Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

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1 Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July 008 Absrac We are concerned wih he opimal decision o sell or buy a sock in a given period wih reference o he ulimae average of he sock price. Sricly speaking, we aim o deermine an opimal selling buying ime so as o maximize minimize he expecaion of he raio of he selling buying price o he ulimae average price over he period. This is an opimal sopping ime problem which can be formulaed as a variaional inequaliy problem. The associaed sopping region corresponds o he opimal selling buying sraegy. We provide a parial differenial equaion approach o sudy he opimal sraegy. I urns ou ha he opimal selling sraegy is bang-bang, which is he same as ha obained by Shiryaev, Xu and Zhou 008 aking he ulimae maximum of he sock price as he benchmark. However, he opimal buying sraegy can be a feedback one subjec o he ype of average and parameer values. Keywords. opimal sraegy, average price, opimal sopping problem, variaional inequaliy AMS subjec classificaions. 35Q80, 60G40, 9B8 Inroducion Assume ha he discouned sock price evolves according o ds = µs d + σs db, This projec is suppored by NUS academic research fund and NUS RMI research fund. The auhors hank seminar paricipans a Oxford Universiy and Tongji Universiy for helpful discussion and commens. All errors are our own. madm@nus.edu.sg. Fax: Elecronic copy available a: hp://ssrn.com/absrac=884

Sricly speaking, we aim o deermine an opimal selling buying ime so as o maximize minimize he expecaion of he raio of he selling buying price o he ulimae average price over he period.

2 where consans µ, + and σ > 0 are he discouned expeced rae of reurn and volailiy, respecively, and {B ; > 0} is a sandard -dimension Brownian moion on a filered probabiliy space S, F, {F } 0, P wih B 0 = 0 almos surely. We are ineresed in he following opimal decision o sell or buy a sock in a given period [0, T ] wih reference o he ulimae average: Buy case: Sell case: min E max E Sν Sν,.,. where E sands for he expecaion, ν is a sopping ime, and he benchmark value is aken as eiher geomeric or arihmeic average price over he period [0,T], namely, { T exp T = log S 0 νdν, geomeric average, T S.3 T 0 νdν, arihmeic average. Problem. and. are moivaed by Shiryaev, Xu and Zhou 008 ha sudied he opimal sock selling sraegy wih reference o he ulimae maximum, ha is, he benchmark is aken as max S ν. They derived a surprising opimal selling sraegy: one eiher sells he sock immediaely or holds i unil expiry. More precisely, if µ > σ /, i is opimal o hold he sock unil expiry; if µ σ /, i is opimal o sell he sock immediaely a ime zero. Naurally one may ask why his leads o such a bang-bang selling sraegy. Wheher do we sill have such a simple sraegy if we insead use he average price as he benchmark? Wha happens for he buy case? This paper aemps o answer hese quesions. To analyze he opimal sraegy, Shiryaev, Xu and Zhou 008 adoped a sochasic analysis approach which was also employed by Graversen, Peskir and Shiryaev 00, Pedersen 003 and Du Toi and Peskir 007 where various models of predicing he maximum of a Brownian moion were sudied. In he presen paper, he arihmeic average involved makes he problems inracable. We will make use of a parial differenial equaion PDE approach o overcome i. The res of he paper is organized as follows. In he subsequen secion, we formulae problem. and. as variaional inequaliy problems also called obsacle problems. In secion 3, we confine o he case of geomeric average in which he problems allow analyical soluions and he opimal sraegy can be readily figured ou. Ineresingly, he opimal selling sraegy is sill bang-bang, whereas he opimal buying sraegy can be a feedback one subjec o parameer values. Moreover, he sraegy only depends on he ime o expiry. Secion 4 is devoed o he case of arihmeic average. Due o lack of analyical soluions, we provide a horough heoreical analysis on he opimal sraegy. The resuling sraegy resembles ha of he geomeric average case, bu depends on he raio of he sock price o he running average in addiion o he ime o expiry in some scenarios. We conclude he paper in secion 5. There is a gap in heir paper abou he scenario of 0 < µ < σ /. Bu he gap can be fixed by virue of he approach adoped in he presen paper. Elecronic copy available a: hp://ssrn.com/absrac=884

T ] wih reference o he ulimae average: Buy case: Sell case: min E max E Sν Sν,.

