This paper deals with the problem of discrete time option pricing using the fractional Black–Scholes model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained. In addition, the relation between scaling and implied volatility smiles is discussed.
Over the last few years, the financial markets have been regarded as complex and nonlinear dynamic systems. A series of studies has found that many financial market time series display scaling laws and long-range dependence. Therefore, it has been proposed that one should replace the Brownian motion in the classical Black–Scholes model [1] by a process with long-range dependence. A simple modification is to introduce fractional Brownian motion (fBm) as the source of randomness. Thus one adds one parameter, HH, to model the dependence structure in the stock prices (for references to these studies see [2], [3], [4], [5], [6], [7] and [8]). The fractional Black–Scholes model is a generalization of the Black–Scholes model, which is based on replacing the standard Brownian motion by a fractional Brownian motion in the Black–Scholes model. Since fractional Brownian motion is not a semimartingale [9], it has been shown that the fractional Black–Scholes model admits arbitrage in a complete and frictionless market [4], [5], [6], [7] and [9]. However, Guasoni [8] has proved that proportional transaction costs of any positive size eliminate arbitrage opportunities from the fractional Black–Scholes model, but he did not give any corresponding option pricing formulas. Therefore, in a more realistic situation of transaction costs, the magnitude of arbitrage returns associated with those trading strategies in [4], [5], [6], [7] and [10] may create an illusion of profit opportunity when, in fact, none exists.
In this paper, on the basis of the points of view of behavioral finance [11] and [12] and econophysics [13] and empirical findings of the long-range dependence in stock returns in [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] and [27], we will study the option pricing problem under transaction costs while the dynamics of stock price StSt satisfies Leland[
wasthefirstwhoexaminedoptionreplicationinthepresenceoftransactioncostsinadiscretetimesetting.
FromthepointofviewofLeland[
],inamodelwheretransactioncostsareincurredeverytimethestockorthebond
is traded, the arbitrage-free argument used by Black and Scholes [
1
] no longer applies. The problem is that, due to the
infinite variation of the geometric Brownian motion, perfect replication incurs an infinite amount of transaction costs.
Hence, he suggested a delta-hedging strategy incorporating transaction costs based on revision at a discrete number of
times.Transactioncostsleadtothefailureoftheno-arbitrageprincipleandthecontinuous-timetradeingeneral:instead
ofnoarbitrage,theprincipleofhedgepricing–accordingtowhichthepriceofanoptionisdefinedastheminimumlevel
ofinitialwealthneededtohedgetheoption–comestothefore.
Mandelbrot (for more details, see the discussions in [
]) proposed the trading time concept and considered the
problemofchoosingtheappropriatetimescalingtouseforanalyzingfinancialmarketdataandpricingoptions.InSection
byusingadelta-hedgingstrategy,initiatedbyLeland[
],wewillshowthatthepriceofEuropeanoptionswithtransaction
costsunderthefractionalBlack–Scholesmodelaredeterminedbythetradingtimeintervalswhichvarywithrespectto
time
.InSection
,fromthepointofviewofbehavioralfinance,wegiveanexplanationfortheimpliedvolatilitysmilein
optionpricing.InSection
,aconclusionisgiven.
Withoutusinganarbitrageargument,inthispaperweobtainaEuropeancalloptionpricingformulawithtransaction
costsforthefractionalBlack–ScholesmodelwithHurstexponent
.Ithasbeenshownthatthetimescaling
and
theHurstexponent
H
playanimportantroleinoptionpricingwithtransactioncosts.Inparticular,for
,theminimal
priceofanoptionundertransactioncostsisobtained,whichcanbeusedastheactualpriceofanoption.Inaddition,we
alsoshowthattheoptionrehedgingtimeinterval
varieswithrespecttotime
andthetimescaling
playsanimportantroleindeterminingtheshapeofIVFs.
