$The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions$

The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions

The paper is devoted to the problem of quantile hedging of contingent claims in the framework of a model defined by the finite number of independent Brownian and fractional Brownian motions. The maximal success probability depending on initial capital is estimated.

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Дата:	2008
Автори:	Bratyk, M., Mishura, Y.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2008
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/4566
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions / M. Bratyk, Y. Mishura // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 27-38. — Бібліогр.: 6 назв.— англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-45662009-12-08T12:00:25Z The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions Bratyk, M. Mishura, Y. The paper is devoted to the problem of quantile hedging of contingent claims in the framework of a model defined by the finite number of independent Brownian and fractional Brownian motions. The maximal success probability depending on initial capital is estimated. 2008 Article The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions / M. Bratyk, Y. Mishura // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 27-38. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4566 en Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The paper is devoted to the problem of quantile hedging of contingent claims in the framework of a model defined by the finite number of independent Brownian and fractional Brownian motions. The maximal success probability depending on initial capital is estimated.
format	Article
author	Bratyk, M. Mishura, Y.
spellingShingle	Bratyk, M. Mishura, Y. The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions
author_facet	Bratyk, M. Mishura, Y.
author_sort	Bratyk, M.
title	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions
title_short	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions
title_full	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions
title_fullStr	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions
title_full_unstemmed	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions
title_sort	generalization of the quantile hedging problem for price process model involving finite number of brownian and fractional brownian motions
publisher	Інститут математики НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/4566
citation_txt	The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions / M. Bratyk, Y. Mishura // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 27-38. — Бібліогр.: 6 назв.— англ.
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fulltext	Theory of Stochastic Processes Vol.14 (30), no.3-4, 2008, pp.27-38 MYKHAYLO BRATYK AND YULIYA MISHURA THE GENERALIZATION OF THE QUANTILE HEDGING PROBLEM FOR PRICE PROCESS MODEL INVOLVING FINITE NUMBER OF BROWNIAN AND FRACTIONAL BROWNIAN MOTIONS The paper is devoted to the problem of quantile hedging of contingent claims in the framework of a model defined by the finite number of independent Brownian and fractional Brownian motions. The maxi- mal success probability depending on initial capital is estimated. 