aa r X i v : . [ q -f i n . M F ] J a n Change of measure under the hard-to-borrow model
Peng Liu ∗ aa Department of Finance and Insurance, Nanjing UniversityJanuary 29, 2020
Abstract
As the Securities and Exchange Commission(SEC) has implemented anew regulation on short-sellings, short-sellers are required to repurchasestocks once the clearing risk rises to a certain level. Avellaneda and Lipkinproposed a fully coupled SDE system to describe the mechanism whichis referred as Hard-To-Borrow(HTB) models. Guiyuan Ma obtained thePDE system for both American and European options. There is a tech-nical error in Guiyuan Ma where two correlated Brownian motion shouldbe converted before change of measure. In this paper, I will provide sup-plement conditions.
One of the most important tasks in mathematical finance is to determine anoption’s value. The foundation of modern option pricing theory was attributedto Black Scholes[1], Merton[2] who first proposed the analytical solution for Eu-ropean options. Black and Scholes[1] assumed the market is complete, meaningthe short-sellings on underlying assets are permitted without costs. However,short-sellings are restricted in many developed markets, or even forbidden inmost emerging ones. Generally, short-selling requires borrowing stocks fromothers where different stakeholders are involved. The broker is going to arrangea buyer once the stock has been indicated by the short-seller. Then the short-seller need return stocks to the clearing firm within a negotiated period. Stockscan be borrowed from a stock-loan desk. The availability of stocks for borrow-ing depends on market conditions. There are many stocks which can be easilyborrowed without any lending fees, which exactly fits one of Black-Scholes as-sumptions, while others could be in short supply. In the latter case, short-sellingthose stocks, referred as hard-to-borrow (HTB) stocks, could be quite costly.Normally, there would not be any unexpected cash flows before the set-tlement. Regulators have paid attention on the clearing risk in short-sellings. ∗ Corresponding author: [email protected], [email protected] buy-in mechanism, has been implemented by the Securities and Ex-change Commission(SEC), clearing firms representing short-sellers are requiredto repurchase stocks to cover shortfalls once the clearing risk reaches to a cer-tain level. To describe the buy-in mechanism associated with hard-to-borrowstocks, Avellaneda and Lipkin[3] proposed a dynamic model (it is referred toas the HTB model) by introducing two coupled SDEs. The HTB model hasattracted attention from different aspects. Guiyuan Ma and Song-ping Zhu [4]studied the early exercise of American option on the HTB model. They con-firmed that it is the lending fee that results in the early exercise of Americancall options and we shall also demonstrate to what extent lending fees have af-fected the early exercise decision. Yong Chen and Jingtang Ma[5] introduced anew expansion approach to deal with the spatial delay term. They comparedthe modified Laplace transform method with the time-stepping finite differencemethod. The numerical comparison indicates that the modified Laplace trans-form method outperforms the modified FDMs. Guiyuan Ma, Song-ping Zhu,Wenting Chen[6] studied European call option pricing problem under the HTBmodel. Through their numerical results, they find that the semi-explicit formulais a good approximate solution when the coupling parameter is small. Whenthe stock price and the buy-in rate are significantly coupled, the PDE approachis preferred to solve the option pricing problem under the full hard-to-borrowmodel.The contribution of this paper is amending