Hindawi Publishing Corporation International Scholarly Research Notices Volume 2014, Article ID 746196, 7 pages http://dx.doi.org/10.1155/2014/746196

Research Article Lookback Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion

Jiao-Jiao Sun, Shengwu Zhou, Yan Zhang, Miao Han, and Fei Wang

College of Sciences, China University of Mining and Technology, Xuzhou 221116, China

Correspondence should be addressed to Shengwu Zhou; [email protected]

Received 22 May 2014; Revised 14 September 2014; Accepted 14 September 2014; Published 29 October 2014

Academic Editor: Francesco Zirilli

Copyright © 2014 Jiao-Jiao Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland’s hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, the effectiveness of the proposed form is verified through a numerical example. Meanwhile, the impact of transaction cost rate and on lookback option value is discussed.

1. Introduction during the drift of the option. In 1979, the lookback option pricing formula was firstly given by Goldman et al.6 [ ]. Then In 1985, the problems of option pricing and replication with the pricing formula was extended by Conze and Viswanathan transaction costs were firstly examined by Leland1 [ ]. Because [7]. They obtained explicit formulas of various European of infinite variance of geometric Brownian motion process, lookback options and also provided some results for the transaction costs will become infinite, thus the arbitrary American counterparts by means of probability method. argument used by Black-Scholes to price options can no With underlying asset price following geometric Brownian longer be used. Barles and Soner [2] considered the difference motion, analytic expressions of discrete lookback option between the maximum utility from final wealth when there value were presented by Heynen and Kat [8]. Many empirical is no option liability and when there is such a liability. They studies have shown that distribution of asset price is not derived the nonlinear Black-Scholes equation satisfied by entirely lognormal, and the probability density function of its the option value by using the utility maximization theory. logarithm yield tends to have the feature of “fat-tailedness,”so Considering both transaction costs and the risk from a geometric Brownian motion processes do not accord with the volatile portfolio, in 2005, Jandackaˇ and Sevˇ coviˇ c[ˇ 3]got realistic environment. For characteristics of self-similarity, another more complicated model. The influence of paying fat tails, and long-term dependence, fractional Brownian transaction costs and discontinuous change of stock price motion (fBm) becomes an effective tool to describe the stock on option value was studied by Amster et al. [4]. Under a price process instead of geometric Brownian motion. fBm is process, Mariani et al. [5]investigatedthe neither a Markov process nor a semimartingale, so stochastic option pricing problem with transaction cost and derived the analysis theory for semimartingale cannot be applied directly. pricing model. Meanwhile they got numerical solutions by In incomplete markets, arbitrage opportunities will exist. A using the finite difference method. considerable number of arbitrage strategies for fBm have As a kind of exotic options, the holder of lookback option been provided by Rogers [9], Shiryaev [10], Salopek [11], couldlookbackontheevolutionprocessofunderlying and Cheridito [12]. In 2012, Gu et al. [13] dealt with the assets during the life of option at maturity. It is one kind problem of discrete time option pricing in the presence of of path-dependent options where the payoff is based on the transaction costs by a time-changed geometric fractional maximum or the minimum of the underlying asset price Brownian motion model. Through a mean self-financing 2 International Scholarly Research Notices

delta-hedging argument, they obtained the pricing formula of transaction cost is given by 𝜅|V𝑡|𝑆𝑡 in either buying for the European in discrete time setting. Feng or selling, where 𝜅>0is the fixed transaction cost [14] discussed the lookback option under fBm model. They rate. derived the explicit solutions of lookback pricing formula. (iii) There exists no arbitrage opportunities and the port- In this paper, we will firstly give the price dynamics model folio’s expected rate is the risk-free interest rate 𝑟, under fBm process. Then a three-dimensional nonlinear where 𝑟 is a constant. mathematical pricing model for lookback put option value 𝛿𝑡 with fixed transaction costs will be established. We reduced (iv) The portfolio is revised every time interval ,where 𝛿𝑡 it to a two-dimensional model through variable substitution. is a finite and fixed time step. Butthereducedmodelisstillnonlinear,soitisnoteasyto get the analytic solution. So we get its numerical solution The price process 𝐵𝑡 of risk-free debt evolves over time as by constructing a Crank-Nicolson numerical scheme for the 𝑟𝑡 transformed model. Finally, we analyze the effectiveness of 𝐵𝑡 =𝑒 ,𝐵0 =1. (2) the numerical scheme and the influence of parameters on lookback put option value. Supposethatthepriceprocess𝑆𝑡 oftheriskyshareevolves over time as 2. The Model Dynamics

Definition 1. Let 𝐻 be a constant belonging to (0, 1).A 𝑑𝑆𝑡 =𝑟𝑆𝑡𝑑𝑡 +𝑡 𝜎𝑆 𝑑𝐵𝐻 (𝑡) ,𝑆0 =𝑠>0, (3) 𝐻 fractional Brownian motion (𝐵 (𝑡))𝑡≥0 of Hurst index 𝐻 is a continuous and centered Gaussian process with covariance where 𝜎 indicates the volatility of the stock price. function