3 PDE formulaion In his secion, we will provide a PDE formulaion for he opimal sopping problems. and.. Le us begin wih he buy case.. Buy case As in.3, we denoe by A he running average over [0, ]. Then, we can wrie he value funcion associaed wih problem. as ϕs, A, =. [ ] min E Sν = min ν T A E S ν E ν,. T ν T where E = E F. Denoe φs, A,. = E { A. Since log S da = A d, geomeric case S A d, arihmeic case. = fs, A, d, i is easy o see ha φ saisfies [cf. Wilmo, Dewynne and Howison 995, Jiang and Dai 004, or Dai and Kwok 006] { L0 φ = 0, 0 < S, A <, 0, T, φs, A, T = A,. where L 0 = σ S SS µs S fs, A, A. I follows from. ha ϕs, A, = min ν T E [S ν φs ν, A ν, ν],.3 which is governed by { max{l0 ϕ, ϕ Sφ} = 0, ϕs, A, T = S, A 0 < S, A <, 0, T,.4 Nex, we will show ha problem. and.4 can be reduced o one-dimensional ime dependen problems. Indeed, by he ransformaion z = A, τ = T, V z, τ = ϕs, A, and Φz, τ = SφS, A,,.5 S.4 reduces o { max{l V, V Φ} = 0, in D, V z, 0 = z,.6 where D = 0, 0, T, L = τ σ z zz σ µz z fz, τ z,.7 3

Since log S da = A d, geomeric case S A d, arihmeic case. = fs, A, d, i is easy o see ha φ saisfies [cf.

4 ha Le us only consider he case of µ σ Φ0, τ = lim Φz, τ = lim T E z 0 + z 0 + S T since he case of µ < σ T = T E expµ σ ν + σb ν dν τ T E e dν σbν, for τ 0, T. 0 T = T lim E z + z 0 + In a similar way, T U 0, τ = lim U z, τ = lim T E S ν z + dν z 0 + z 0 + S T T exp µ σ = T E ν + σb ν dν exp µ σ T + σb T T = T E exp µ σ ν T + σb ν B T dν τ T E e σb ν dν Φ0, τ, for τ 0, T, 0 is similar. Noe S ν dν S where Bν = B T ν B T is also a sandard Brownian moion independen of F. In addiion, we have he equaliy if and only if µ = σ. The proof is complee. References [] Brezis, H., and A. Friedman 976: Esimaes on he suppor of soluions of parabolic variaional inequaliies, Illinois Journal of Mahemaics, 0:8-97. [] Dai, M., Y.K. Kwok and L.X. Wu 004, Opimal shouing policies of opions wih srike rese righs, Mahemaical Finance, 43: [3] Dai, M., and Y.K. Kwok 006: Characerizaion of opimal sopping regions of American pah dependen opions, Mahemaical Finance, 6:63-8. [4] Du Toi, J., and G. Peskir 007: The rap of complacency in predicing he maximum, The Annals of Probabiliy, 35: [5] Friedman, A. 98: Variaional Principles and Free-boundary Problems, Wiley, New York. [6] Graversen, S.E., G. Peskir, and A.N. Shiryaev 00: Sopping Brownian moion wihou anicipaion as close as possible o is ulimae maximum, Theory of Probabiliy and Is Applicaion, 45,

τ, for τ 0, T, 0 is similar. Noe S ν dν S where Bν = B T ν B T is also a sandard Brownian moion independen of F. In addiion, we have he equaliy if and only if µ = σ. The proof is complee.

5 [7] Jiang, L., and M. Dai 004: Convergence of binomial ree mehod for European/American pah-dependen opions, SIAM Journal on Numerical Analysis, 43: [8] Pedersen, J.L. 003: Opimal predicion of he ulimae maximum of Brownian moion, Sochasics and Sochasics Repors, 75, [9] Shiryaev, A., Z.Q. Xu and X.Y. Zhou 008: Thou shal buy and hold, o appear. [0] Wilmo, P., J. Dewynne, and S. Howison 995: The Mahemaics of Financial Derivaives: A Suden Inroducion. Cambridge Universiy Press. 6

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

To appear in Mathematical Finance 1 Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Min Dai Yifei Zhong Dept of Math, National University of Singapore, Singapore First version: