non-uniform interval splitting.Moreover, to obtain the asymptotically complet replication he modified thestrategy, which is called Lepinettes strategy, and proved that for α > α = 0 then the portfolio value of strategy converges in probability to thepayoff plus two positive functions depending on payoff. To improve a rate ofconvergence E. Lepinette in [12] also used a non-uniform interval splitting andproved that for strategy suggested in [11] with α = 0 the approximation errormultiplied by n β weakly converges to a centered mixed Gaussian variable as n → ∞ .Another way to enlarge application of Lelands strategy is to consider thehedging problem with transaction costs in the models where the value ofvolatility depends on time and on the price of the stock, so-called the localvolatility models. E.Lepinette and T.Tran [13] extended results obtained in[12] to this models. The proof of the result is really complicated, since the ex-istence of a solution of a non-uniform parabolic Cauchy problem is nontrivial,if we adjust the volatility as well as in work [11].To extend the Lelands approach many others authors considered differ-ent situations including more general contingent claims, more general priceprocesses and etc. see [14,15]. Thus Lelands strategy has great importancein option pricing and the hedging problem due to it is easily implemented inpractice.Our goal is to extend this hedging methods for the hedging problem for thefinancial markets with transaction costs. To this end we use the approximativehedging approach proposed Leland, Kabanov, Safarian, Pergamenshchikov,Lepinette [6,8,10,11]. Note that is all this paper the hedging strategy is basedon the delta-strategy. But for Asian option one need to change basic strategy,i.e. to pass frpm delta-strategy to Asian hedging strategy constructed in [17].In this paper we study assymptotic property for the portfolio value withtransaction cost in the Black-Scholes model with risky asset without drift andrisk-free asset with interest rate r = 0. We use the modification of Lelandsstrategy. Main result of our study are obtined sufficient conditions, whichprovide assymptotic hedging. r = 0, i.e. the bond price is constant overtime B t = 1 throughout this article. Let ( Ω, F , ( F t ) ≤ t ≤ , P ) be the standardfiltered probability space with F t = σ ( W s , ≤ s ≤ t ) and W is a Wiener Serguei Pergamenchtchikov, Alena Shishkova process. The asset price process S t given by dS t = σS t dW t , ≤ t ≤ S t = S e σW t − σ t/ . Remark that S t is a martingale under measure P . The model is consideredon the interval [0 ,
1] where 1 is a maturity of the Asian option with payofffunction f = (cid:90) S u du − K + . Definition 1
The financial strategy ( Π t ) ≤ t ≤ = ( β t , γ t ) ≤ t ≤ is called anadmissible self-financing strategy if it is F t -adapted, integrable with t (cid:90) ( | β t | + γ t ) dt < ∞ and the portfolio value is V t = β t + γ t S t = V + t (cid:90) γ u dS u . Here V is an initial capital, β t and γ t are quantity of the risk-free asset andrisk asset respectively.Suppose an investor operating on a (B,S)– market solves the following”investment problem”: using a self-financing portfolio at some predeterminedpoint in time 1, in the future bring its capital to f . Obviously, the implemen-tation of this goal depends on the initial capital x invested in the portfolioand on the investor strategy ( Π t ) ≤ t ≤ of portfolio reorganization used by theinvestor. Definition 2
For a given x > f a self-financing strategy is called a( x, f ) - hedge if ∀ ω ∈ Ω, V Π = x, V Π ≥ f a.s. edging problems for Asian options with transactions costs 7 dS t = σS t dW t , ≤ t ≤ κ . Here κ is a positive constant defined by marketmoderators. We assume that the investor plans to revise his portfolio at dates( t i ) = i/n , where n is the number of revisions.Under the presence of proportional transaction costs, it was proposed by[6] and then generalized by [8] that the volatility should be adjusted as (cid:98) σ = σ + σκn / − α (cid:112) /π (2)in order to create an artificial increase in the option price C ( t, S t ) to com-pensate possible trading fees. This form is inspired from the observation thatthe trading cost κ n S t i | C x ( t i , S t i ) − C x ( t i − , S t i − ) | in the interval of time[ t i − , t i ] can be approximate by κ n S t i − C xx ( t i − , S t i − ) | ∆S t i | ≈ κ n σS t i − C xx ( t i − , S t i − ) E | ∆W t i | . (3)For simplicity, we assume that the portfolio is revised at uniform grig t i = i/n, i = 1 , ..., n of the option life interval [0 , E | ∆W t i / ( ∆t i ) / | = (cid:112) /π one approximates the last term in (3) by κ n σ (cid:112) /π ( ∆t i ) / S t i − C xx ( t i − , S t i − ) , which is the cost paid for portfolio readjustment in [ t i − , t i ]. Hence, by thestandard argument of Black-Scholes (BS) theory, the option price inclusive oftrading cost should satisfy C t ( t i − , S t i − ) ∆t i + 12 σ S t i − C xx ( t i − , S t i − ) ∆t i + κ n σ (cid:112) /π ( ∆t i ) / S t i − C xx ( t i − , S t i − ) = 0 . Since ∆t i = 1 /n , one deduces that C t ( t i − , S t i − ) + 12 ( σ + κ n σ (cid:112) n /π ) S t i − C xx ( t i − , S t i − ) = 0 , which implies that the option price inclusive trading cost should be evaluatedby the following modified-volatility version of the Black-Scholes PDE (cid:98) C t ( t, x ) + 12 (cid:98) σ x (cid:98) C xx ( t, x ) = 0 , (cid:98) C (1 , x ) = max( x − K, , where the adjusted volatility (cid:98) σ is defined by (2).To compensate transaction costs caused by hedging activities, the optionseller is suggested to follow the Leland strategy defined by the piecewise process γ nt = n (cid:88) i =1 (cid:98) C x ( t i − , S t i − ) ( t i − ,t i ] ( t ) . Serguei Pergamenchtchikov, Alena Shishkova
Then the portfolio value corresponding to this strategy at time t defined by V nt = V + (cid:90) t γ nu dS u − κ n n (cid:88) i =1 S t j | γ nt i − γ nt i − | . Definition 3
Strategy γ nt is called hedging if V n P −−−−→ n →∞ f . f isto choose the admissible self-financing strategy ( β t , γ t ) such that V = V + (cid:90) γ u dS u ≥ f , a.s.To construct a hedging strategy in the case of model (1) apply the representa-tion theorem for quadratic integrated martingale to the following martingale M t = E ( f |F t ) . (4)We will find the square integrable process ( α t ) ≤ t ≤ adapted w.r.t. F t suchthat for all t ∈ [0 , M t = M + (cid:90) t α s dW s . (5)Clearly that dM t = α t dW t . (6)For coefficients α t we use the following formula (cid:104) M, W (cid:105) t = (cid:90) t α s ds, therefore α t = ddt (cid:104) M, W (cid:105) t . Also the portfolio value satisfies the equality dV t = γ t dS t = γ t σS t dW t . (7) edging problems for Asian options with transactions costs 9 Equating (6) and (7), we obtain the formulas for strategy Π = ( β t , γ t ) ≤ t ≤ γ t = α t /σS t , (8) β t = E f + (cid:90) t α s dW s − γ t S t , (9)In our case the martingale has the following form M t = E ( f ) |F Wt ) = E (cid:32)(cid:18)(cid:90) S v dv − K (cid:19) + |F Wt (cid:33) , (10)If v ≥ t then S v = S t exp (cid:8) σ ( W v − W t ) − σ ( v − t ) / (cid:9) . It means that we can represent the integral in the equality (10) as (cid:90) S v dv = ξ t + S t η t , where ξ t = t (cid:90) S v dv, η t = (cid:90) t exp (cid:8) σ ( W v − W t ) − σ ( v − t ) / (cid:9) dv, Note that ξ t is measurable w.r.t. F t , and η t is independent on F t . Hence M t = G ( t, ξ t , S t ) , (11)here G ( t, x, y ) = E ( x + yη t − K ) + . Theorem 1