a technical error during changeof measure. Avellaneda and Lipkin [3] claim that the correlation of two Brown-ian motion is irrelevant. The situation where two Brownian motion W ( t ) , Z ( t )are correlated has not been discussed which is contrary to the footprint of []. Inthe following part, I will provide a mathematical proof that the change of mea-sure depends on the covariance between W ( t ) and Z ( t ). To change of measureunder multi-dimension Girsanov theorem, multi-dimension Brownian motion B ( t ) = ( B ( t ) , B ( t )) are introduced to convert the correlated Brownian mo-tions W ( t ) , Z ( t ) into a two-dimension Brownian motion. My future work willfocus on fast simulation for the HTB model. Avellaneda and Lipkin[3] modelled the buy-in mechanism with two coupledSDEs. dS t S t = σdW t + γλ t dt − γdN λ t ( t ) dx t = κdZ t + α ( x − x t ) dt + β dS t S t , x t = ln ( λ t λ ) (2.1)The first equation describes the logarithmized price process with a Brow-nian motion W t and a compensated Poisson process γλ t dt − γdN λ t ( t ). Short-sellers would suffer from ”squeezes” triggered by buy-ins. The profit or loss2uring a time interval (t,t+dt) affected by buy-ins can be represented as P N L = − dS t − ξγS t = − S t ( σdW t + λ t γdt ) P rob. { ξ = 0 } = 1 − λ t dt + o ( dt ) P rob. { ξ = 1 } = λ t dt + o ( dt ) (2.2)The second Mean-Reverting process characterizes the logarithmized Pois-son intensity λ of the previous equation and another Brownian motion Z t isintroduced. Two Brownian motions W t , Z t with volatility σ, κ respectively arecorrelated with coefficient ρ = cov ( W t , Z t ) while another parameter β deter-mines how deeply two equations are coupled. γ is the price elasticity of demanddue to buy-ins. α determines the reverting speed and x is the long-term levelof x . The measure change of the HTB models was initially proposed by Avel-laneda and Lipkin. Avellaneda and Lipkin[3] assumed that short-sellers do notbenefit from the downward jumps because short-sellers are no longer short bythe time the buy-in is completed. Since jumps and buy-ins occur with frequency λ t , the expected economic gain is λ t γS t . λ t γ can be viewed as the cost-of-carryfor borrowing the stock. Statistically the economic costs of paying rent or risk-ing buy-ins are equivalent. In particular, the cost of carrying (or financing)stock can be quantified in terms of λ and the interest rate.An arbitrage-free pricing measure associated with the physical process wasintroduced. After change of measure, the first equation of 2.1 should take theform: dS t S t = σdW t + rdt − γdN λ t ( t ) (3.1)where r is the instantaneous interest rate. The absence of the drift term λ t γ inthis last equation is due to the fact that, under an arbitrage pricing measure,the price process adjusted for dividends and interest is a martingale. The risk-neutral measure was formally obtained by Guiyuan Ma[6]. Toconduct measure transform, Guiyuan Ma defined two new processes as f W t = W t + Z t γλ l − rσ dl (3.2) f Z t = Z t + Z t αz ( l, x l , S l ) κ dl (3.3)3he risk-neutral measure was defined by the Randon-Nikodym derivative d Q d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = exp {− Z t [ γλ l − rσ + αz ( l, x l , S l ) κ ] dW l − Z t [ ( γλ l − r ) σ + α z ( l, x l , S l ) κ ] dl } (3.4)Guiyuan Ma explained that z ( t, x, S ) is an arbitrary function for the uncer-tainty raised by buy-ins. Any source of uncertainty needs to be compensated bythe associated market price of risk or risk premium. In the classic BlackScholesmodel[1],[2], the market price of risk for the underlying is µ − rσ . In the Hes-ton model[7], an additional source uncertainty is