1. Introduction The problem of hedging of contingent claims is well known in the case of complete non-arbitrage models. Consider an investor, who wants to ensure that a claim H will be hedged and operates with an asset, whose price is modeled by a semimartingale. Necessary and sufficient condition for such successful hedging is the availability of capital H0 = EP ∗(H), where EP ∗ is the expectation w.r.t. the unique martingale measure P ∗, i.e. the measure w.r.t. which the price process Xt is a martingale. If investor is unwilling or unable to use the whole amount H0 and wants or is able to use a certain amount ν < H0, it follows from the absence of arbitrage that he cannot supply the replication of claim H in all possible scenario, i.e. he cannot hedge the claim H with probability 1. In this case the problem of quantile hedging for investor can be reduced to maximization of success probability, i.e. the probability to hedge the claim H . In paper [4] the general principles concerning such type of hedging are considered in the case where the price process is a semimartingale. In paper [6] the special case of jump-diffusion market is considered, the stochastic differential equation for hedging strategy is deduced, the hedging strategy and the price of European option are obtained. 2000 Mathematics Subject Classifications 91B28, 60G48, 60G15. Key words and phrases. Quantile hedging, incomplete market, fractional Brownian motion. 27 28 MYKHAYLO BRATYK AND YULIYA MISHURA In paper [2] the model with so called long-range dependence, more ex- actly, with the unique fractional Brownian and independent Brownian mo- tion is considered and maximal possible success probability on the available initial capital ν < H0 is estimated. The market is complete there. In this paper our aim is to consider incomplete market with several fractional Brownian motions and independent Brownian motions and to construct the set on which the investor can hedge the contingent claim and find the dependence of the estimation of maximal success probability on the initial capital ν. For technical simplicity we restrict ourselves to the case of two pairs of such processes. In Section 2 we sketch the solution of the quantile hedging problem on the complete non-arbitrage financial market. In Section 3 we prove the incompleteness of the price process model defined by two Wiener processes and two fractional Brownian motions and estimate the successful probability for quantile hedging for this case. In Section 4 we consider the results of Section 3 for European call option and illustrate the dependence of the probability of successful hedge on the initial capital ν. Let Bi t (i = 1, 2) be Wiener processes and BHi t (i = 1, 2) be normalized fractional Brownian motions (fBm) with Hurst index Hi consequently on a probability space (Ω, F, P ). The last means that BHi t is a continuous Gaussian process with zero mean and stationary increments, which has the covariance function of the form EBHi t B Hi s = 1 2 ( s2Hi + t2Hi − \|s− t\|2Hi ) , i = 1, 2. we suppose everywhere that Hi ∈ (1/2, 1), since exactly this case corre- sponds to the long-range dependence of the model. Consider the so-called mixed model, where the price process is defined as Xt = X0 exp { mt+ σ1B 1 t + σ2B 2 t + μ1B H1 t + μ2B H2 t } = = X0 exp { mt+ σ1 ( B1 t + μ1 σ1 BH1 t ) + σ2 ( B2 t + μ2 σ2 BH2 t )} = X0 exp { mt+ σ1 ( B1 t + δ1B H1 t ) + σ2 ( B2 t + δ2B H2 t )} , (1) where δi = μi σi , i = 1, 2, X0 > 0 is some constant. The processes B1 t , B 2 t , B H1 t , BH2 t are supposed to be mutually indepen- dent. The absence of arbitrage in this model follows from existence of at least one martingale measure for the process (1), i.e. such measure w.r.t. which the discounted price process will be a martingale. The set of martingale measures for the price process model is constructed in Section 3. QUANTILE HEDGING ON INCOMPLETE MARKET 29 2. Quantile hedging on the complete non-arbitrage market We consider complete stochastic basis (Ω, F, Ft, t ≥ 0, P ). The problem of quantile hedging is to maximize the probability P {VT ≥ H} over all self- financing strategies ξ with initial capital equal to fixed V0 ≤ ν, admissible in the sense that the capital process Vt = V0 + t∫ 0 ξsdXs (2) for the strategy ξ is non-negative