𝐻 𝐻 1 2𝐻 2𝐻 2𝐻 𝐸[𝐵 (𝑡) 𝐵 (𝑠)]= (𝑡 +𝑠 − |𝑡−𝑠| ). (1) 3. The Valuation of Lookback Put Option with 2 Transaction Costs under fBm For 𝐻 = 0.5, the fBm is then a standard Brownian motion. 𝐻 By Definition 1 we obtain that a standard fBm 𝐵 (𝑡) has the In this section, we adopt Leland hedging strategy to derive a following properties. model for lookback option price. The following lemma gives Fractional Ito’sˆ differential rule which takes great value in 𝐻 𝐻 (i) 𝐵 (0) = 0 and 𝐸[𝐵 (𝑡)] = 0 for all 𝑡≥0; deriving the formula. 𝐵𝐻(𝑡) 𝐵𝐻(𝑡 + (ii) has homogeneous increments; that is, Lemma 2 (see [18]). Consider the stochastic differential equa- 𝑠) −𝐻 𝐵 (𝑠) 𝐵𝐻(𝑡) 𝑠, 𝑡 ≥0 has the same law of for ; tion 𝐻 𝐻 2 2𝐻 (iii) 𝐵 (𝑡) is a Gaussian process and 𝐸[(𝐵 (𝑡)) ]=𝑡 , 𝑡≥0 𝐻 ∈ (0, 1) ,forall ; 𝑑𝑆𝑡 =𝜇(𝑡, 𝜔) 𝑑𝑡 +𝜎 (𝑡, 𝜔) 𝑑𝐵𝐻 (𝑡) . (4) 𝐻 (iv) 𝐵 (𝑡) has continuous trajectories. Here for 𝑡≥0, 𝜇(𝑡, 𝜔) and 𝜎(𝑡, 𝜔) : [0, 𝑇] ×𝑅→ are The fBm has long-term dependency for 𝐻 ∈ (0.5, 1) two stochastic processes. Assume that a two-variable function and is antisustainable for 𝐻 ∈ (0, 0.5).For𝐻 ≠ 0.5, + 𝑓(𝑡,𝑡 𝑆 ):[0,𝑇]×𝑅 →𝑅has uniformly continuous partial it is neither a Markov process nor semimartingale, so the 𝜕𝑓/𝜕 𝑡 𝜕𝑓/𝜕𝑆 𝜕2𝑓/𝜕𝑆2 𝐻 ∈ (0, 1) classical Ito’sˆ calculus cannot be used to define a fully derivatives , 𝑡, 𝑡 .Thenfor ,we stochastic calculus for fBm. Lin [15] and Decreusefond and have ust¨ unel¨ [16] developed the stochastic calculus of variations 𝜕 𝜕 under fBm model, but Shiryaev [10]investigatedthatthere 𝑑𝑓 (𝑡, 𝑆 𝑡)= 𝑓(𝑡,𝑆𝑡)𝑑𝑡+ 𝑓(𝑡,𝑆𝑡)𝜇(𝑡, 𝜔) 𝑑𝑡 exists arbitrage chance in the market. Hu and Øksendal [17] 𝜕𝑡 𝜕𝑆𝑡 presented the fractional Ito’scalculusbasedontheWickˆ integration driven by fBm. Bender [18] constructed fractional 𝜕 + 𝑓(𝑡,𝑆𝑡)𝜎(𝑡, 𝜔) 𝑑𝐵𝐻 (𝑡) (5) Ito’sˆ formula with this type of calculus. For more details about 𝜕𝑆𝑡 fBm, interested readers may refer to Hu and Peng [19]. 𝜕2 +𝐻 𝑓(𝑡,𝑆)𝜎2 (𝑡, 𝜔) 𝑡2𝐻−1𝑑𝑡. Consider a financial market with two primitive securities, 2 𝑡 𝜕𝑆𝑡 namely, a risky asset 𝑆𝑡 and a risk-free bond 𝐵𝑡. We will need the following assumptions. Consider a European-type lookback strike put option. Let (i) The securities trading is carried out continuously with 𝑉𝑡 =𝑉(𝑆𝑡,𝐽𝑡,𝑡)which denotes the value of the option at time 𝑡 short selling allowed; The corresponding payoff 𝐽𝑡 = max0≤𝑢≤𝑡𝑆𝑢,where𝑆𝑡 satisfies (ii) Suppose V𝑡 sharesofthetradedstockarebought(V𝑡 > (3). The following theorem gives the model for the value of 0) and sold (V𝑡 <0)at the price 𝑆𝑡.Thenaproportion lookback put option with transaction costs. International Scholarly Research Notices 3

Theorem 3. The pricing model for the value of lookback then putoptionwithtransactioncostsunderfractionalBrownian 𝑑𝐽 𝑆𝑛 −𝐽𝑛 (𝑡) motion process is given as 𝑛𝐽𝑛−1 (𝑡) 𝑛 = 𝑡 𝑛 , 𝑛 𝑑𝑡 𝑡 2 (13) 𝜕𝑉𝑡 1 2 2 𝜕 𝑉𝑡 𝜕𝑉𝑡 + 𝜎̃ 𝑆 +𝑟𝑆 −𝑟𝑉 =0, lim 𝐽𝑛 (𝑡) = max 𝑆𝜏 =𝐽𝑡. 𝑡 2 𝑡 𝑛→∞ 0≤𝜏≤𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