introduced by the stochasticvolatility and an additional market price of volatility risk is defined through anarbitrary function, i.e., λ ( t, S, v ) In the HTB model, the new buy-in processalso brings in an additional source of uncertainty and the corresponding mar-ket price of buy-in risk is represented by the function z ( t, x, S ). HTB modeloperates in an incomplete market where it is impossible to perfectly hedge aportfolio composed of hard-to-borrow stocks and a unique risk-neutral measuredoes not exist. The market price of buy-in risk in the HTB model should bedetermined by market data. Avellaneda and Lipkin[3] claimed that the correlation of two Brownianmotion W t , Z t in 2.1 is irrelevant. The mathematical result obtained by GuiyuanMa[6] also neglect the situation where two Brownian motion are correlated.The contribution of my work is to give supplement conditions before change ofmeasure.For convenience, we denote Γ( l ) = γλ l − rσ ,Θ( l ) = αz ( l,x l ,S l ) κ To change of measure under multi-dimension Girsanov theorem, multi-dimension Brownian motion B ( t ) = ( B ( t ) , B ( t )) are introduced to convertthe correlated Brownian motions W ( t ) , Z ( t ) into a two-dimension Brownianmotion. Proposition 1 B ( t ) , B ( t ) are two independent Brownian motion, Y ( t ) = B ( t ) X ( t ) , X ( t ) are two correlated Brownian motion with covariance ρ Proof X ( t ) = B ( t ) X ( t ) = ρB ( t ) + p − ρ B ( t ) (4.1) dX ( t ) dX ( t ) = ρ dB ( t ) dB ( t ) + 2 ρ p − ρ dB ( t ) dB ( t )+ (1 − ρ ) dB ( t ) dB ( t )= ρ dt + (1 − ρ ) dt = dt (4.2) dX ( t ) dX ( t ) = ρ dB ( t ) dB ( t ) + p − ρ dB ( t ) dB ( t )= ρdt (4.3)4 irsanov theorem [8] Let Y ( t ) ∈ R n be an It ˆ o process of the form dY ( t ) = β ( t, ω ) dt + θ ( t, ω ) dB ( t ) where B ( t ) ∈ R m , β ( t, ω ) ∈ R n and θ ( t, ω ) ∈ R n × m . Suppose there existprocesses u ( t, ω ) ∈ W m H and α ( t, ω ) ∈ W n H such that θ ( t, ω ) u ( t, ω ) = β ( t, ω ) − α ( t, ω ) Put M t = exp {− Z t u ( s, ω ) dB s − Z t u ( s, ω ) ds } ; t ≤ T (4.4) and dQ ( ω ) = M T ( ω ) dP ( ω ) on F ( m ) T (4.5) Assume that M t is a martingale( ω.r.t. F ( n ) t and P) . Then Q is a probabilitymeasure on F ( m ) T ,the process ˆ B ( t ) := Z t u ( s, ω ) ds + B ( t ); t ≤ T (4.6) is a Brownian motion ω.r.t .Q and in terms of ˆ B ( t ) the process Y(t) has thestochastic integral representation dY ( t ) = α ( t, ω ) dt + θ ( t, ω ) d ˆ B ( t ) (4.7) If n=m and θ ∈ R n × n is invertible, then the process u ( t, ω ) satisfying isgiven by u ( t, ω ) = θ − ( t, ω )[ β ( t, ω ) − α ( t, ω )] (4.8) Proposition 2
The stochastic process Y ( t ) = ( f W t , f Z t ) ( − < ρ < , Journal of Political Economy , 81:637–657, 1973.[2] Robert C. Merton. Theory of Rational Option Pricing. The Bell Journal ofEconomics and Management Science , 4:141–183, 1973.[3] Marco Avellaneda and Mike Lipkin. A dynamic model for Hard-to-BorrowStocks. Risk , (June):92–97, 2009.[4] Guiyuan Ma and Song Ping Zhu. Pricing American call options under ahard-to-borrow stock model. European Journal of Applied Mathematics ,29(3):494–514, 2018.[5] Yong Chen and Jingtang Ma. Numerical methods for a partial differentialequation with spatial delay arising in option pricing under hard-to-borrowmodel. Computers and Mathematics with Applications , 76(9):2129–2140,2018.[6] Guiyuan Ma, Song Ping Zhu, and Wenting Chen. Pricing European call op-tions under a hard-to-borrow stock model. Applied Mathematics and Com-putation , 357:243–257, 2019.[7] Steven L. Heston. A Closed-Form Solution for Options with StochasticVolatility with Applications to Bond and Currency Options. Review of Fi-nancial Studies , 6:327–343, 1993.[8] Oksendal Bernt.