a.s. for all 0 ≤ t ≤ T , and VT evidently is a capital at maturity date. In paper [4] this optimization problem is solved with the help of Neumann- Pearson lemma. Below we sketch the solution of this problem. We call the set A = {VT ≥ H} the success set corresponding to chosen strategy. Define such measure Q∗ that dQ∗ dP ∗ = H H0 . (3) Then the optimization problem is reduced to maximization of probability P {A} over all FT -measurable sets A satisfying the condition V0 H0 = EP ∗ [HIA] H0 = EQ∗ [IA] ≤ α := ν H0 . (4) Put ā = inf { a : Q∗ [ dP dP ∗ > ā ·H] ≤ α } . Suppose that Q∗ [ dP dP ∗ = ā ·H ] = 0. (5) Then by the Neumann-Pearson lemma the optimal set is A = Aā := { dP dP ∗ > ā ·H } . (6) Moreover, due to the choice of ā for initial capital ν there exists a strategy which permits to hedge almost surely the claim H̄ = HIAā, i.e. to hedge H with probability P (Aā). It is the strategy that maximizes the probability P (VT ≥ H). If we consider the simplest case, where the claim H depends only on the final asset price, i.e. H = HT = f (XT ), then in terms of the Neumann- Pearson lemma, the problem is reduced to finding such constant ā that the probability P of the set A = Aa := { dP dP ∗ > a ·H} is maximal given that Q∗ (A) ≤ α = ν H0 . 30 MYKHAYLO BRATYK AND YULIYA MISHURA Unfortunately, as it is shown in [2], in case of a so-called mixed Brownian- fractional-Brownian motion model this set has rather complicated structure, in particular it depends on the trajectory of the Wiener process Wt on the whole interval [0, T ], and thus the probabilities Q∗ (A) and P (A) are hardly computable, which does not permit to use directly the approach, introduced in [4] even for the simplest payoff functions f (XT ) (e.g., European call option). Using the procedure of re-discounting, another type of set was obtained in [2], and this permits us to avoid aforementioned difficulty. Put Ãa = { a < XT X0 · f (XT ) } , (7) and let the claim H be hedged on this set, if the initial capital is equal to ν. The set Ãā is not necessarily the maximal success probability set for initial capital ν, nevertheless P ( Ãā ) is lower estimate for the maximal probability. Therefore, one can guarantee successful hedge of the claim with probability not smaller than P ( Ãā ) . 3. Quantile hedging in incomplete case In the incomplete case the equivalent martingale measure is no longer unique. The following lemma states the incompleteness of the market de- fined by price process (1). Note that the processes Bi t + δiB Hi t , i = 1, 2, consequently Xt, are not semimartingales w.r.t. the filtration generated by processes B1 t , B 2 t , B H1 t , BH2 t . Nevertheless, as it is shown in [3], for Hi > 3 4 , i = 1, 2, the random process Y i t = Bi t+δiB Hi t is equivalent in distribution to Wiener process. Moreover, according to [5], there exists the representation Y i t = W i t − t∫ 0 s∫ 0 rδi (s, u) dW i uds, (8) where W i t is Wiener process w.r.t. P , r = rδi is Volterra kernel, that is, the unique solution of equation r (t, s) + s∫ 0 r (t, x) r (s, x) dx = δ2 iHi (2Hi − 1) · \|t− s\|2Hi−2, (9) 0 ≤ s ≤ t ≤ T, which satisfies t∫ 0 s∫ 0 (rδi (s, u))2 duds <∞, i = 1, 2. Thus, Y i t is a semimartin- gale w.r.t. the natural filtration F Y , generated by the processes Y i t , i = 1, 2. In what follows we fix this filtration and we will consider only it. QUANTILE HEDGING ON INCOMPLETE MARKET 31 Lemma 3.1. Let Hi > 3 4 , i = 1, 2, m̃1(s) is predictable process with respect to natural filtration F Y , m̃2(s) = m − m̃1(s) and the following conditions hold: E ∫ t 0 m2 i (s)ds <∞ and E exp ( − T∫ 0 (m̃i(s) σi + 1 2 σi − s∫ 0 rδi(s, u)dW i u ) dW i s− 1 2 T∫ 0 (m̃i(s) σi + 1 2 σi − s∫ 0 rδi(s, u)dW i u )2 ds ) = 1, i = 1, 2. Then the model for the process Xt of the form (1) is incomplete