𝑉(𝑆𝑇,𝐽𝑇,𝑇)=𝐽𝑇 −𝑆𝑇,(0≤𝑆𝑇 ≤𝐽𝑇 <∞), (6) We use 𝐽𝑛(𝑡) instead of path dependence variable 𝐽𝑡;since𝑆𝑡 ≤ 󵄨 𝐽𝑡,wecanget 𝜕𝑉 󵄨 𝑡 󵄨 =0. (0≤𝑡≤𝑇) , 󵄨 𝑛−1 𝜕𝐽𝑡 󵄨𝑆 =𝐽 (𝑆 /𝐽 ) 𝑆 −𝐽 𝑡 𝑡 𝑡 𝑛 𝑡 𝑛 󳨀→ 0 (𝑛󳨀→∞) . (14) 2 2 2𝐻−1 𝑛𝑡 where 𝜎̃ =2𝜎(𝐻𝑡 −𝐿𝑒(𝐻)sign(𝑉𝑆𝑆)), 𝐿𝑒(𝐻) = 𝐻−1 √2/𝜋(𝜅/𝜎)(𝛿𝑡) , 𝛿𝑡 is the hedging time interval. Consequently, from (11)

2 Proof. For the option purchaser, a hedging portfolio Π𝑡 is 𝜕 𝑉 V ≈ (𝑆 ,𝐽,𝑡)𝜎𝑆𝛿𝐵 (𝑡) . constructed: one long position of lookback put option and 𝑡 2 𝑡 𝑡 𝑡 𝐻 (15) 𝜕𝑆𝑡 short Δ 𝑡 shares of stock; then the value of the portfolio at 𝑡 Π =𝑉−Δ 𝑆 the current time is 𝑡 𝑡 𝑡 𝑡. Assume that the trade The transaction costs is as follows: 𝑡 𝑡+𝛿𝑡 𝛿𝑡 occurs only at times and where denotes the hedging 󵄨 󵄨 time interval. Then over a small time interval 𝛿𝑡,thechange 󵄨 󵄨 󵄨𝜕2𝑉 󵄨 𝜅 󵄨V 󵄨 𝑆 =𝜅󵄨 (𝑆 ,𝐽,𝑡)𝜎𝑆2𝛿𝐵 (𝑡)󵄨 in value of the portfolio will be 󵄨 𝑡󵄨 𝑡 󵄨 2 𝑡 𝑡 𝑡 𝐻 󵄨 󵄨 𝜕𝑆𝑡 󵄨 󵄨 󵄨 (16) 𝛿Π =𝛿𝑉−Δ 𝛿𝑆 −𝜅󵄨V 󵄨 𝑆 . 󵄨 2 󵄨 𝑡 𝑡 𝑡 𝑡 󵄨 𝑡󵄨 𝑡 (7) 2 󵄨𝜕 𝑉 󵄨 =𝜅𝜎𝑆 󵄨 (𝑆 ,𝐽,𝑡)𝛿𝐵 (𝑡)󵄨 . 𝑡 󵄨 𝜕𝑆2 𝑡 𝑡 𝐻 󵄨 The stochastic partial differential equation (3)canbeshown 󵄨 𝑡 󵄨 as According to the Definition 1,weknowthat 𝛿𝑆 =𝑟𝑆𝛿𝑡 + 𝜎𝑆 𝛿𝐵 (𝑡) . 𝑡 𝑡 𝑡 𝐻 (8) 2𝐻 𝛿𝐵𝐻 (𝑡) =𝐵𝐻 (𝑡+𝛿𝑡) −𝐵𝐻 (𝑡) ∼ 𝑁 (0, (𝛿𝑡) ). (17) ByusingdiscreteformofLemma2,wecanget Therefore 2 𝜕𝑉𝑡 𝜕𝑉𝑡 2 2𝐻−1 2 𝜕 𝑉𝑡 𝛿𝑉 =( +𝑟𝑆 +𝐻𝜎 𝑡 𝑆 )𝛿𝑡 󵄨 2 󵄨 𝑡 𝑡 𝑡 2 󵄨 󵄨 2 󵄨𝜕 𝑉 󵄨 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝐸(𝜅󵄨V 󵄨 𝑆 )=𝐸[𝜅𝜎𝑆 󵄨 (𝑆 ,𝐽,𝑡)𝛿𝐵 (𝑡)󵄨] 󵄨 𝑡󵄨 𝑡 𝑡 󵄨 2 𝑡 𝑡 𝐻 󵄨 (9) 󵄨 𝜕𝑆𝑡 󵄨 𝜕𝑉 𝜕𝑉 +𝜎𝑆 𝑡 𝛿𝐵 (𝑡) + 𝑡 𝛿𝐽 . 󵄨 󵄨 (18) 𝑡 𝜕𝑆 𝐻 𝜕𝐽 𝑡 2 󵄨𝜕2𝑉 󵄨 𝑡 𝑡 = √ 𝜅𝜎𝑆2(𝛿𝑡)𝐻 󵄨 𝑡 󵄨 +𝑂(𝛿𝑡) . 𝑡 󵄨 2 󵄨 𝜋 󵄨 𝜕𝑆𝑡 󵄨 Substituting (9)into(7)yields From the assumption (iii), we have 𝐸(𝛿Π𝑡)=𝑟Π𝑡𝛿𝑡.Thus 𝜕𝑉 𝜕2𝑉 𝜕𝑉 𝛿Π =( 𝑡 +𝐻𝜎2𝑆2𝑡2𝐻−1 𝑡 )𝛿𝑡+( 𝑡 −Δ )𝛿𝑆 𝑡 𝜕𝑡 𝑡 𝜕𝑆2 𝜕𝑆 𝑡 𝑡 𝜕𝑉 𝜕2𝑉 𝑡 𝑡 𝑟(𝑉 −Δ 𝑆 )𝛿𝑡=( 𝑡 +𝐻𝜎2𝑆2𝑡2𝐻−1 𝑡 )𝛿𝑡 𝑡 𝑡 𝑡 𝜕𝑡 𝑡 𝜕𝑆2 𝜕𝑉 󵄨 󵄨 𝑡 + 𝑡 𝛿𝐽 −𝜅󵄨V 󵄨 𝑆 . (19) 𝑡 󵄨 𝑡󵄨 𝑡 󵄨 2 󵄨 𝜕𝐽𝑡 𝜕𝑉 2 󵄨𝜕 𝑉 󵄨 + 𝑡 𝛿𝐽 − √ 𝜅𝜎𝑆2(𝛿𝑡)𝐻 󵄨 𝑡 󵄨 . (10) 𝑡 𝑡 󵄨 2 󵄨 𝜕𝐽𝑡 𝜋 󵄨 𝜕𝑆𝑡 󵄨