The function G ( t, x, y ) has the continuous derivatives ∂∂t G ( t, x, y ) , ∂∂x G ( t, x, y ) , ∂∂y G ( t, x, y ) , ∂ ∂y G ( t, x, y ) . The proof see in [17].Since for any t > W t + u − W t ) u ≥ is Wiener process thendistribution of the random variable η t coincides with the distribution of thefollowing random variable˜ η v = (cid:90) v exp { σW u − σ u/ } du, (12)Therefore G ( t, x, y ) = E ( x + y ˜ η v − K ) + . (13) Taking into account Theorem 1 and applying Ito’s formula to the function G ( t, x, y ) we obtain M t = M + (cid:90) t (cid:18) G (cid:48) t ( v, ξ v , S v ) + G (cid:48) x ( v, ξ v , S v ) + σ S v G (cid:48)(cid:48) y y ( v, ξ v , S v ) (cid:19) dv + ˜ M t , (14)where ˜ M t = σ (cid:90) t G (cid:48) y ( v, ξ v , S v ) S v dW v ,G (cid:48) t = ∂G/∂t and other partial derivative similarly. The quadratic characteristicis calculated by the formula (cid:104) M, W (cid:105) t = P − lim n →∞ n (cid:88) j =1 E (cid:0)(cid:0) M t j − M t j − (cid:1) (cid:0) W t j − W t j − (cid:1) |F t j − (cid:1) . We have that (cid:90) t (cid:18) G (cid:48) t ( v, ξ v , S v ) + G (cid:48) x ( v, ξ v , S v ) + σ S v G (cid:48)(cid:48) y y ( v, ξ v , S v ) (cid:19) dv = 0 , since it is the continuous martingale. Then (cid:104) M, W (cid:105) t = σ (cid:104) ˜ M , W (cid:105) t = σ (cid:90) t G (cid:48) y ( v, ξ v , S v ) S v dv. Next, we find the formula for calculating martingale coefficients in (5) α t = σG (cid:48) y ( t, ξ t , S t ) S t . (15)Using(15) in formulas (9) and (8), we obtain the hedging strategy γ t = G (cid:48) y ( t, ξ t , S t ) . For the obtained strategy V = f . Moreover G ( t, x, y ) is the unique solutionof the following equation (cid:40) G (cid:48) t ( t, x, y ) + yG (cid:48) x ( t, x, y ) + σ y G (cid:48) yy ( t, x, y ) = 0 G (1 , x, y ) = ( x − K ) + . (16)3.2 With transaction costsSuppose that traders have to pay for a successful transaction some fee whichis proportional to the trading volume. We assume that the cost proportion κ n = κ n − α . To compensate the transaction cost Leland [6] suggested tocorrect the volatility. The new parameter ˆ σ we have to put in the PDE (16) edging problems for Asian options with transactions costs 11 and calculate the strategy again with a new volatility. Applying the Lelandapproach we modify the strategy as follows γ nt = n (cid:88) i =1 ˆ G (cid:48) y ( t j − , ξ t j − , S t j − ) χ ( t j − ,t j ] ( t ) , where ˆ G (cid:48) y ( t, x, y ) is the solution of the equation (16) with parameter ˆ σ . More-over ˆ G (cid:48) y ( t, x, y ) has the following formˆ G (cid:48) y ( t, x, y ) = (cid:90) ∞ b z ˆ q ( v, z ) dz, b = ( K − x ) + y here ˆ q ( v, z ) is a density of random variable ˜ η v with new parameter ˆ σ and givenby ˆ q ( v, z ) = E (cid:32) ϕ , (ˆ a ( t, z ))ˆ K ( v, ˆ a ( t, z )) (cid:33) , ˆ K ( v, ˆ a ( t, z )) = ˆ σ (cid:90) v u exp { ˆ σ ˜ W u − ˆ σ u σu ˆ a ( t, z ) } , ˜ W u = W u − uW . This form of density has been received in the article [17]. The portfolio valueat t with the initial capital V = ˆ G (0 , ξ , S ) has the form V nt = ˆ G (0 , ξ , S ) + (cid:90) t γ nu dS u − κ n J n , (17)where the total trading volume is given by J n = n (cid:88) j =1 S t j | γ nt j − γ nt j − | . In order to keep the hedging strategy it is necessary to satisfy the followingcondition V n P −−−−→ n →∞ f . For this we need to consider a hedging error V n − f . By Ito formula we have G ( t, ξ t , S t ) = G (0 , ξ , S ) + (cid:90) t (cid:18) G (cid:48) t ( u, ξ u , S u ) + G (cid:48) x ( u, ξ u , S u ) S u + σ S u G (cid:48)(cid:48) yy ( u, ξ u , S u ) (cid:19) du + (cid:90) t G (cid:48) y ( u, ξ u , S u ) σS u dW u , since G (cid:48) t ( t, ξ t , S t ) + G (cid:48) x ( t, ξ t , S t ) S t + σ S t G (cid:48)(cid:48) yy ( t, ξ t , S t ) = 0 then G ( t, ξ t , S t ) = G (0 , ξ , S )+ (cid:90) t G (cid:48) y ( u, ξ u , S u ) σS u dW u = G (0 , ξ , S )+ (cid:90) t G (cid:48) y ( u, ξ u , S u ) dS u . Condition of replication is V = f = (cid:18)(cid:90) S u du − K (cid:19) + . Since by the construction of the strategy V = G (1 , ξ , S ) then f = G (0 , ξ , S ) + (cid:90) t γ u dS u , where γ t = G (cid:48) y ( t, ξ t , S t ). Thus, taking into account (17) we have V n − f = ˆ G (0 , ξ , S ) − G (0 , ξ , S ) + (cid:90) t ( γ nu − γ u ) dS u − κ n J n . Since G (1 , ξ , S ) = ˆ G (1 , ξ , S ) = ( x − K ) + , i.e. the same boundary con-dition we can write the following equalityˆ G (0 , ξ , S ) − G (0 , ξ , S ) = ( G (1 , ξ , S ) − G (0 , ξ , S )) − ( ˆ G (1 , ξ , S ) − ˆ G (0 , ξ , S ))= (cid:90) G (cid:48) y ( u, ξ u , S u ) dS u + ˆ σ − σ (cid:90) ˆ G (cid:48)(cid:48) yy ( u, ξ u , S u ) S u du − (cid:90) ˆ G (cid:48) y ( u, ξ u , S u ) dS u , By Ito’s formula we haveˆ G (1 , ξ , S ) = ˆ G (0 , ξ , S )+ σ − ˆ σ (cid:90) ˆ G (cid:48)(cid:48) yy ( u, ξ u , S u ) S u du + (cid:90) ˆ G (cid:48) y ( u, ξ u , S u ) dS u and G (1 , ξ , S ) = G (0 , ξ , S ) + (cid:90) G (cid:48) y ( u, ξ u , S u ) dS u . Then V n − f = (cid:90) ( G (cid:48) y ( t, ξ t , S t ) − ˆ G (cid:48) y ( t, ξ t , S t )) dS t + ˆ σ − σ (cid:90) ˆ G (cid:48)(cid:48) yy ( t, ξ t , S t ) S t dt + (cid:90) ( γ nt − γ t ) dS t − κ n J n Finally we obtain V n − f = (cid:90) ( γ nt − ˆ γ t ) dS t + ˆ σ − σ (cid:90) ˆ G (cid:48)(cid:48) yy ( t, ξ t , S t ) S t dt − κ n J n , because (cid:82) G (cid:48) y ( t, ξ t , S t ) dS t = (cid:82) γ t dS t and ˆ γ t = ˆ G (cid:48) y ( t, ξ t , S t ). edging problems for Asian options with transactions costs 13 The option cost is defined as C = G (0 , , S ) = + ∞ (cid:90) b ( zS − K ) + q (1 , z ) dz, b = K/S . Recall that G ( t, x, y ) = E ( x + yη v − K ) + = + ∞ (cid:90) b ( x + yz − K ) + q ( v, z ) dz, where b = ( K − x ) + /y, v = 1 − t and q ( v, z ) is the density of the randomvariable η v = v (cid:90) exp (cid:26) σW u − σ u (cid:27) du and given by q ( v, z ) = E ϕ , ( a ( v, z )) K ( v, a ) . Here ϕ , ( a ) is the Gaussian density and a ( v, z ) has an implisit form z = (cid:90) exp (cid:26) σW u − σuW − σ u + σua ( v, z ) (cid:27) du. After we have introduced the transaction costs and changed the volatility asˆ σ = σ + σ (cid:114) π κ n √ n we obtain that the cost of option is equal (cid:98) C = + ∞ (cid:90) b ( zS − K ) + (cid:98) q (1 , z ) dz. There are three variants of changes in value of option.1) Case (cid:98) σ → σ if κ n = κ n − / and κ → (cid:98) C → C .
2) Case (cid:98) σ = σ + σ (cid:113) π κ n √ n with κ n = 1 /κ . In this case, the hedgingwill be, but the value of the option will increase by a constant σ (cid:113) π .3) Case (cid:98) σ → + ∞ if κ n = o ( n − α ) and α ≥ /
5. Then we obtain the strategy ”buy and hold” (cid:98) C → S . It is proved in Proposition 1.
Proposition 1
Let ρ ( u ) = exp (cid:110)(cid:98) σW u − (cid:98) σ u (cid:111) and ρ ( u ) → as (cid:98) σ → ∞ . Then (cid:98) η = (cid:90) ρ ( u ) du P −−−−→ (cid:98) σ →∞ and (cid:98) C −−−−→ (cid:98) σ →∞ S . Proof
First of all we will prove that (cid:98) η P −−−−→ (cid:98) σ →∞
0. Represent it like (cid:98) η = (cid:98) η , + (cid:98) η , , where (cid:98) η , = δ (cid:90) ρ ( u ) du, (cid:98) η , = (cid:90) δ ρ ( u ) du. We choose δ so that it tends to zero not very quickly, for example δ = 1 / √ (cid:98) σ .Then E (cid:98) η , = δ (cid:90) E ρ ( u ) du = δ −−−−→ (cid:98) σ →∞ . For the second termvwe can use estimatemax δ ≤ u ≤ ρ ( u ) ≤ exp (cid:26)(cid:98) σ max ≤ u ≤ W u − (cid:98) σ δ (cid:27) ≤ exp (cid:26)(cid:98) σ max ≤ u ≤ W u − (cid:98) σ / δ (cid:27) −−−−→ (cid:98) σ →∞ E (cid:98) η , −−−−→ (cid:98) σ →∞
0. We have (cid:98) C = E ( S (cid:98) η − K ) + = E ( S (cid:98) η − K ) + { (cid:98) η >K/S } + E ( S (cid:98) η − K ) + { (cid:98) η ≤ K/S } = E ( S (cid:98) η − K ) { (cid:98) η >K/S } = S E (cid:98) η { (cid:98) η >K/S } − K P ( (cid:98) η > K/S )The last probabilities tends to zero therefore (cid:98) C = S E (cid:98) η { (cid:98) η >K/S } . edging problems for Asian options with transactions costs 15 If represent the indicator as { (cid:98) η >K/S } = 1 − { (cid:98) η ≤ K/S } then E (cid:98) η { (cid:98) η >K/S } = E (cid:98) η − E (cid:98) η { (cid:98) η ≤ K/S } = 1 − E (cid:98) η { (cid:98) η ≤ K/S } Since (cid:98) η is bounded and goes to zero then by Lebesgue’s theorem on majorizedconvergence E (cid:98) η { (cid:98) η ≤ K/S } → q ( v, z ) To exlore the distribution of the random variable ˜ η v we introduce the notationof Brownian bridge. Definition 4
Coming from zero and coming to a ∈ R the Brownian bridge( B at ) ≤ t ≤ T is the Gaussian process such that B at = W t − tW + ta, where a – some constant.Conditional distributions are calculated for a fixed finite value of the Wienerprocess using this process, i.e. for any function L : C [0 , → R and for anyBorel set Γ P ( L ( W t ) ≤ t ≤ ∈ Γ | W t = a ) = P ( L ( B at ) ≤ t ≤ ∈ Γ ) . Proposition 2
For any ≤ t ≤ the random variable ˜ η v has a distributiondensity.Proof Let Q - some bounded function R → R . In our case E Q (˜ η v ) = E ( E ( Q (˜ η v ) | W )) = (cid:90) R Q ( F ( v, a )) ϕ ( a ) da, where F ( v, a ) = (cid:90) v exp { σW u − σuW − σ u/ σua } du, Next we make the change of variable z = F ( v, a ), i.e. we introduce the function a = a ( v, z ) as z = F ( v, a ( v, z )) . (18)It means that a (cid:48) z ( v, z ) = 1 K ( v, a ( v, z )) , where K ( v, a ) = F (cid:48) a ( v, a ) = σ (cid:90) v u exp { σW u − σuW − σ u/ σua } du. (19) Then E Q (˜ η v ) = (cid:90) ∞ Q ( z ) q ( v, z ) dz, here q ( v, z ) = E ϕ ( a ) K ( v, a ) , ϕ ( · ) ∼ N (0 , T ) . Thus the density of the random variable ˜ η v has the form q ( v, z ) = ϕ ( a ) K ( v, a ) . (20)Next we will use the following propositions. Proposition 3
For v ∗ = min( σ v, and some constants ˜ c > and κ > q ( v, z ) ≤ ˜ cσ v ∗ (cid:16) exp (cid:110) − κσ v (ln( z/v )) (cid:111) { z>v } + { z ≤ v } (cid:17) , | q z ( v, z ) | ≤ ˜ cσ v ∗ (cid:16) exp (cid:110) − κσ v (ln( z/v )) (cid:111) { z>v } + { z ≤ v } (cid:17) , Proposition 4
For v ∗ = min( σ v, and some constants ˜ c > and κ > | q v ( v, z ) | ≤ ˜ cσ v ∗ (cid:16) { z ≤ v } + exp (cid:110) − κσ v (ln( z/v )) (cid:111) { z>v } (cid:17) . See proofs in Appendix.