since there exists a family of the martingale measures P ∗ depending on m̃1 and m̃2, with Radon-Nykodym derivatives of the form dP ∗ dP ∣∣∣ FY t = 2∏ i=1 exp ⎛⎝− t∫ 0 ⎛⎝m̃i(s) σi + 1 2 σi − s∫ 0 rδi (s, u) dW i u ⎞⎠ dW i s− −1 2 t∫ 0 ⎛⎝m̃i(s) σi + 1 2 σi − s∫ 0 rδi (s, u) dW i u ⎞⎠2 ds ⎞⎠ . (10) Proof. The price of asset Xt can be rewritten in the form Xt = X0 exp ( mt+ σ1Y 1 t + σ2Y 2 t ) = = X0 exp ⎛⎝mt+ σ1W 1 t − σ1 t∫ 0 s∫ 0 rδ1 (s, u) dW 1 uds + +σ2W 2 t − σ2 t∫ 0 s∫ 0 rδ2 (s, u) dW 2 uds ⎞⎠ = = X0 2∏ i=1 exp ⎛⎝σiW i t − 1 2 σ2 i t+ t∫ 0 (m̃i(s) + 1 2 σ2 i )ds −σi t∫ 0 s∫ 0 rδi (s, u) dW i uds ⎞⎠ . Denote Ŵ i t = W i t + t∫ 0 ⎛⎝m̃i(s) σi + 1 2 σi − s∫ 0 rδi (s, u) dW i u ⎞⎠ ds, i = 1, 2. (11) 32 MYKHAYLO BRATYK AND YULIYA MISHURA By the Girsanov theorem this process is Wiener process w.r.t. the measure P ∗ of the form (10) and then w.r.t. P ∗ the process Xt has the form Xt = X0 2∏ i=1 exp ( σiŴ i t − 1 2 σ2 i t ) , (12) i.e. it is a martingale. (We suppose everywhere that non-risky asset identi- cally equals 1.) Correspondingly to the choice of m̃1 and m̃2 we obtain different martin- gale measures. Thus the martingale measure is not unique and the market is incomplete. � Remark. Considering another filtration than F Y , we can obtain another martingale measures for the process Xt, not defined in Lemma 3.1. Example 3.1. Let we define the processes B̃1 t (α) = B1 t cosα− B2 t sinα, B̃2 t (α) = B1 t sinα +B2 t cosα. These processes are uncorrelated and, thus, independent. We have B1 t (α) = B̃1 t cosα + B̃2 t sinα, B2 t (α) = −B̃1 t sinα + B̃2 t cosα, and Xt = X0 exp { mt+ σ1 ( B1 t + δ1B H1 t ) + σ2 ( B2 t + δ2B H2 t )} = = X0 exp { mt+ σ1 ( B̃1 t (α) cosα + B̃2 t (α) sinα + δ1B H1 t ) + +σ2 ( −B̃1 t (α) sinα + B̃2 t (α) cosα + δ2B H2 t )} = = X0 exp { mt+ (σ1 cosα− σ2 sinα)B̃1 t (α) + (σ1 sinα + σ2 cosα)B̃2 t (α)+ +μ1B H1 t + μ2B H2 t } = = X0 exp { mt+ σ̃1B̃ 1 t (α) + σ̃2B̃ 2 t (α) + μ1B H1 t + μ2B H2 t } = = X0 exp { mt+ σ̃1 ( B̃1 t (α) + μ1 σ̃1 BH1 t ) + σ̃2 ( B̃2 t (α) + μ2 σ̃2 BH2 t )} = = X0 exp { mt+ σ̃1 ( B̃1 t (α) + δ̃1B H1 t ) + σ̃2 ( B̃2 t (α) + δ̃2B H2 t )} = QUANTILE HEDGING ON INCOMPLETE MARKET 33 = X0 exp { mt+ σ̃1Ỹ 1 t + σ̃2Ỹ 2 t } , (13) where σ̃1 = σ1 cosα − σ2 sinα, σ̃2 = σ1 sinα + σ2 cosα, δ̃i = μi σ̃i , Ỹ i t = B̃i t(α) + δ̃iB Hi t , i = 1, 2. Note that in general case we have δ̃i = δi, i = 1, 2. Thus, the processes Ỹ i t and Y i t have different Volterra kernels, which are the solutions of (9). It means that corresponding martingale measures for the process Xt, which satisfy (10), are different. Remark. Although the martingale measures, mentioned above, are differ- ent, the following equality holds: σ2 1 + σ2 2 = σ̃2 1 + σ̃2 2. (14) If the contingent claim in incomplete model is not attainable, it is pos- sible to hedge the claim almost surely by means of superhedging strategy. As it is mentioned in [4], in such case the least amount of capital, which is needed to be on the safe side, is given by inf ⎧⎨⎩V0\| ∃ξ : (V0, ξ) admissible, V0 + T∫ 0 ξsdXs ≥ H P − a.s. ⎫⎬⎭. (15) Thus, the least needed amount is equal to the largest arbitrage-free price: U0 := sup P ∗∈P E∗[H ] <∞ (16) where P is the set of all equivalent martingale measures P ∗ satisfying the conditions of lemma 3.1. If investor is not able to use whole amount U0 and is willing to use only amount ν < U0, then he cannot hedge the claim H with probability 1 because of the absence of arbitrage. But if the following inequality holds: sup P ∗∈P EP ∗ [ HIÃa ] ≤ ν, (17) where Ãa has the form introduced in (7), then an investor with initial capital ν is able to hedge HIÃa almost surely, i.e., to hedge H with probability P ( Ãa ) . We make use of the form (7) of the set Ãa. Remark. When a increases, the sets Ãa decrease. So, we look for minimal a, for which the