In order to eliminate risk, let Δ 𝑡 =𝜕𝑉𝑡/𝜕𝑆𝑡.Sointhesame Consequently interval 𝛿𝑡,thetradedsharesofstockare 󵄨 󵄨 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕2𝑉 2 󵄨𝜕2𝑉 󵄨 V = (𝑆 ,𝐽 ,𝑡+𝛿𝑡)− (𝑆 ,𝐽,𝑡) 𝑡 +𝐻𝜎2𝑆2𝑡2𝐻−1 𝑡 − √ 𝜅𝜎𝑆2(𝛿𝑡)𝐻−1 󵄨 𝑡 󵄨 𝑡 𝑡+𝛿𝑡 𝑡+𝛿𝑡 𝑡 𝑡 𝑡 2 𝑡 󵄨 2 󵄨 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑡 𝜕𝑆𝑡 𝜋 󵄨 𝜕𝑆𝑡 󵄨 (20) 2 2 (11) 𝜕 𝑉 𝜕 𝑉 𝜕𝑉𝑡 ≈ (𝑆 ,𝐽,𝑡)𝜎𝑆𝛿𝐵 (𝑡) + 𝑡 𝛿𝐽 . +𝑟𝑆 −𝑟𝑉 =0. 2 𝑡 𝑡 𝑡 𝐻 𝑡 𝑡 𝜕𝑆 𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡𝜕𝐽𝑡 𝑡

Since 𝐽𝑡 is not differentiable, we need to approximate it as The above equation can be simplified as

1 𝑡 1/𝑛 𝜕𝑉 1 𝜕2𝑉 𝜕𝑉 𝑛 𝑡 + 𝜎̃2𝑆2 𝑡 +𝑟𝑆 𝑡 −𝑟𝑉 =0. 𝐽𝑛 (𝑡) =[ ∫ (𝑆𝜏) 𝑑𝜏] , (0≤𝜏≤𝑡) ; (12) 𝑡 2 𝑡 𝑡 (21) 𝑡 0 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 4 International Scholarly Research Notices