Recall that an option seller should increase volatility in order to compensatefor transaction costs. Choose a new volatility parameterˆ σ = σ + σ √ nκ n (cid:114) π , κ n = κ n − α . (21)Then the following theorem holds. Theorem 2
For α = 1 / in the equation (21) the portfolio value V n convergesin probability to the payout function f as n → ∞ .Proof We have an expression for a hedging error V n − f = (cid:90) ( γ nt − ˆ γ t ) dS t + ˆ σ − σ (cid:90) ˆ G (cid:48)(cid:48) yy ( t, ξ t , S t ) S t dt − κ n J n . Since γ nt = n (cid:88) i =1 ˆ G (cid:48) y ( t j − , ξ t j − , S t j − ) χ ( t j − ,t j ] ( t ) edging problems for Asian options with transactions costs 17 and ˆ γ t = ˆ G (cid:48) y ( t, ξ t , S t ) uniformly continuous on the segment [0 , n → ∞ and it remains only to verify thatˆ σ − σ (cid:90) ˆ G (cid:48)(cid:48) yy ( t, ξ t , S t ) S t dt − κ n J n P −−−−→ n →∞ . First, well evaluate κ n J n . We introduce the notation H ( t j , ξ t j , S t j ) = ˆ G y ( t j , ξ t j , S t j ) . Then κ n J n = κ n n (cid:88) j =1 S t j | H ( t j , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j − ) | . Add and subtract the term | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) | and rep-resent κ n J n as κ n J n = A (1) n + A (2) n , where A (1) n = κ n n (cid:88) j =1 S t j | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) | and A (2) n = κ n n (cid:88) j =1 S t j (cid:0) | H ( t j , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j − ) | − | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) | (cid:1) . Using the fact || x | − | y || ≤ | x − y | , we obtain | A (2) n | ≤ κ n n (cid:88) j =1 S t j | H ( t j , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j ) | = B n . In the section 8.1 we proved that P − lim n →∞ B n = 0, see Lemma 1. Thus, furtherwe need to consider only A (1) n . Recall that according to the Taylor formula wecan write H ( t j − , ξ t j − , S t j ) = H ( t j − , ξ t j − , S t j − )+ H y ( t j − , ξ t j − , S t j − )( S t j − S t j − )+ o ( n − )and represent A (1) n as A (1) n = κ n n (cid:88) j =1 S t j (cid:0) | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) | − | H y ( t j − , ξ t j − , S t j − ) || S t j − S t j − | (cid:1) + κ n n (cid:88) j =1 S t j | H y ( t j − , ξ t j − , S t j − ) || S t j − S t j − | . Next we denote D (2) n = κ n n (cid:88) j =1 S t j (cid:0) | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) | − | H y ( t j − , ξ t j − , S t j − ) || S t j − S t j − | (cid:1) and in Lemma 2 we will prove P − lim n →∞ D (2) n = 0. Thus, we have κ n J n ≈ κ n n (cid:88) j =1 S t j | H y ( t j − , ξ t j − , S t j − ) || S t j − S t j − | and by Lemma 3 we obtain that κ n J n P −−−−→ n →∞ κ n √ nσ (cid:114) π (cid:90) ˆ G (cid:48)(cid:48) yy ( t, ξ t , S t ) S t dt. Then ˆ σ − σ (cid:90) ˆ G (cid:48)(cid:48) yy ( t, ξ t , S t ) S t dt − κ n J n P −−−−→ n →∞ To build a hedging strategy, we need to calculate the coefficients ( α t ) ≤ t ≤ .First we compute the function G ( t, x, y ) , for this we simulate L random vari-ables η jt . We take the time step which is equal dt = 1 /N, where N – number of partitions. The mathematical expectation is calculatedby the Monte Carlo method. We get the computational formula η t = 1 − tN N (cid:88) k =1 exp (cid:26) σW k (1 − t ) − σ (1 − t ) k N (cid:27) . (22)Then for the function G ( t, x, y ) we obtain the expression G ( t, x, y ) ≈ L L (cid:88) j =1 ( x + yη jt − K ) + . (23)To calculate the partial derivative G (cid:48) y ( t, x, y ) we use the following formula ∂∂y G ( t, x, y ) = G ( t, x, y + δ ) − G ( t, x, y ) δ , δ = 0 , . (24)Before proceeding to the calculation of the coefficients ( α t ) ≤ t ≤ we write thecalculation formulas for ( ξ t ) ≤ t ≤ and ( S t ) ≤ t ≤ . ξ t = S tN N (cid:88) k =1 exp (cid:26) σW k ( t ) − σ tk N (cid:27) edging problems for Asian options with transactions costs 19 and S t = S exp (cid:26) σW t − σ t (cid:27) . Next, we find the coefficients ( α t ) ≤ t ≤ and build a strategy Π = ( β t , γ t ) ≤ t ≤ .Consider the implementation of the asset process and hedging strategiesfor σ = 0 .
05 with S = 100 and N = 100, we obtain the following results. Fig. 1
Asset price in a market with volatility σ = 0 . Fig. 2
The quantity ( γ t ) ≤ t ≤ of risky asset ( S t ) ≤ t ≤ in a hedging strategy for an Asianmarket option with volatility σ = 0 . Let us compare the value of the terminal portfolio and the payoff functionfor a different number of partitions N with parameters σ = 0 . S = 100, K = 50. Fig. 3
The quantity ( β t ) ≤ t ≤ of riskless asset ( B t ) ≤ t ≤ in a hedging strategy for anAsian market option with volatility σ = 0 . Table 1
The terminal portfolio X and the pauoff f N
20 50 100 200 500 1000 X f C ≈ L L (cid:88) j =1 (cid:32) S N (cid:88) k =1 dt ∗ exp { σW k (1) − σ t k / } (cid:33) + . Consider the simulation results for S = 100, t ∈ [0 , L = 500000 and n = 1000.In Fig. 4time changes on the abscissa axis, and corresponding values onthe ordinate axis. We see that at every moment in time, the trajectory of theoption value almost repeats the trajectory of investor capital, which is naturalfor the hedging task. The size of the terminal portfolio exceeds the payofffunction, which confirms that the strategy is hedging.Investigating the behavior of the option value depending on the initial stockprice S , strike price K and volatility σ , we obtain the following results. Table 2
The dependence of the value of an Asian option on the volatility parameter with K = S σ C edging problems for Asian options with transactions costs 21 Fig. 4
Graphs of option value, investor’s capital and Asian option payoff function.
Table 3
The dependence of the value of an Asian option on the volatility parameter with K = S / σ C σ = 0 .
05 and S = 100 , K = 70 for a different number of portfoliorevisions, we obtain the following result.We have investigated the behavior of the hedging error V n − f withdifferent portfolio revision numbers ”n” and different parameters σ . Let S = K = 100, κ = 0 . Table 4
The hedging error when σ = 0 . n
20 50 100 200 500 1000 V n − f -0.3264 -0.1479 -0.0693 -0.0097 0.0026 0.0061An analysis of the numerical results showed that the value of the optionincreases if the strike price is less than the initial value of the stock. Volatil-ity also affects the value of the option, it increases with increasing volatility,but not significantly. The portfolio revealed an inverse proportion between thenumber of risky and risk-free assets. As a result of the experiment, the influ- Fig. 5
The value of the option in the market with transaction costs and without costs.
Table 5
The hedging error when σ = 0 . n
20 50 100 200 500 1000 V n − f -0.7106 -0.4065 -0.3307 -0.1938 -0.0801 -0.0213ence of the number of revisions of the portfolio n on the value of the option infinancial markets with transaction costs was confirmed, it was revealed thatwith the growth of n the value of the option also increases. The cost of anoption in financial markets without costs does not depend on the number ofrevisions. Also, a numerical experiment showed that in markets with trans-action costs, the hedging error decreases with an increase in the number ofportfolio revisions. It was also revealed that hedging error is greater with highmarket volatility. Lemma 1
Let B n = κ n n (cid:88) j =1 S t j | H ( t j , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j ) | , (25) κ n −→ n →∞ then P − lim n →∞ B n = 0 (26) edging problems for Asian options with transactions costs 23 Proof
We can represent H ( t j , ξ t j , S t j ) = H ( t j , ξ t j , S t j ) − H ( t j − , ξ t j , S t j ) (cid:124) (cid:123)(cid:122) (cid:125) h j + H ( t j − , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j ) (cid:124) (cid:123)(cid:122) (cid:125) h (1) j + H ( t j − , ξ t j − , S t j ) . Then B n = D ( t ) n + D ( x ) n = κ n n (cid:88) j =1 d ( t ) j + κ n n (cid:88) j =1 d ( x ) j , where d ( t ) j = S t j h j , d ( x ) j = S t j h (1) j . It is necessary to show that ∀ µ > n →∞ P ( D ( t ) n > µ ) = 0 and lim n →∞ P ( D ( x ) n > µ ) = 0 . (27)Recall that H ( t, x, y ) = ˆ G y ( t, x, y ) = ∞ (cid:90) b z ˆ q ( v, z ) dz, where ˆ q ( v, z ) – the density of the random densityˆ η v = v (cid:90) exp { ˆ σW u − ˆ σ u/ } du and b = (cid:18) K − xy (cid:19) + . We introduce the stopping time τ = inf { t > ξ t ≥ K } ∧ . (28)Clear that always 0 < τ ≤ τ , all coefficients b = 0. Tocompensate b and S t we introduce the following sets Γ ε,M = { ξ > K } ∩ { τ ≤ − ε } ∩ { M − ≤ min ≤ t ≤ S t ≤ max ≤ t ≤ S t ≤ M } ˜ Γ δ,M = { ξ ≤ K − δ } ∩ { M − ≤ min ≤ t ≤ S t ≤ max ≤ t ≤ S t ≤ M } moreover lim M →∞ lim ε → P ( Γ cε,M ) = 0 , lim M →∞ lim δ → P ( ˜ Γ cδ,M ) = 0 . (29) Then we represent probabilities as P ( D ( t ) n > µ ) = P ( D ( t ) n > µ, ξ > K ) + P ( D ( t ) n > µ, ξ ≤ K ) ≤ P ( D ( t ) n > µ, Γ ε,M ) + P ( Γ cε,M ) + P ( D ( t ) n > µ, ˜ Γ δ,M ) + P ( ˜ Γ cδ,M )(30)and P ( D ( x ) n > µ ) = P ( D ( t ) n > µ, ξ > K ) + P ( D ( x ) n > µ, ξ ≤ K ) ≤ P ( D ( x ) n > µ, Γ ε,M ) + P ( Γ cε,M ) + P ( D ( x ) n > µ, ˜ Γ δ,M ) + P ( ˜ Γ cδ,M )(31)Taking into account Proposition 5 and Proposition 6 we obtain the equalities(27). Proposition 5
For fixed ε > and M > n →∞ P ( D ( t ) n > µ, Γ ε,M ) = 0; (32)lim n →∞ P ( D ( x ) n > µ, Γ ε,M ) = 0 . (33) Here Γ ε,M = { ξ > K } ∩ { τ ≤ − ε } ∩ { M − ≤ min ≤ t ≤ S t ≤ max ≤ t ≤ S t ≤ M } Proof
First we divide D ( t ) n and D ( x ) n into two amounts D ( t ) n = κ n n (cid:88) j =1 d ( t ) j + κ n n (cid:88) j = n +1 d ( t ) j ,D ( x ) n = κ n n (cid:88) j =1 d ( x ) j + κ n n (cid:88) j = n +1 d ( x ) j , n = [(1 − ε ) n ] . We choose n from the condition t n ≥ − ε . In this case, we have( t j ) nj = n +1 > τ and ( ξ t j ) nj = n +1 ≥ K. Therefore b j = 0 , j = n + 1 , n and since H ( t, x, y ) = ∞ (cid:90) b z ˆ q ( v, z ) dz = ∞ (cid:90) z ˆ q ( v, z ) dz = E ˆ η v = E v (cid:90) exp { ˆ σW u − ˆ σ u/ } du = v (cid:90) E exp { ˆ σW u − ˆ σ u/ } du = v edging problems for Asian options with transactions costs 25 we obtain d ( t ) j = S t j | H ( t j , ξ t j , S t j ) − H ( t j − , ξ t j , S t j ) | ≤ M ∆v j ≤ Mn , j = n + 1 , κ n n (cid:88) j = n +1 d ( t ) j −→ n →∞ d ( x ) j = S t j | H ( t j − , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j ) | = 0 , j = n + 1 , . Thus, on the set Γ ε,M we have D ( t ) n = κ n n (cid:88) j =1 d ( t ) j ; D ( x ) n = κ n n (cid:88) j =1 d ( x ) j . Next we evaluate d ( t ) j on the interval [ t , t n ], on which all t j < − ε . d ( t ) j = S t j (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:90) b tj z ˆ q ( v j , z ) dz − ∞ (cid:90) b tj − z ˆ q ( v j − , z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ M ∞ (cid:90) z | ˆ q ( v j , z ) − ˆ q ( v j − , z ) | dz ≤ M ∞ (cid:90) z (cid:12)(cid:12)(cid:12)(cid:12) v j (cid:90) v j − ˆ q v ( u, z ) du (cid:12)(cid:12)(cid:12)(cid:12) dz ≤ (cid:90) z (cid:12)(cid:12)(cid:12)(cid:12) v j (cid:90) v j − ˆ q v ( u, z ) du (cid:12)(cid:12)(cid:12)(cid:12) dz + ∞ (cid:90) z (cid:12)(cid:12)(cid:12)(cid:12) v j (cid:90) v j − ˆ q v ( u, z ) du (cid:12)(cid:12)(cid:12)(cid:12) dz. Naking into account Proposition 4, we can estimate (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) v j v j − ˆ q v ( u, z ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ cε (cid:90) v j − v j exp {− θ (ln( z/v )) } du, because we can evaluate 1 /v ≤
1. If 0 < z ≤
1, then we can use a uniformestimate for ˆ q v ( v, z ) ˆ q v ( v, z ) ≤ cv ≤ cε and if z >
1, we use inequality ln( z/v ) ≥ ln( z/ε ). Thenˆ q v ( v, z ) ≤ cε exp (cid:8) − θ (ln( z/ε )) (cid:9) . Therefore, d ( t ) j ≤ M c nε + M cnε ∞ (cid:90) z exp (cid:8) − θ (ln( z/ε )) (cid:9) dz, and the integral I = ∞ (cid:90) z exp (cid:8) − θ (ln( z/ε )) (cid:9) dz is converge. Then D ( t ) n ≤ M κ n n (cid:88) j =1 d ( t ) j ≤ κ n n (cid:88) j =1 M cnε (1 / I ) = κ n M cn nε (1 / I ) P −−−−→ n →∞ n →∞ P ( D ( t ) n > µ, Γ ε,M ) = 0 . Let’s estimate d ( x ) j on the interval [ t , t n ]. Since H ( t, x, y ) = ∞ (cid:82) b z ˆ q ( v, z ) dz ,then H (cid:48) x ( t, x, y )( t, x, y ) = − by ˆ q ( v, b )and we can evaluate this derivative using Proposition 3. | H x ( t, x, y ) | ≤ bq ( v, b ) y ≤ KM cε We used a uniform estimate for ˆ q ( v, b ). Thus the derivative H x ( t, x, y ) is uni-formly bounded. This means that the function H ( t, x, y ) satisfies the conditionsof Lipschitz and moreover | H ( t j − , ξ t j , S t j ) − H ( t j − , ξ t j − , S t j ) | ≤ | H x ( t, x, y ) || ξ t j − ξ t j − | ≤ KM ε | ξ t j − ξ t j − |≤ KM ε (cid:90) t j t j − S u du ≤ KM ε ∆t j = KM nε . Then D ( x ) n ≤ κ n M n (cid:88) j =1 KM nε ≤ κ n KM cn nε P −−−−→ n →∞ . that is lim n →∞ P ( D ( x ) n > µ, Γ ε,M ) = 0 . Proposition 6
For fixed δ > and M > n →∞ P ( D ( t ) n > µ, ˜ Γ δ,M ) = 0; (34)lim n →∞ P ( D ( x ) n > µ, ˜ Γ δ,M ) = 0 . (35) Here ˜ Γ δ,M = { ξ ≤ K − δ } ∩ { M − ≤ min ≤ t ≤ S t ≤ max ≤ t ≤ S t ≤ M } edging problems for Asian options with transactions costs 27 Proof
We can use the following estimate D ( x ) n ≤ κ n M n (cid:88) j =1 | H x ( t, x, y ) || ξ t j − ξ t j − | , since | H x ( t, x, y ) | ≤ b ˆ q ( v, b ) y ≤ KM cv (cid:0) exp (cid:8) − θ (ln( b/v )) (cid:9) { z>v } + { z ≤ v } (cid:1) . It’s obvious that ξ t j ≤ ξ ≤ K − δ on the set ˜ Γ δ,M , so b ∗ = δM ≤ b t j ≤ KM.
Similar to the proof of the Proposition 6 we split D ( x ) n into two amounts D ( x ) n = κ n n (cid:88) j =1 d ( x ) j + κ n n (cid:88) j = n +1 d ( x ) j . If 0 < v < b ∗ , i.e. 1 − b ∗ < t <
1, then we use estimate | H x ( t, x, y ) | ≤ KM cv exp (cid:8) − θ (ln( b ∗ /v )) (cid:9) . If b ∗ < v <
1, i.e. 0 ≤ t ≤ − b ∗ , then | H x ( t, x, y ) | ≤ KM cb ∗ . Thus, D ( x ) n ≤ κ n cM Kb ∗ n (cid:88) j =1 | ξ t j − ξ t j − | + M Kκ n n (cid:88) j = n +1 cv j exp {− θ (ln( b ∗ /v j )) }| ξ t j − ξ t j − |≤ cM Kn κ n nb ∗ + M Kκ n n (cid:88) j = n +1 cv j exp {− θ (ln( b ∗ /v j )) } ∆t j Note that n (cid:88) j = n +1 cv j exp {− θ (ln( b ∗ /v j )) } ∆t j −−−−→ n →∞ (cid:90) − b ∗ cv exp {− θ (ln( b ∗ /v )) } dt. Consider J = (cid:90) − b ∗ cv exp {− θ (ln( b ∗ /v )) } dt. Given that v = 1 − t , make a variable change˜ a = 11 − t , then dt = d ˜ a/ ˜ a and we obtain J = (cid:90) ∞ /b ∗ ˜ a exp {− θ (ln( b ∗ ˜ a )) } d ˜ a = 1 b ∗ (cid:90) ∞ exp {− θ (ln ˆ a ) } d ˆ a = 1 b ∗ (cid:90) ∞ exp {− θ (ln y ) + y } dy The last integral I = b ∗ (cid:82) ∞ exp {− θ (ln y ) + y } dy converges. Thus, D ( x ) n ≤ κ n cKM (cid:18) n nb ∗ + I (cid:19) P −−−−→ n →∞ n →∞ P ( D ( x ) n > µ, ˜ Γ δ,M ) = 0 . Consider D ( t ) n = κ n n (cid:88) j =1 d ( t ) j . Recall that on the set ˜ Γ δ,M all ξ t j ≤ ξ ≤ K − δ , then b t j = K − ξ t j S t j ≥ b ∗ , δ = δM . Similarly, we evaluate d ( t ) j . d ( t ) j = S t j (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ∞ b tj z ˆ q ( v j , z ) dz − (cid:90) ∞ b tj − z ˆ q ( v j − , z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ M (cid:90) ∞ b ∗ z (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) v j v j − q v ( u, z ) du (cid:12)(cid:12)(cid:12)(cid:12) dz ≤ M (cid:90) ∞ b ∗ z (cid:32)(cid:90) v j − v j | ˆ q v ( u, z ) | du (cid:33) dz = M (cid:90) v j − v j (cid:18)(cid:90) ∞ b ∗ z | ˆ q v ( u, z ) | dz (cid:19) du Next, we use the estimate for the derivative of the density ˆ q v ( u, z ), obtainedin Proposition 4, (cid:90) v j − v j (cid:18)(cid:90) ∞ b ∗ z | ˆ q v ( u, z ) | dz (cid:19) du ≤ (cid:90) v j − v j cu (cid:18)(cid:90) ∞ b ∗ z exp {− θ (ln( z/u )) } dz (cid:19) du Make the change of variable y = z/u (cid:90) v j − v j (cid:18)(cid:90) ∞ b ∗ z | ˆ q v ( u, z ) | dz (cid:19) du ≤ (cid:90) v j − v j cu (cid:32)(cid:90) ∞ b ∗ /u y exp {− θ (ln y ) } dy (cid:33) du = (cid:90) v j − v j cu I ( u ) du edging problems for Asian options with transactions costs 29 We consider separately the integral I ( u ) = (cid:90) ∞ b ∗ /u y exp {− θ (ln y ) } dy. We use the change of variable x = ln y and obtain I ( u ) = (cid:90) ∞ ln( b ∗ /u ) exp {− θx + 2 x } dx. Select a parabola so that c ∗ = sup x (exp {− θx / x } ), then we can write anestimate I ( u ) ≤ c ∗ (cid:90) ∞ ln( b ∗ /u ) exp {− θx / } dx = c ∗ (cid:90) ∞ ln( b ∗ /u ) exp (cid:8) − θx / (cid:9) exp (cid:8) − θx / (cid:9) dx ≤ c ∗ exp (cid:8) − θ (ln( b ∗ /u )) / (cid:9) (cid:90) ∞−∞ exp (cid:8) − θx / (cid:9) dx If u ≥ b ∗ , then I ( v ) ≤ c ∗ (cid:90) + ∞−∞ exp (cid:8) − θx / (cid:9) dx. If 0 < u < b ∗ , then ln( b ∗ /u ) > I ( u ) ≤ c ∗ exp (cid:8) − θ (ln( b ∗ /u )) / (cid:9) (cid:90) ∞ exp (cid:8) − θx / (cid:9) dx. Then d ( t ) j ≤ M (cid:90) v j − v j cu I ( u ) du ≤ M (cid:90) cu I ( u ) du ≤ M J (cid:90) b ∗ exp (cid:8) − θ (ln( b ∗ /u )) / (cid:9) cu du + M J (cid:90) b ∗ cu du, with constants J = c ∗ (cid:90) + ∞ exp (cid:8) − θx / (cid:9) dx ; J = c ∗ (cid:90) + ∞−∞ exp (cid:8) − θx / (cid:9) dx. It is clear that the integral I = (cid:90) b ∗ cu du converges. Consider the integral I = (cid:90) b ∗ exp (cid:8) − θ (ln( b ∗ /u )) / (cid:9) cu du. Let’s make a change y = b ∗ /u , then a change z = ln y , so I = cb ∗ (cid:90) ∞ exp (cid:8) − θ (ln y ) / (cid:9) dy = cb ∗ (cid:90) ∞ exp (cid:8) − θz / z (cid:9) dz < + ∞ Thus, it was shown that d ( t ) j is limited by some constant d ( t ) j ≤ l ( M, J , J , I , I ) . Therefore, D ( t ) n = κ n n (cid:88) j =1 d ( t ) j ≤ κ n l ( M, J , J , I , I ) n P −−−−→ n →∞ n →∞ P ( D ( t ) n > µ, ˜ Γ δ,M ) = 0 . . Lemma 2
Let D ( y ) n = κ n n (cid:88) j =1 d ( y ) j , where d ( y ) j = S t j | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) − H y ( t j − , ξ t j − , S t j − )( S t j − S t j − ) | . Then P − lim n →∞ D ( y ) n = 0 . (36) Proof
Represent d ( y ) j as d ( y ) j ≤ (cid:12)(cid:12)(cid:12)(cid:12) S t j S tj (cid:90) S tj − ( H y ( t j − , ξ t j − , u ) − H y ( t j − , ξ t j − , S t j − )) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) S t j S tj (cid:90) S tj − u (cid:90) S tj − H yy ( t j − , ξ t j − , a ) da du (cid:12)(cid:12)(cid:12)(cid:12) . (37)Let’s find the derivative H yy ( t, x, y ). Recall that H ( t, x, y ) = ∞ (cid:90) b z ˆ q ( v, z ) dz b = ( K − x ) + y . If x ≥ K , then H ( t, x, y ) = 1 − t and H y ( t, x, y ) = H yy ( t, x, y ) = 0. If x < K ,then H y ( t, x, y ) = − b ˆ q ( v, b ) b (cid:48) y = b y ˆ q ( v, b ) = ( K − x ) y ˆ q ( v, b ) edging problems for Asian options with transactions costs 31 and H yy ( t, x, y ) = − K − x ) y ˆ q ( v, b ) + ( K − x ) y ˆ q z ( v, b ) b (cid:48) y = − b y ˆ q ( v, b ) − b y ˆ q z ( v, b )It is necessary to show that for ∀ µ > n →∞ P ( D ( y ) n > µ ) = 0 . As before in the proof of Lemma 1 we introduce the stopping time τ and sets Γ ε,M = { ξ > K } ∩ { τ ≤ − ε } ∩ { M − ≤ min ≤ t ≤ S t ≤ max ≤ t ≤ S t ≤ M } Γ δ,M = { ξ ≤ K − δ } ∩ { M − ≤ min ≤ t ≤ S t ≤ max ≤ t ≤ S t ≤ M } , with lim M →∞ lim ε → P ( Γ cε,M ) = 0 , lim M →∞ lim δ → P ( Γ cδ,M ) = 0 . (38)Represent the probability P ( D ( y ) n > µ ) as P ( D ( y ) n > µ ) = P ( D ( y ) n > µ, ξ > K ) + P ( D ( y ) n > µ, ξ ≤ K ) ≤ P ( D ( y ) n > µ, Γ ε,M ) + P ( Γ cε,M ) + P ( D ( y ) n > µ, Γ δ,M ) + P ( Γ cδ,M ) . Consider P ( D ( y ) n > µ, Γ ε,M ). Analogically we split D ( y ) n into two sums D ( y ) n = κ n n (cid:88) j =1 d ( y ) j + κ n n (cid:88) j = n +1 d ( y ) j , where n = [(1 − ε ) n ], i.e. t n ≤ − ε . Starting from j = n + 1 all moments t j > τ and ξ t j ≥ K . So, b t j = 0 and d ( y ) j = 0. Thus, D ( y ) n = κ n n (cid:88) j =1 d (2) j ≤ M κ n n (cid:88) j =1 | H ( t j − , ξ t j − , S t j ) − H ( t j − , ξ t j − , S t j − ) − H y ( t j − , ξ t j − , S t j − )( S t j − S t j − ) |≤ M κ n n (cid:88) j =1 (cid:12)(cid:12)(cid:12)(cid:12) S tj (cid:90) S tj − u (cid:90) S tj − H yy ( t j − , ξ t j − , a ) da du (cid:12)(cid:12)(cid:12)(cid:12) . Using Proposition 3, we can estimate 1 /v ≤ | H yy ( t, x, y ) | = (cid:12)(cid:12)(cid:12)(cid:12) b y ˆ q ( v, b ) + b y ˆ q z ( v, b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) b y cv exp {− θ (ln( b/v )) } + b y cv exp {− θ (ln( b/v )) } (cid:12)(cid:12)(cid:12)(cid:12) . On the set Γ ε,M we change the exponent by 1 and v = 1 − t by ε , also takeinto account that b ≤ K/y ≤ KM . For some constant c ∗ we have | H yy ( t, x, y ) | ≤ cy ε + cy ε ≤ c ∗ M ε . Then d ( y ) j ≤ M (cid:12)(cid:12)(cid:12)(cid:12) S tj (cid:90) S tj − u (cid:90) S tj − H yy ( t j − , ξ t j − , a ) da du (cid:12)(cid:12)(cid:12)(cid:12) ≤ M c ∗ ε (cid:12)(cid:12)(cid:12)(cid:12) S tj (cid:90) S tj − ( u − S t j − ) du (cid:12)(cid:12)(cid:12)(cid:12) = cM ε | S t j − S t j − | Note that E ( S t j − S t j − ) = σ E t j (cid:90) t j − S u du ≤ σ e σ ( t j − t j − ) = cn , Since E S u = E exp { σW u − σ u } = e σ u ≤ e σ . Therefore, D ( y ) n ≤ M κ n n (cid:88) j =1 cM ε | S t j − S t j − | Let ˜ c = cM /ε , then P (cid:16) D ( y ) n > µ, Γ ε,M (cid:17) ≤ P κ n ˜ c n (cid:88) j =1 | S t j − S t j − | > µ, Γ ε,M ≤ P κ n ˜ c n (cid:88) j =1 | S t j − S t j − | > µ edging problems for Asian options with transactions costs 33 Further, by Chebyshev’s inequality, we obtain P ( κ n ˜ c n (cid:88) j =1 | S t j − S t j − | > µ ) ≤ κ n ˜ c (cid:80) n j =1 E | S t j − S t j − | µ P −−−−→ n →∞ . Thus, lim n →∞ P ( D ( y ) n > µ, Γ ε,M ) = 0 . (39)Consider P ( D ( y ) n > µ, Γ δ,M ). On the set Γ δ,M b ∗ = δM ≤ b ≤ KM, then | H yy ( t, x, y ) | = (cid:12)(cid:12)(cid:12)(cid:12) b y ˆ q ( v, b ) + b y ˆ q z ( v, b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K M | ˆ q ( v, b ) + ˆ q z ( v, b ) |≤ K M (cid:12)(cid:12)(cid:12)(cid:12) cv exp (cid:8) − θ (ln( b/v )) (cid:9) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K M (cid:12)(cid:12)(cid:12)(cid:12) cv exp (cid:8) − θ (ln( b ∗ /v )) (cid:9) (cid:12)(cid:12)(cid:12)(cid:12) . Let ˜ c = K M c . By making a variable change z = ln( b ∗ /v ), we obtain | H yy ( t, x, y ) | = ˜ cb ∗ (cid:18) b ∗ v (cid:19) exp (cid:8) − θ (ln( b ∗ /v )) (cid:9) ≤ ˜ cb ∗ exp {− θz + 4 z }≤ ˜ cb ∗ exp { sup z ∈ R ( − θz + 4 z ) } = ˜ cb ∗ Then d ( y ) j ≤ ˜ cb ∗ | S t j − S t j − | So, D ( y ) n ≤ κ n ˜ cb ∗ n (cid:88) j =1 | S t j − S t j − | Similarly to the first part of the proof by Chebyshev inequality, we obtain P ( D ( y ) n > µ, Γ δ,M ) ≤ κ n c (cid:80) nj =1 E | S t j − S t j − | µ −−−−→ n →∞ . Thus, lim n →∞ P ( D ( y ) n > µ, Γ δ,M ) = 0 . (40)As a result, from the expressions (38),(39) and (40) we obtain (36). Lemma 3