inequality (17) holds. Let consider 34 MYKHAYLO BRATYK AND YULIYA MISHURA any of martingale measures P ∗ of the form of (10) and find the minimal a, for which the following inequality holds: EP ∗ [ HIÃa ] ≤ ν. (18) In this case an investor with initial capital ν is able to hedge HIÃa al- most surely, i.e., to hedge H with probability P ( Ãa ) . Theorem 3.1. Let the function f (x) satisfy the condition ∀z ∈ R : λ ({ x f (x) = z }) = 0, (19) where λ is the Lebesgue measure. Then the probability of successful hedge of the claim H = f(XT ) is at least P ( Ãā ) , where ā is determined by the equation∫ Cā f ( X0e √ Tσy− 1 2 σ2T ) 1√ 2π e− y2 2 dy = ν (20) and Ca = ⎧⎨⎩y ∣∣∣∣∣∣a < exp (√ Tσy − 1 2 σ2T ) f ( X0 exp (√ Tσy − 1 2 σ2T )) ⎫⎬⎭ , (21) where σ = √ σ2 1 + σ2 2 . (22) Proof. With respect to any of the martingale measures P ∗ of the form of (10) the price process XT has the form (12) and it can be re-written in the following way: XT = X0 2∏ i=1 exp ( σiŴT − σ2 i 2 · T ) = = X0 exp( √ Tσ1ξ1 − σ2 1 2 T + √ Tσ2ξ1 − σ2 2 2 T ), = X0 exp( √ Tσξ − σ2 2 T ), (23) where ξ1, ξ2 ∼ N (0, 1) and ξ = σ1ξ1 + σ2ξ2 ∼ N (0, σ2 1 + σ2 2). Remark. For another representation for Xt in the form of (13) the equality (14) holds and, thus, the representation (23) for XT is true. QUANTILE HEDGING ON INCOMPLETE MARKET 35 Then XT X0f(XT ) = exp (√ Tσy − 1 2 σ2T ) f ( X0 exp (√ Tσy − 1 2 σ2T )) and EP ∗ [ HIÃa ] = ∫ Ca f ( X0 exp (√ Tσy − 1 2 σ2T )) 1√ 2π e− y2 2 dy . (24) Thanks to assumptions of the theorem for all a ∈ R we have λ ⎛⎝⎧⎨⎩y ∣∣∣∣∣∣a < exp (√ Tσy − 1 2 σ2T ) f ( X0 exp (√ Tσy − 1 2 σ2T )) = a ⎫⎬⎭ ⎞⎠ = 0, thus it follows from (24) that EP ∗ [ HIÃa ] is continuous and non-decreasing function of a. This implies that for ā = inf { a ∣∣EP ∗ [ HIÃa ] ≤ ν } one has EP ∗ [ HIÃā ] = ν, which is (20). The set Ãā is not necessarily the maximal success probability set for initial capital ν, nevertheless P ( Ãā ) is lower estimate for the maximal probability. Therefore, one can guarantee successful hedge of the claim with probability not smaller than P ( Ãā ) . � 4. Quantile hedging for European call option Let consider European call option f (XT ) = (XT −K)+ , (25) where K > 0, and find the representation for the set Ãa = { a < XT X0·f(XT ) } in this case. Note that for XT ≤ K the inequality a < XT X0·f(XT ) = ∞ holds. For XT ≥ K, the inequality a < XT X0·f(XT ) takes the form: XT X0 (XT −K) > a, XT > (XT −K) aX0, KaX0 > XT (aX0 − 1) . (26) If aX0 ≤ 1, then (26) is always true, i.e. Ãa = Ω. Thus, when a ≤ 1 X0 we have that Ãa = Ω, but this cannot be true, because by the problem 36 MYKHAYLO BRATYK AND YULIYA MISHURA formulation it is impossible to hedge the claim almost surely, but on Ãa it will be hedged. If aX0 > 1, i.e. a > 1 X0 , then Ãa = { XT < KaX0 aX0−1 } , at that K < KaX0 aX0−1 . That is, the set Ãa can be presented in the following form: Ãa = { Ω, a ≤ 1 X0 ,{ XT < KaX0 aX0−1 } , a > 1 X0 . (27) Theorem 4.1. For European call option (25) the maximal probability of successful hedge is at least Φ (U), where U is given by the equation X0 ( Φ ( U − σ √ T ) − Φ ( L− σ √ T )) −K (Φ (U) − Φ (L)) = ν, (28) where L = ln K X0 + σ2 2 T σ √ T . (29) and Φ - is standard normal distribution function. Proof. The function f (XT ) = (XT −K)+, where K > 0, clearly satisfies the assumptions of Theorem 3.1. Then there exists ā, for which (20) holds or, equivalently, the inequality (18) turns to equality. Since in this case the set Ãa has form (27), then the left-hand side (18) takes the form: EP ∗ [HIÃ] = ∫ Ã (XT −K)+ dP ∗ = ∫ K<XT< KaX0 aX0−1 (XT −K) dP ∗. (30) Accounting for (23) the inequality K < XT < KaX0 aX0−1 can be rewritten as: K < X0e σ √ Tξ−σ2 2 T < K aX0 aX0 − 1 ; ln K X0 + σ2 2 T σ √ T < ξ < ln Ka aX0−1 + σ2T 2 σ √ T . (31) Denote: U = U (a) = ln Ka aX0−1 + σ2 2 T σ √ T . (32) The inequality (31) takes the form L < ξ < U (a), and equation (20) can be rewritten as: EP ∗ [HIÃ] = U(ā)∫ L ( X0e σ √ Ty−σ2 2 T −K ) 1√ 2π e− y2 2 dy = QUANTILE HEDGING ON INCOMPLETE MARKET 37 = U(ā)∫ L X0e σ √ Ty−σ2 2 T− y2 2√ 2π dy −K (Φ (U (ā)) − Φ (L)) = = X0 ( Φ ( U (ā) − σ √ T ) − Φ ( L− σ √ T )) −K (Φ (U (ā)) − Φ (L)) = ν, which means the equality (28). Now compute P ( Ãā ) : P ( Ãā ) = P ( X0e σ √ Tξ−σ2 2 T < KāX0 āX0 − 1 ) = (33) = P ( ξ < ln Kā āX0−1 + σ2T 2 σ √ T ) = Φ (U (ā)) If we find U (ā) from equation (28), we can find P ( Ãā ) . Note that to compute P ( Ãā ) it is enough to find U = U (ā), and it is not needed to evaluate ā. � Example 4.1. For European call option (25) formulae (28), (33) allow for wishful value of P ( Ãa ) to compute U = Ua and, correspondingly, the necessary capital ν, or vice versa: having ν to compute the success prob- ability P ( Ãa ) . Note that hedge always succeeds with probability at least P (XT < K) = Φ (L), where L is given by formula (29), thus for XT < K there is nothing to pay for the claim, which means that it will be hedged with the void strategy. On the other hand, when the success probability P ( Ãa ) increases from Φ (L) to 1 the necessary capital ν increases from 0 to option’s fair value H0. Let H = 0, 8, X0 = 1, T = 10, m = 2, σ1 = σ2 = μ2 = μ2 = 1 2 √ 2 , then for different (depending on K ) European calls, fixing P ( Ãa ) , we can compute corresponding values ν. 5. Conclusion The model defined by two independent Brownian and two fractional Brownian motions is considered. But all obtained results may be general- ized for the model defined by the finite number of independent Brownian and fractional Brownian motions. The set of martingale measures for the price process model is constructed, which proves the absence of arbitrage and incompleteness of the financial market. The set Ãā depending on the available initial investor’s capital ν on which the contingent claim is hedged 38 MYKHAYLO BRATYK AND YULIYA MISHURA K Φ (L) P ( Ãa ) ν H0 5 0,964733 0,99 0,055677 0,23375 2 0,890456 0,9 0,000807 0,418561 0,99 0,210488 1 0,785402 0,9 0,053051 0,570805 0,99 0,352732 0,5 0,63765 0,9 0,141527 0,709281 0,99 0,486208 0,1 0,252797 0,9 0,305201 0,912955 0,99 0,685882 0,01 0,016919 0,9 0,373309 0,990063 0,99 0,76209 with probability 1 is constructed. The maximal success probability of con- tingent claim hedging is estimated from below by P (Ãā). For different parameters of the European call option the probability P (Ãā) is computed. References 1. Androshchuk T., Mishura Y., Mixed Brownian- fractional Brownian model: absence of arbitrage and related topics, Stochastics: An International Jour- nal of Probability and Stochastic Processes, v. 78, N5 (2006), 281–300. 2. Bratyk M., Mishura Y., Quantile hedging with rediscounting on complete financial market. Prykladna statystyka. Aktuarna i finansova matematyka N 2, (2007), 46–57. 3. Cheridito P., Regularizing fractional Brownian motion with a view towards stock price modeling, PhD thesis, Zurich, (2001), 157–173. 4. Föllmer H., Leukert P., Quantile hedging. Finance Stochast. 3, (1999), 251–273. 5. Hitsuda M., Representation of Gaussian processes equivalent to Wiener process, Osaka Journal of Mathematics 5, (1968), 299-312. 6. Krutchenko R. N., Melnikov A. V., Quantile hedging for a jump-diffusion financial market model, Trends in Mathematics, (2001), 215–229. Department of Mathematics, The University of ”Kyiv-Mohyla Aca- demy”, Kyiv, Ukraine. E-mail address: mbratyk@ukr.net Department of Probability Theory and Mathematical Statistics, Kyiv National Taras Shevchenko University, Kyiv, Ukraine E-mail address: myus@univ.kiev.ua

The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions

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