Here Time derivatives and spatial derivatives are, respectively, approximated by difference method 2 𝜅 𝐿𝑒 (𝐻) = √ (𝛿𝑡)𝐻−1, 𝜋 𝜎 𝑢𝑛+1 −𝑢𝑛 (22) 𝜕𝑢 𝑖 𝑖 (𝑥𝑖,𝜏𝑛)≈ , 2 2 2𝐻−1 𝜕𝜏 𝑘 𝜎̃ =2𝜎 (𝐻𝑡 −𝐿𝑒(𝐻) sign (𝑉𝑆𝑆)) . 𝜕𝑢 1 𝑢𝑛 −𝑢𝑛 𝑢𝑛+1 −𝑢𝑛+1 (𝑥 ,𝜏 )≈ ( 𝑖+1 𝑖−1 + 𝑖+1 𝑖−1 ), The boundary condition is 𝜕𝑥 𝑖 𝑛 2 2ℎ 2ℎ 󵄨 𝜕𝑉 󵄨 𝑡 󵄨 =0, 𝜕2𝑢 󵄨 (23) (𝑥 ,𝜏 ) 𝜕𝐽𝑡 󵄨 𝑖 𝑛 𝑆𝑡=𝐽𝑡 𝜕𝑥2 whose financial significance is that the value of option is not 1 𝑢𝑛 −2𝑢𝑛 +𝑢𝑛 𝑢𝑛+1 −2𝑢𝑛+1 +𝑢𝑛+1 ≈ ( 𝑖+1 𝑖 𝑖−1 + 𝑖+1 𝑖 𝑖−1 ). sensitive to the maximum when the underlying asset price 2 ℎ2 ℎ2 reaches the maximum [20]. According to the definition of the (29) lookback put option with floating , the formula canbegivenas Substituting the above equations into the model (26)gives standard Crank-Nicolson scheme. Since the corrected volatil- 𝑉(𝑆𝑇,𝐽𝑇,𝑇)=𝐽𝑇 −𝑆𝑇. (24) ity (27) has first- and second-order derivatives, its solution Hence the result is proved. canbeobtainedthroughnonlineariteration.Butitwillcon- sume too much time. We will use standard central difference Since the parameter 𝜎̃ is nonlinear in the model (6), it’s scheme to approximate it. Hence we get the new numerical difficult to get its analytic solution. Then we will give its scheme as follows: numerical scheme in the next section. 𝑛+1 𝑛 𝑢𝑖 −𝑢𝑖 𝑘 4. Crank-Nicolson Scheme 1 𝑢𝑛 −2𝑢𝑛 +𝑢𝑛 𝑢𝑛+1 −2𝑢𝑛+1 +𝑢𝑛+1 − 𝜌𝑛 [ 𝑖+1 𝑖 𝑖−1 + 𝑖+1 𝑖 𝑖−1 ] Firstly we should reduce the three-dimensional nonlin- 2 1,𝑖 ℎ2 ℎ2 ear mathematical model (6)intoacorrespondingtwo- dimensional model so that it can be solved more easily. Let 𝑛 𝑛 𝑛+1 𝑛+1 1 𝑛 𝑢𝑖+1 −𝑢𝑖−1 𝑢𝑖+1 −𝑢𝑖−1 + 𝜌2,𝑖 [ + ]=0, 𝐽𝑡 2 2ℎ 2ℎ 𝑥=ln ,𝑉(𝑆𝑡,𝐽𝑡,𝑡)=𝑆𝑡𝑢 (𝑥, 𝜏) ,𝜏=𝑇−𝑡;(25) 𝑆𝑡 (30) 𝑛 2 2𝐻−1 𝑛 𝑛 consequently 𝜌1,𝑖 =2𝜎 (𝐻(𝑇𝑛 −𝜏 ) −𝐿𝑒(𝐻) sign (Δ 𝑖 −∇𝑖 )) , (31) 𝜕𝑢 1 𝜕𝑢 1 𝜕2𝑢 𝜌𝑛 =𝑟+𝜌𝑛 , +(𝑟+ 𝜎̂2) − 𝜎̂2 =0, 2,𝑖 1,𝑖 𝜕𝜏 2 𝜕𝑥 2 𝜕𝑥2 with (0<𝑥<∞,0≤𝜏≤𝑇) , 𝑢𝑛 −2𝑢𝑛 +𝑢𝑛 𝑢𝑛 −𝑢𝑛 (26) 𝑛 𝑖+1 𝑖 𝑖−1 𝑛 𝑖+1 𝑖−1 𝑥 Δ 𝑖 = ,∇𝑖 = . (32) 𝑢 (𝑥, 𝜏)|𝜏=0 =𝑒 −1, (0≤𝑥<∞) , ℎ2 2ℎ 󵄨 𝜕𝑢󵄨 So the numerical scheme equals 󵄨 =0, (0≤𝜏≤𝑇) , 𝜕𝑥 󵄨𝑥=0 𝑛 𝑛+1 𝑛 𝑛+1 𝑛 𝑛+1 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑎𝑖 𝑢𝑖+1 +𝑏𝑖 𝑢𝑖 +𝑐𝑖 𝑢𝑖−1 =𝛼𝑖 𝑢𝑖+1 +𝛽𝑖 𝑢𝑖 +𝛾𝑖 𝑢𝑖−1, where (33) 𝑖=1,2,...,𝑀−1, 2 2 2𝐻−1 𝜎̂ =2𝜎 (𝐻(𝑇−𝜏) −𝐿𝑒(𝐻) sign (𝑢𝑥𝑥 −𝑢𝑥)) . (27) where Then the Crank-Nicolson scheme will be constructed for the 𝑘 𝑘 nonlinear problem (26). Consider the problem to a finite 𝛼𝑛 = 𝜌𝑛 − 𝜌𝑛 , 𝑖 2ℎ2 1,𝑖 4ℎ 2,𝑖 domain Ω=[0,𝑥max]×[0,𝑇]. Discrete it using the following method and construct a group of grid points (𝑥, 𝜏) 𝑖= (𝑥 ,𝜏𝑛). 𝑘 𝛽𝑛 =1− 𝜌𝑛 , Let 𝑖 ℎ2 1,𝑖 𝑢 =𝑢(𝑥,𝜏 ), 𝑥 = 𝑖ℎ, 𝑡 =𝑛𝑘, 𝑘 𝑘 𝑖,𝑛 𝑖 𝑛 𝑖 𝑛 𝛾𝑛 = 𝜌𝑛 + 𝜌𝑛 , (28) 𝑖 2 1,𝑖 2,𝑖 𝑖=0,1,2,...,𝑀, 𝑛=0,1,2,...,𝑁. 2ℎ 4ℎ 𝑘 𝑘 𝑎𝑛 =− 𝜌𝑛 + 𝜌𝑛 , With the time step 𝑘 = 𝑇/𝑁 and spatial step ℎ=𝑥max/𝑀. 𝑖 2ℎ2 1,𝑖 4ℎ 2,𝑖 International Scholarly Research Notices 5