Let β ( t ) – continuous consistent function almost surely. Then √ n n (cid:88) j =1 β ( t j − ) | S t j − S t j − | P −−−−→ n →∞ (cid:114) π σ (cid:90) S t β ( t ) dt. Proof
All auxiliary constants will be denoted by the letter c . We single outthe martingale term1 √ n n (cid:88) j =1 β ( t j − ) | S t j − S t j − | = 1 √ n n (cid:88) j =1 β ( t j − ) E (cid:18) | S t j − S t j − | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) + 1 √ n M n , (41)where M n = n (cid:88) j =1 η j ,η j = β ( t j − ) | S t j − S t j − | − E (cid:18) β ( t j − ) | S t j − S t j − | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) . In Proposition 7 we have established that1 √ n M n P −−−−→ n →∞ . Now we consider the first term of the equality (41). It’s clear that S t j − S t j − = σ (cid:90) t j t j − S u dW u , so β ( t j − ) E (cid:18) | S t j − S t j − | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = σβ ( t j − ) E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t j t j − S u dW u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33) = σβ ( t j − ) E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t j t j − (cid:2) S t j − + ( S u − S t j − ) (cid:3) dW u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33) = σβ ( t j − ) E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) S t j − ( W t j − W t j − ) + (cid:90) t j t j − ( S u − S t j − ) dW u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33) . Introduse the notation g j = (cid:90) t j t j − ( S u − S t j − ) dW u . (42)Then β ( t j − ) E (cid:18) | S t j − S t j − | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = σβ ( t j − ) E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) S t j − ∆W t j + g j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = σβ ( t j − ) E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) S t j − ∆W t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) + σβ ( t j − ) E (cid:18) ν j (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = σ (cid:114) π β ( t j − ) | S t j − | √ n + σβ ( t j − ) E (cid:18) ν j (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) , (43) edging problems for Asian options with transactions costs 35 where ν j = (cid:12)(cid:12)(cid:12)(cid:12) S t j − ∆W t j + g j (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) S t j − ∆W t j (cid:12)(cid:12)(cid:12)(cid:12) . By module property (cid:12)(cid:12) | a | − | b | (cid:12)(cid:12) ≤ | a − b | we have | ν j | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S t j − ∆W t j + g j (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) S t j − ∆W t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | g j | . (44)By the equality (42) and Jensen’s inequalities we get E (cid:0) | g j | (cid:12)(cid:12) F t j − (cid:1) = E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t j t j − ( S u − S t j − ) dW u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33) = (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t j t j − ( S u − S t j − ) dW u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33)(cid:33) ≤ (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) E (cid:32)(cid:90) t j t j − ( S u − S t j − ) dW u (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) F t j − = (cid:118)(cid:117)(cid:117)(cid:116) E (cid:32)(cid:90) t j t j − ( S u − S t j − ) du (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33) = (cid:115)(cid:90) t j t j − E (cid:18) ( S u − S t j − ) (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) du. For the integrand we can write the estimate E (cid:18) ( S u − S t j − ) (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) ≤ c S t j − ( u − t j − ) (45)This estimate is valid by virtue of the following reasoning. E (cid:18) ( S u − S t j − ) (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = E (cid:32) σ (cid:90) ut j − S v dW v (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) F t j − = σ E (cid:32)(cid:90) ut j − S v dv (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:33) = σ (cid:90) ut j − E (cid:18) S v (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) dv. (46)By definition of a risky asset in this model, we have S v = S exp { σW v − vσ / } . Obviously, the following equality holds. S v = S t j − exp { σ ( W v − W t j − ) − σ ( v − t j − ) } . Then E (cid:18) S v (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = S t j − E (cid:18) exp { σ ( W v − W t j − ) − σ ( v − t j − ) } (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) = S t j − E (cid:0) exp { σ (cid:112) v − t j − η − σ ( v − t j − ) } (cid:1) = S t j − exp { σ ( v − t j − ) }≤ S t j − exp { σ ( t j − t j − ) } ≤ S t j − e, where η ∼ N (0 , E (cid:0) | g j | (cid:12)(cid:12) F t j − (cid:1) ≤ c (cid:115) S t j − (cid:90) t j t j − ( u − t j − ) du = cn S t j − . (47)Taking into account the equality (43), rewrite (41) without martingale term1 √ n n (cid:88) j =1 β ( t j − ) | S t j − S t j − | = σ (cid:114) π n (cid:88) j =1 β ( t j − ) 1 n S t j − + σ √ n n (cid:88) j =1 β ( t j − ) E (cid:18) | ν j | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) . We note here that due to inequalities (44) and (47) σ √ n n (cid:88) j =1 β ( t j − ) E (cid:18) | ν j | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) ≤ σ √ n n (cid:88) j =1 | β ( t j − ) | E (cid:18) | g j | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) ≤ cσn √ n n (cid:88) j =1 | β ( t j − ) | S t j − . Since 1 n n (cid:88) j =1 | β ( t j − ) | S t j − −−−−→ n →∞ (cid:90) | β ( t ) | S t dt .. , then cσ √ n (cid:90) | β ( t ) | S t dt P −−−−→ n →∞ . Hence, , in respect Proposition 7, we obtain1 √ n n (cid:88) j =1 β ( t j − ) | S t j − S t j − | P −−−−→ n →∞ σ (cid:114) π (cid:90) β ( t ) S t dt. The lemma is proved.
Proposition 7
Let M n = (cid:80) nj =1 η j , where η j = β ( t j − ) | S t j − S t j − | − E (cid:18) β ( t j − ) | S t j − S t j − | (cid:12)(cid:12)(cid:12)(cid:12) F t j − (cid:19) . For an arbitrary continuous consistent function β ( t ) , ≤ t ≤ √ n M n P −−−−→ n →∞ . (48) Proof