𝑘 𝑏𝑛 =1+ 𝜌𝑛 , Substituting the boundary conditions (37)and(39)into(35) 𝑖 2 1,𝑖 ℎ and using chasing method gives this format’s solution. The 𝑠𝑢𝑁 𝑛 𝑘 𝑛 𝑘 𝑛 value of the lookback put option at the starting time is 0 . 𝑐 =− 𝜌 − 𝜌 . 𝐽 𝑖 2ℎ2 1,𝑖 4ℎ 2,𝑖 Itcanbeseenthatitdoesnotcontainthevariable 𝑡 in thenewnumericalschemesothatitavoidsthedifficultyof (34) determining the variable 𝐽𝑡 in the iterative process. The format (33) is a six-point implicit scheme, whose local 2 2 truncation error is 𝑂(ℎ +𝑘 ).Onlyalittlecomputationcan 5. Numerical Example achieve satisfactory accuracy. For numerical convenience, we 𝑛 𝑛 𝑛 𝑛 𝑇 In this section, we will firstly study the convergence of the define a vector 𝑢 =[𝑢1,𝑢2,...,𝑢𝑀−1] .Thentheformat(33) canbegivenasbelow: numerical scheme to explain its validity and discuss the influence of various parameters on the lookback option’s 𝑛+1 𝑛 𝑛 value. For simplicity, we will only consider valuation of 𝐴 (𝑛) 𝑢 =𝐵(𝑛) 𝑢 +𝑝 , 𝑛=0,1,2,...,𝑁−1. (35) lookback put option for 𝐻 ∈ (0.5, 1).

Here Example 1. Consider a European put option in half a year 𝑛 𝑛 when the underlying asset is a stock, whose price follows 𝑏1 𝑎1 0 0 ⋅⋅⋅ 0 [ ] Fractional Brown Motion (3),andtheoptionvalueissatisfied [ ] [𝑐𝑛 𝑏𝑛 𝑎𝑛 0 ⋅⋅⋅ 0 ] by the model (6). Suppose that the stock price is $30, the [ 2 2 2 ] [ ] volatility is 20% per annum, the risk-free interest rate is 5%, [ 𝑛 𝑛 𝑛 ] [ 0𝑐 𝑏 𝑎 ⋅⋅⋅ 0 ] the Hurst parameter is 0.6, the hedging time interval is 0.005, [ 3 3 3 ] 𝐴 (𝑛) = [ ] , and the number of time steps is taken as 100, spatial steps [ ] [ dd d ] taken as 300 in the numerical scheme (35). With our usual [ ] [ 𝑛 𝑛 𝑛 ] notation, this means that [ 0 0 ⋅⋅⋅ 𝑐𝑀−2 𝑏𝑀−2 𝑎𝑀−2] [ ] 𝑇 = 0.5, 𝑟 = 0.05, 𝜎 =0.2, 𝑛 𝑛 [ 0 0 ⋅⋅⋅ 0 𝑐𝑀−1 𝑏𝑀−1 ] 𝑆𝑡 = 30, 𝐻 = 0.6, 𝛿𝑡 = 0.005, (40) 𝛽𝑛 𝑎𝑛 0 0 ⋅⋅⋅ 0 1 1 (36) [ ] 𝑀 = 300, 𝑁 = 100. [ ] [𝛾𝑛 𝛽𝑛 𝑎𝑛 0 ⋅⋅⋅ 0 ] [ 2 2 2 ] [ ] For different values of Hurst parameter, Table 1 shows [ 𝑛 𝑛 𝑛 ] [ 0𝛾 𝛽 𝑎 ⋅⋅⋅ 0 ] the price for lookback put option with transaction cost [ 3 3 3 ] 𝐵 (𝑛) = [ ] , through Crank-Nicolson numerical approach and proves its [ ] [ dd d ] convergence of this method. From Table 1, we can also see [ ] that, with the increase of Hurst parameter, the magnitude [ ] [ 00⋅⋅⋅𝛾𝑛 𝛽𝑛 𝛼𝑛 ] of decrease for lookback put option value is increasing. This [ 𝑀−2 𝑀−2 𝑀−2] implies the self-similarity of fBm model for 𝐻 ∈ (0.5, 1). 𝑛 𝑛 [ 00⋅⋅⋅0 𝛾𝑀−1 𝛽𝑀−1] Then we study the influence of transaction cost rates on lookback put option value. Other values of parameters 𝑛 𝑛 𝑛 𝑛 𝑛+1 𝑛 𝑛 𝑛 𝑛+1 𝑇 𝑝 =[𝛾1 𝑢0 −𝑐1 𝑢0 ,0,...0,𝛼𝑀−1𝑢𝑀 −𝑎𝑀−1𝑢𝑀 ] . remain unchanged. Figure 1 displays that the value of the option is decreasing with the increase of transaction costs. 𝑛 We also need to know the value of the boundary points 𝑢𝑀 This is mainly because in this paper, for the long position 𝑛 and 𝑢0. So discrete the boundary conditions by second-order of options, we only consider the purchase price. Transaction Gear formula. rates’ increase makes a unit of an asset’s cost increase, directly Hence leading to the option hedging costs’ raising. So option value will decrease. Besides, with the initial stock price increased, 󵄨 3𝑢𝑛 −4𝑢𝑛 +𝑢𝑛 4𝑢𝑛 −𝑢𝑛 𝜕𝑢󵄨 0 1 2 𝑛 1 2 thelookbackputoptionvalueincreases. 󵄨 = =0, that is, 𝑢0 = . 𝜕𝑥󵄨𝑥=0 2ℎ 3 Figure 2 shows the effect of time 𝑇and stock (37) price 𝑆𝑡 on lookback put option value. From it we can see that longer maturity time can produce larger option value. When the price of underlying asset 𝑆𝑡 reaches 0, the lookback Moreover, when the expiration time is shorter, it grows put option must be exercised. Therefore another boundary moreobviously.Thisismainlybecausethattheoptionvalue condition can be given as increases over time. Then we discuss the impact of risk-free rate and volatility −𝑟(𝑇−𝑡) 𝑉 (0, 𝐽𝑡,𝑡) =𝑒 𝐽𝑡. (38) change on lookback put option value. From Figure 3,itcan be seen that the option value decreases with risk-free rate According to variable substitution (25), we have increasing. One factor is that the expected rate of return risesasrisk-freerategrows.Anotherfactoristhatrisk-free 𝑛 −𝑟𝜏 𝑢𝑀 =𝑒 . (39) rate’s growth leads to the present value of future cash flows 6 International Scholarly Research Notices

Table 1: Convergence for lookback put option with transaction cost in the fBm Process.

𝐻 Number of time steps 100 300 500 1000 2000 3000 4000 5000 0.6 2.6325 2.6281 2.6271 2.6264 2.6261 2.6259 2.6259 2.6258 0.7 2.5247 2.5186 2.5174 2.5164 2.5159 2.5158 2.5157 2.5157 0.8 2.4008 2.3940 2.3926 2.3916 2.3910 2.3909 2.3908 2.3907 0.9 2.2697 2.2625 2.2610 2.2599 2.2594 2.2592 2.2591 2.2590

25 12 20 10 15 8 10 6 5 4 0

2 Lookback put option value 0.8 0.6 1 0 0.8 Lookback put option value option put Lookback 8 Risk-fr0.4 0.6 ee r 0.2 0.4 6 100 ate 0 0 0.2 Transaction rate 80 Volatility 4 60 2 40 20 Figure 3: Lookback put option value along with risk-free rate and 0 Stock price 0 volatility. ×10−3

Figure 1: Lookback put option value along with transaction costs and stock price. empirical price to reflect the rationality of the model pro- posed. China Guodian warrants was listed on May 22, 2008, andexercisedonMay21,2010,withthestrikeprice7.47yuan andtherightratio1.00.Fromthehistoricaldataofthestock 15 price for China Guodian power (600795) during the time interval from March 18, 1997, to November 1, 2009, it can be estimated that the volatility 𝜎 = 0.5937. Assume that interest 10 rate 𝑟=2%andtransactioncostrate𝜅 = 0.001. Table 2 below shows the statistical characteristics of the 5 logarithmic rate of return for China Guodian warrants. It canbeobservedthatthevalueofkurtosisforthewarrantsis 0 greater than that for the standard normal distribution and the Lookback put option value option put Lookback 1 probability to obtain corresponding statistic 𝐽-𝐵 is less than 100 0.05, which indicates share earnings is distributed the feature Exp 80 iration0.5 time 60 aiguille large remaining part. 40 20 0 0 Stock price By comparison between theoretical and empirical prices, Figure 4 shows pricing errors under the pricing model of Figure 2: Lookback put option value along with expiration time and lookback option with transaction costs based on fractional stock price. Brownian motion process (𝐻 = 0.7) and Black-Scholes model, respectively. As can be seen from Figure 4, the pricing errors under FBM model are significantly less than BS model. dropping. On the other hand, the larger the volatility, the This indicates that the new model involved in this paper is larger the option value. It is mainly because that volatility’s more realistic. increase amplifies the probability of making large gains in the future. 6. Conclusions Finally, we illustrate the advantages of the model in this paper from the data in financial market. Taking China In this paper, the lookback option pricing problem, with Guodian warrants for example, we will use the new model proportional transaction costs under the Fractional Brown presented in this paper and Black-scholes model to reprice Motion model, was studied. By using hedging principle, the warrants and obtain the error between theoretical and the nonlinear partial differential equations, satisfied by International Scholarly Research Notices 7

Table 2: The basic statistical characteristics of China Guodian [3] M. Jandackaˇ and D. Sevˇ coviˇ c,ˇ “On the risk-adjusted pricing- warrants (data sources: China Tai’an database). methodology-based valuation of vanilla options and explana- tion of the volatility smile,” Journal of Applied Mathematics,vol. Statistics Rates of return for China Guodian warrants 2005, no. 3, pp. 235–258, 2005. − Mean value 0.00025 [4] P. Amster, C. G. Averbuj, M. C. Mariani, and D. Rial, “A Black- Maximum 0.288731 Scholes option pricing model with transaction costs,” Journal Minimum −0.15535 of Mathematical Analysis and Applications,vol.303,no.2,pp. Variance 0.00168 688–695, 2005. Median 0.002021 [5] M. C. Mariani, I. SenGupta, and P. Bezdek, “Numerical solu- tions for option pricing models including transaction costs and Skewness 0.905533 stochastic volatility,” Acta Applicandae Mathematicae,vol.118, Kurtosis 10.6741 pp. 203–220, 2012. 𝐽 𝐵 - statistics 875.5861 [6]M.B.Goldman,H.B.Sossin,andM.A.Gatto,“Pathdependent Probability 0.001 options: buy at the low, sell at the high,” Journal of ,vol. 34, pp. 1111–1127, 1979. [7] A. Conze and R. Viswanathan, “Path dependent options, the 2 case of lookback options,” The Journal of Finance,vol.46,no. 5, pp. 1893–1907, 1991. 1.5 [8]R.C.HeynenandH.M.Kat,“Lookbackoptionswithdiscrete and partial monitoring of the underlying price,” Applied Math- 1 ematical Finance,vol.2,pp.273–284,1995. [9] L. C. Rogers, “Arbitrage with fractional Brownian motion,” 0.5 Mathematical Finance,vol.7,no.1,pp.95–105,1997. [10] A. N. Shiryaev, “On arbitrage and replication for fractal models,” 0 Research Report 30, Department of Mathematical Sciences, University of Aarhus, MaphySto, Aarhus, Denmark, 1998.

Warrant price error price −0.5 [11] D. M. Salopek, “Tolerance to arbitrage,” Stochastic Processes and Their Applications, vol. 76, no. 2, pp. 217–230, 1998. −1 [12] P.Cheridito, “Arbitrage in fractional Brownian motion models,” Finance and Stochastics,vol.7,no.4,pp.533–553,2003. −1.5 [13]H.Gu,J.R.Liang,andY.X.Zhang,“Time-changedgeometric 0 0.5 1 1.5 fractional Brownian motion and option pricing with transac- Deal time tion costs,” Physica A: Statistical Mechanics and Its Applications, BS model vol. 391, no. 15, pp. 3971–3977, 2012. FBM model [14] D. Y. Feng, “Study of lookback option pricing in fractional Brownian motion environment,” The Journals of North China Figure 4: Comparison of pricing errors of China Guodian warrants University of Technology,vol.32,pp.67–72,2009. under two different models. [15] S. J. Lin, “Stochastic analysis of fractional Brownian motion,” Stochastic and Stochastic Reports,vol.55,pp.422–437,1995. [16] L. Decreusefond and A. S. ust¨ unel,¨ “Stochastic analysis of the fractional Brownian motion,” Potential Analysis,vol.10,no.2, the option value, have been derived. As for the equations, we pp.177–214,1999. have constructed the Crank-Nicolson scheme and verified its [17]Y.HuandB.Øksendal,“Fractionalwhitenoisecalculusand effectiveness by using the Matlab software. It can be seen that applications to finance,” Infinite Dimensional Analysis, Quan- themodelwehavebuiltupismoreinlinewiththerealityof tum Probability and Related Topics,vol.6,no.1,pp.1–32,2003. the market environment through Matlab software simulation. [18] C. Bender, “An Itoˆ formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter,” Stochastic Processes and Their Applications,vol.104,no.1,pp. Conflict of Interests 81–106, 2003. The authors declare that there is no conflict of interests [19] Y. Z. Hu and S. G. Peng, “Backward stochastic differential equation driven by fractional Brownian motion,” SIAM Journal regarding the publication of this paper. on Control and Optimization,vol.48,no.3,pp.1675–1700,2009. [20] C. H. John, Options, Futures, and other Derivatives, Simon & References Schuster, New York, NY, USA, 2005.

[1] H. E. Leland, “Option pricing and replication with transaction costs,” Journal of Finance,vol.40,pp.1283–1301,1985. [2] G. Barles and H. M. Soner, “Option pricing with transaction costs and a nonlinear Black-Scholes equation,” Finance and Stochastics, vol. 2, no. 4, pp. 369–397, 1998. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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