We consider two cases. In the first case, suppose that for some constant L sup ≤ t ≤ | β ( t ) | ≤ L. By virtue of the martingality property, we obtain E ( M n ) = E n (cid:88) j =1 η j = n (cid:88) j =1 EE ( η j |F t j − ) = E n (cid:88) j =1 E ( η j |F t j − ) . (49) edging problems for Asian options with transactions costs 37 Introduce the notation η j = β ( t j − ) ς j , where ς j = | S t j − S t j − | − E (cid:0) | S t j − S t j − | (cid:12)(cid:12) F t j − (cid:1) . Then E ( η j |F t j − ) = β ( t j − ) E ( ς j |F t j − ) ≤ L E ( | S t j − S t j − | (cid:12)(cid:12) F t j − ) ≤ L S t j − ∆ π . Then we can evaluate the equality (49) E ( M n ) ≤ L E n (cid:88) j =1 S t j − ∆ π = L πn n (cid:88) j =1 E S t j − ≤ c . Therefore, we obtain E √ n | M n | = 1 √ n E | M n | ≤ √ n ( E M n ) / ≤ c √ n P −−−−→ n →∞ β ( t ) – continuous function. We introducethe stopping time τ L = inf { t ≥ | β ( t ) | ≥ L } ∧ . For continuous function β ( t ) the following equality holds P ( max ≤ t ≤ | β ( t ) | < ∞ ) = 1 . It means that β ( t ) is limited and therefore, P ( τ L < −−−−→ L →∞ . (51)To prove convergence (48), it must be shown that the next probability is zero. P (cid:18) √ n | M n | > δ (cid:19) = P (cid:18) √ n | M n | > δ, τ L = 1 (cid:19) + P (cid:18) √ n | M n | > δ, τ L < (cid:19) . On the set { τ L = 1 } β ( t ) = ˜ β L ( t ) , where ˜ β L ( t ) = β ( t ) {| β ( t ) |≤ L } and M n = ˜ M ( L ) n = n (cid:88) j =1 ˜ β L ( t j − ) ς j . Thus, we found ourselves in the conditions of the first case considered above.Therefore, P (cid:18) √ n | M n | > δ, τ L = 1 (cid:19) = P (cid:18) √ n | ˜ M ( L ) n | > δ, τ L = 1 (cid:19) ≤ P (cid:18) √ n | ˜ M ( L ) n | > δ (cid:19) ≤ δ √ n E | ˜ M ( L ) n | . By (50), last expectation tends to zero. Then ∀ L > n →∞ P (cid:18) √ n | M n | > δ (cid:19) = lim n →∞ (cid:18) P (cid:18) √ n | ˜ M ( L ) n | > δ (cid:19) + P ( τ L < (cid:19) ≤ P ( τ L < . Pass to the limit by L → ∞ , with (51) we obtain (48).8.2 Proof of the density properties Proof (Proposition 3)We need to look at the asymptotic behavior q ( v, z ) when v → z > q ( v, z ), it is necessary toestimate the function K ( v, a ) from below. K ( v, a ) = σ (cid:90) v u exp { σW u − σuW − σ u/ σua ( v, z ) } du = σ (cid:90) v u exp { σW u + γu } du, where γ = σa ( v, z ) − σW − σ / . Next we make the change of variables s = uσ to use the scale invarianceproperty of Wiener process. Then1 σ (cid:90) σ v s exp { σW s/σ + γs/σ } ds = 1 σ (cid:90) σ v s exp { ¯ W s + γs/σ } ds, here ¯ W s = σW s/σ is also the Wiener process. Further we suppose that σ ≥ K ( v, a ) as follow K ( v, a ) ≥ σ (cid:90) v ∗ s exp { ¯ W s − W s/σ − s/ sa ( v, z ) /σ } ds ≥ σ exp {− max ≤ s ≤ | ¯ W s | − | W | − | a ( v, z ) |} v ∗ where v ∗ = min( σ v, β = max ≤ s ≤ | ¯ W s | + | W | then K ( v, a ) ≥ v ∗ σ exp {− β − | a ( v, z ) |} . Substituting this estimate in q ( v, z ), we have q ( v, z ) ≤ cσ v ∗ E exp { β − a ( v, z ) / } , (52)where c = e/ √ π and exp { sup a ( | a | − a / } = e . Next obtain the lower boundfor a ( v, z ).The function a ( v, z ) is specified implicitly as follows z = (cid:90) v exp { σW u − σuW − σ u/ σua ( v, z ) } du, v = 1 − t. edging problems for Asian options with transactions costs 39 Consider β ∗ = max ≤ u ≤ v | W u − uW | ≤ max ≤ u ≤ v | W u | + v | W | ≤ √ v ( max ≤ u ≤ v | W u | / √ v + √ v | W | ) ≤ √ v ( max ≤ u ≤ v | W u | / √ v + | W | ) = √ vβ ∗ ,β ∗ = max ≤ u ≤ v | W u | / √ v + | W | and E e Nβ ∗ < ∞ . Therefore for z the following estimate is hold z ≤ v exp { σ √ vβ ∗ } exp { σv | a ( v, z ) |} , Clearly that ln (cid:16) zv (cid:17) ≤ σ √ vβ ∗ + σv | a ( v, z ) | from which it follows that | a ( v, z ) | ≥ σv (cid:0) ln( z/v ) − σ √ vβ ∗ (cid:1) . (53)Clearly that ln( z/v ) is large for v → E exp { β − a ( v, z ) / } = E exp { β − a ( v, z ) / } ( { β ∗ ≤ L } + { β ∗ >L } ) ≤ E exp { β − a ( v, z ) / } { β ∗ ≤ L } + E exp { β } { β ∗ >L } ≤ E e β exp {− a ( v, z ) / } { β ∗ ≤ L } + ( E e β ) / ( P ( β ∗ > L )) / Let c = max( E e β , ( E e β ) / ). Using Markov’s inequality we obtain P ( β ∗ > L ) = P ( e δ ∗ β ∗ > e δ ∗ L ) ≤ exp {− δ ∗ L } E exp { δ ∗ β ∗ } = c exp {− δ ∗ L } , Consider c = E exp { δ ∗ β ∗ } = ∞ (cid:88) m =0 δ m m ! E β m ∗ . and besides E β m ∗ ≤ m E (cid:18) | W | m + ( max ≤ u ≤ v | W u | / √ v ) m (cid:19) = 2 m (cid:18) (2 m − v m E max ≤ u ≤ v | W u | m (cid:19) ≤ m (cid:32) m m ! + 1 v m (cid:18) m m − (cid:19) m E | W v | m (cid:33) ≤ c m m ! + 2 m (2 m − v m v m ≤ c m m ! . Hence E exp { δ ∗ β ∗ } ≤ ∞ (cid:88) m =0 cδ m ∗ m and this series will converge if we choose δ ∗ < / Then the expectation take the form E exp { β − a ( v, z ) / } ≤ c (exp {− a ( v, z ) / } { β ∗ ≤ L } + c exp {− δ ∗ L / } )If β ∗ ≤ L then inequality (53) will take the form | a ( v, z ) | ≥ σv (cid:0) ln( z/v ) − σ √ vβ ∗ (cid:1) ≥ σv (cid:0) ln( z/v ) − σ √ vL (cid:1) . The constant L must be chosen so that ln( z/v ) − σL ¿0. Let L = 12 σ √ v ln (cid:16) zv (cid:17) . Then | a ( v, z ) | > σv ln( z/v )and E exp { β − a ( v, z ) / } ≤ c (cid:18) exp (cid:26) − σ v (ln( z/v )) (cid:27) + c exp (cid:26) − δ ∗ σ v (ln( z/v )) (cid:27)(cid:19) ≤ c (1 + c ) exp (cid:26) − δ ∗ σ v (ln( z/v )) (cid:27) . Thus for the constants c ∗ = c (1 + c ) c and κ = δ ∗ / σ we have the following estimate for the density q ( v, z ) ≤ c ∗ σ v ∗ exp (cid:110) − κσ v (ln( z/v )) (cid:111) ∀ z > v. Next, consider the derivative of q ( v, z ) w.r.t. z . Let L ( v, a ( v, z )) = ϕ , ( a ) K ( v, a )First of all, we prove that q z ( v, z ) = ∂∂z E ( L ( v, a ( v, z ))) = E ∂∂z L ( v, a ( v, z )) . Let ˜ L ( v, z ) = L ( v, a ( v, z ))and ξ ∆ ( z ) = ˜ L ( v, z + ∆ ) − ˜ L ( v, z ) ∆ . Then q z ( v, z ) = q ( v, z + ∆ ) − q ( v, z ) ∆ = E ξ ∆ ( z ) . Be the derivative definition we obtain ξ ∆ ( z ) −−−→ ∆ → ∂∂z ˜ L ( v, z ) . edging problems for Asian options with transactions costs 41 Also we can write ∀ ∆ > | ξ ∆ ( z ) | = (cid:12)(cid:12) ∆ (cid:90) z + ∆z ∂∂u ˜ L ( v, u ) du (cid:12)(cid:12) ≤ sup z ∈ R + sup ≤ v ≤ (cid:12)(cid:12) ∂∂z ˜ L ( v, z ) (cid:12)(cid:12) < ∞ Hence we can use Lebesgue’s theoremlim ∆ → E ξ ∆ ( z ) = E lim ∆ → ξ ∆ = E ∂∂z ˜ L ( v, z ) . Thus we obtain that q z ( v, z ) = E (cid:18) ∂L ( v, a ( v, z )) ∂z (cid:19) = E ( L (cid:48) a ( v, a ( v, z )) a (cid:48) z . It’s clear that L (cid:48) a ( v, a ) = ϕ (cid:48) , ( a ) K ( v, a ) − ϕ , ( a ) K (cid:48) a ( v, a ) K ( v, a ) K (cid:48) a ( v, a ) = σ (cid:90) v u exp (cid:8) σW u + σuW + σua − σ u/ (cid:9) du ≤ σK ( v, a ) a (cid:48) z = 1 K ( v, a ) . Therefore we obtain that | q (cid:48) z ( v, z ) | ≤ E (cid:18) | ϕ (cid:48) , ( a ) | + σϕ , ( a ) K ( v, a ) (cid:19) ≤ (1 + σ ) E (cid:18) | ϕ (cid:48) , ( a ) | + ϕ , ( a ) K ( v, a ) (cid:19) . Since K ( v, a ) ≥ v ∗ σ exp {− β − | a |} then | q (cid:48) z ( v, z ) | ≤ (1 + σ ) σ v ∗ √ π E exp { β + 2 | a | − a ( v, z ) / } (1 + | a | )We can write the follows(1 + | a | ) exp { | a | − a / } = (1 + | a | ) exp {− a / } exp { a / | a | − a / }≤ e exp {− a / } (1 + | a | ) exp {− a / }≤ c exp {− a / } , c = e sup x (1 + x ) e − x / . Hence | q (cid:48) z ( v, z ) | ≤ c σ v ∗ E exp { β − a ( v, z ) / } , c = 2 c/ √ π. Thus we have obtained the estimate of q (cid:48) z ( v, z ) similar to the estimate of q ( v, z ).Taking into account that in this case the constant c = max( E e β , ( E e β ) / ) also carries no information about σ we can analogically write the estimate for c ∗ = c (1 + c ) c and κ = δ/ | q (cid:48) z ( v, z ) | ≤ c ∗ σ v ∗ exp (cid:110) − κσ v (ln( z/v )) (cid:111) ∀ z > v. Let ˜ c = max( c ∗ , c ∗ ) then we obtain the desired estimates. Proof (Proposition 4)We need to look at the asymptotic behavior q v ( v, z ) when v → z > L ( v, a ( t, z )) = ϕ , ( a ) K ( v, a )First of all, we prove that q v ( v, z ) = ∂∂v E ( L ( v, a ( v, z ))) = E ∂∂v L ( v, a ( v, z )) . Let ˜ L ( v, z ) = L ( v, a ( v, z ))and ξ ∆ ( z ) = ˜ L ( v + ∆, z ) − ˜ L ( v, z ) ∆ . Then q v ( v, z ) = q ( v + ∆, z ) − q ( v, z ) ∆ = E ξ ∆ ( z ) . Be the derivative definition we obtain ξ ∆ ( z ) −−−→ ∆ → ∂∂v ˜ L ( v, z ) . Moreover ∀ ∆ > | ξ ∆ ( z ) | = (cid:12)(cid:12) ∆ (cid:90) v + ∆v ∂∂u ˜ L ( u, z ) du (cid:12)(cid:12) ≤ sup z ∈ R + sup ≤ v ≤ (cid:12)(cid:12) ∂∂v ˜ L ( v, z ) (cid:12)(cid:12) := ξ ∗ ( z )and E ξ ∗ ( z ) < ∞ . Hence we can use Lebesgue’s theoremlim ∆ → E ξ ∆ ( z ) = E lim ∆ → ξ ∆ = E ∂∂v ˜ L ( v, z ) . Thus we obtain that q v ( v, z ) = E (cid:18) ∂L ( v, a ( v, z )) ∂v (cid:19) . Next we need to calculate ∂L ( v, a ( v, z )) ∂v = L (cid:48) v ( v, a ) + L (cid:48) a ( v, a ) a (cid:48) v . edging problems for Asian options with transactions costs 43 Introduce the notation F ( v, a ) = (cid:90) v exp { σW u − σuW − σ u/ σua } du and P ( v, a ) = exp { σW v − σvW − σ v/ σva } . Find a (cid:48) v from the equality z = F ( v, a ( v, z )). Differentiating by v we obtain0 = F v ( v, a ) + F a ( v, a ) a (cid:48) v hence a (cid:48) v = − F v ( v, a ) F a ( v, a ) = − P ( v, a ) K ( v, a )It’s clear that L (cid:48) a ( v, a ) = ϕ (cid:48) , ( a ) K ( v, a ) − ϕ , ( a ) K (cid:48) a ( v, a ) K ( v, a ) L (cid:48) v ( v, a ) = − ϕ , ( a ) K (cid:48) v ( v, a ) K ( v, a ) , at that K (cid:48) v ( v, a ) = σvP ( v, a ).Then q (cid:48) v ( v, z ) = E (cid:20) − ϕ , ( a ) K (cid:48) v ( v, a ) K ( v, a ) − P ( v, a ) K ( v, a ) (cid:18) ϕ (cid:48) , ( a ) K ( v, a ) − ϕ , ( a ) K (cid:48) a ( v, a ) K ( v, a ) (cid:19)(cid:21) = E (cid:20) − ϕ , ( a ) σvP ( v, a ) K ( v, a ) − P ( v, a ) ϕ , ( a ) K ( v, a ) + P ( v, a ) ϕ , ( a ) K (cid:48) a ( v, a ) K ( v, a ) (cid:21) = E (cid:20) P ( v, a ) ϕ , ( a ) (cid:18) − σvK ( v, a ) + aK ( v, a ) + K (cid:48) a ( v, a ) K ( v, a ) (cid:19)(cid:21) We can estimate | q v ( v, z ) | ≤ E (cid:18) (1 + | a | ) ϕ , ( a ) P ( v, a ) ( σv + 1) K ( v, a ) + | K (cid:48) a ( v, a ) | K ( v, a ) (cid:19) Seeing that K ( v, a ) = σ v (cid:82) uP ( u, a ) du and K (cid:48) a ( v, a ) = σ v (cid:90) u P ( u, a ) du ≤ σvK ( v, a ) we have | q v ( v, z ) | ≤ E (cid:18) (1 + | a | ) ϕ , ( a ) P ( v, a ) ( σv + 1) K ( v, a ) (cid:19) . We can do the following transformation(1 + | a | ) ϕ , ( a ) = (1 + | a | ) 1 √ π exp {− a / } ≤ ce − a / , where c = √ π sup x (1 + x ) exp {− x / } . Then | q v ( v, z ) | ≤ (1 + σ ) E (cid:32) P ( v, a ) K ( v, a ) · ce − a / K ( v, a ) (cid:33) . (54)Next we will estimate separately P ( v, a ) /K ( v, a ) and 1 /K ( v, a ). So, P ( v, a ) K ( v, a ) = P ( v, a ) σ v (cid:82) uP ( u, a ) du = 1 σ v (cid:82) u P ( u,a ) P ( v,a ) du . Consider P ( u, a ) P ( v, a ) = exp { σW u − σuW − σ u/ σua − σW v + σvW + σ v/ − σva } = exp {− σ ( W v − W u ) + σ ( v − u ) W + σ ( v − u ) / − σ ( v − u ) a } . Then σ v (cid:90) u P ( u, a ) P ( v, a ) du = σ v (cid:90) u exp {− σ ( W v − W u )+ σ ( v − u ) W + σ ( v − u ) / − σ ( v − u ) a } du Make the change of variable t = v − uσ v (cid:90) u P ( u, a ) P ( v, a ) du = σ v (cid:90) ( v − t ) exp {− σ ¯ W t + σtW + σ t/ − σta } dt, here ¯ W t = W v − W v − t . Next we make the change of variable s = tσ to usethe scale invariance property of Wiener process. σ v (cid:90) u P ( u, a ) P ( v, a ) du = 1 σ σ v (cid:90) ( σ v − s ) exp {− W ∗ s + sW /σ + s/ − sa/σ } ds, edging problems for Asian options with transactions costs 45 where W ∗ s = σ ¯ W s/σ . Find the lower bound for the last expression. σ v (cid:90) u P ( u, a ) P ( v, a ) du ≥ σ v ∗ (cid:90) ( v ∗ − s ) ds exp {− max ≤ s ≤ W ∗ s − | W | − | a |}≥ v ∗ σ exp {− max ≤ s ≤ W ∗ s − | W | − | a |} . Thus P ( v, a ) K ( v, a ) ≤ σ v ∗ exp {− max ≤ s ≤ W ∗ s − | W | − | a |} Consider analogically K ( v, a ) = σ v (cid:90) u exp { σW u − σuW − σ u/ σua } du = 1 σ σ v (cid:90) s exp { ˆ W s − sW /σ − s/ sa/σ } ds ≥ v ∗ σ exp {− max ≤ s ≤ ˆ W s − | W | − | a |} , where ˆ W s = σW s/σ . Finally we obatin | q v ( v, z ) | ≤ (1 + σ ) E (cid:32) P ( v, a ) K ( v, a ) · ce − a / K ( v, a ) (cid:33) ≤ σ v ∗ E exp { max ≤ s ≤ W ∗ s + 2 | W | + 2 | a | + max ≤ s ≤ ˆ W s − a / } Introduce notation γ ∗ = γ + γ + γ with conponents γ = max ≤ s ≤ W ∗ s γ = max ≤ s ≤ ˆ W s γ = 2 | W | It’s clear that E exp { N γ ∗ } < + ∞ , ∀ N since E e γ e γ e γ ≤ (cid:0) E e γ (cid:1) / (cid:0) E e γ e γ (cid:1) / ≤ (cid:0) E e γ (cid:1) / (cid:0) E e γ (cid:1) / (cid:0) E e γ (cid:1) / . and E exp { N max | W t |} < + ∞ . Then for c ∗ = c c (cid:0) E e γ (cid:1) / (cid:0) E e γ (cid:1) / (cid:0) E e γ (cid:1) / | q v ( v, z ) | ≤ σ c ∗ v ∗ E exp { γ ∗ + 2 | a | − a / − a / }≤ σ ˆ cv ∗ E exp { γ ∗ − a / } , here ˆ c = c ∗ exp { sup x (2 x − x / } .Next we obtain the lower bound for a ( v, z ) similar to the proof of Propo-sition3 | a ( v, z ) | ≥ σv (cid:0) ln( z/v ) − σ √ vβ ∗ (cid:1) , (55)where β ∗ = max ≤ u ≤ v | W u | / √ v + | W | and E exp { N β ∗ < + ∞} . Also we representthe expectation as E exp { γ ∗ − a ( v, z ) / } = E exp { γ ∗ − a ( v, z ) / } ( { β ∗ ≤ L } + { β ∗ >L } ) ≤ E exp { γ ∗ − a ( v, z ) / } { β ∗ ≤ L } + E exp { γ ∗ } { β ∗ >L } ≤ E e γ ∗ exp {− a ( v, z ) / } { β ∗ ≤ L } + ( E e γ ∗ ) / ( P ( β ∗ > L )) / Let c = max( E e γ ∗ , ( E e γ ∗ ) / ). Using Markov’s inequality we obtain P ( β ∗ > L ) = P ( e δ ∗ β ∗ > e δ ∗ L ) ≤ exp {− δ ∗ L } E exp { δ ∗ β ∗ } = c exp {− δ ∗ L } . Then E exp { γ ∗ − a ( v, z ) / } ≤ c (exp {− a ( v, z ) / } { β ∗ ≤ L } + c exp {− δ ∗ L / } )If β ∗ ≤ L then inequality (55) will take the form | a ( v, z ) | ≥ σv (cid:0) ln( z/v ) − σ √ vβ ∗ (cid:1) ≥ σv (cid:0) ln( z/v ) − σ √ vL (cid:1) . The constant L must be chosen so that ln( z/v ) − σL ¿0. Let L = 12 σ √ v ln (cid:16) zv (cid:17) . Then | a ( v, z ) | > σv ln( z/v )and E exp { γ ∗ − a ( v, z ) / } ≤ c (cid:18) exp (cid:26) − σ v (ln( z/v )) (cid:27) + c exp (cid:26) − δ ∗ σ v (ln( z/v )) (cid:27)(cid:19) ≤ c (1 + c ) exp (cid:26) − δ ∗ σ v (ln( z/v )) (cid:27) . Thus for the constants ˜ c = c (1 + c )ˆ c and κ = δ ∗ / σ we have the following estimate for the derivative of density w.r.t. vq v ( v, z ) ≤ ˜ cσ v ∗ exp (cid:110) − κσ v (ln( z/v )) (cid:111) . edging problems for Asian options with transactions costs 47 References
1. Black F. and Scholes M., The pricing of options and corporate liabilities Statistics ofrandom processes.
Journal Polit.Econ. Ann.Math.Stat. Journal Math.Soc.Jpn. J.Econ.Theory J.Econ.Theory Journal of Finance Dissertation, Universitat der Bundeswehr Munchen, Institut fur Mathematik undDatenverarbeitung (1993) .8. Kabanov Yu.M. and Safarian M. On Leland’s strategy of option pricing with transactioncosts.
Finance and Stochastics Annals of Applied Probability
Annals of Applied Probability PhD thesis,Universite de Franche-Comte Besanson (2008).12. Lepinette E. Modified Leland’s strategy for constant transaction costs rate.
Mathemat-ical Finance Applied Mathematical Finance (2013).14. Nguyen H.T. Approximate hedging with transaction costs and Leland’s algorithm instochastic volatility markets. PhD thesis,Universite de Rouen (2014).15. Denis E. and Kabanov Yu. Mean square error for the Leland-Lott hedging strat-egy:convex pay-offs.
Finance Stoch. (2009).16. Liptser R.S. and Shiryaev A.N. Statistics of random processes.
425 p (2001).17. Shishkova A.A. Calculation of Asian options for the Black Scholes model.
Journal ofTomsk State University. Mathematics and mechanics Mathematical Fi-nance
V.3, Numerical Methods in Finance
Banking Finance Risk Journal of Computational Finance
V.5, (2001).23. Fu M.,Madan D.,Wang T. Pricing continuous Asian options: a comparison of Monte-Carlo and Laplace transform invertion methods. Journal of Computational Finance
V.2(1999).24. Seghiouer H., Lidouh A., Nqi F.Z. Pricing Asian options by Monte-Carlo Method underMPI Environment.
Int.Journal of Math.Analysis
V.2, Journal of Applied Probability Finance and Stochastics
V. 4, Journal ofComputational finance
V.4, Quant.Finance
V.4 Finance and Stochastics.
V. 12 Journal of Banking and Finance
V. 32 J.Comput.Appl. Math.
V. 172 Insurance:Math. Econ.
V. 26 J. Comput.Appl.Math.
V. 185 Quant. finance
V.16 Journal of Financial andQuantitative Analysis
V.38 Quant. Finance
V.3 Quant. Finance
V.17 Appl.Math.Comput.
V. 228, 411–422 (2014).39. Kabanov Yu., Pergamenshchikov S. In the insurance business risky investments aredangerous: the case of negative risk sums.
Finance and Stochastics
V. 20 The Journal of the IAA