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反弹走廊

Bouncing Corridor

专题
Finance / 金融
难度
L4

题目详情

金融数学题：反弹走廊。

英文原题

Let BtB_{t} be a Brownian Motion and uu and dd two positive real numbers. We consider an option which pays 1 if BtB_{t} reaches uu and touched the down barrier before. The option is knocked out and pays zero if it touches the up barrier first.

pays 1 if ∃t0:Bt0=u;∃t1∈[0,t0]:Bt1=−d\mathrm{pays~1~if~}\exists t_{0}:B_{t_{0}} = u;\exists t_{1}\in [0,t_{0}]:B_{t_{1}} = -d pays 0 if ∃t0:Bt0=u;∀t∈[0,t0],Bt>−d\mathrm{pays~0~if~}\exists t_{0}:B_{t_{0}} = u;\forall t\in [0,t_{0}],B_{t} > - d

payment is made when the barrier is touched. Calculate the price of this option with rates r>0r > 0 . Generalize to an option paying 1 after nn bounces (pays 1 if the

option touches uu then uu then −d- d etc... n times, pays 0 if uu is touched first).

解析

payoff：若先碰到 −d-d，之后再到 uu 则支付 1（到达 uu 时支付），若先到 uu 则 0。

由强马尔可夫性质：先到 −d-d 的贴现概率乘以上移后的单侧命中贴现概率（从 −d-d 到 uu 等价于从 0 到 u+du+d），得到

价格=e−2r(u+d)⋅sinh⁡(2r d)sinh⁡(2r(u+d)).\boxed{\text{价格}=e^{-\sqrt{2r}(u+d)}\cdot\frac{\sinh(\sqrt{2r}\,d)}{\sinh(\sqrt{2r}(u+d))}}.

更多 bounces 可用同样的分解反复迭代，得到几何型衰减因子（每多一次往返通常再乘一个 e−2r(u+d)e^{-\sqrt{2r}(u+d)} 量级的因子）。

英文解析

We define τ\tau the first hitting time of uu or −d- d . τ\tau is a stopping time. If uu is touched first the payoff is zero. If −d- d is touched first the option becomes similar to the one sided barrier case with an upper barrier at (u+d)(u + d) . The price of that option was calculated in 3.5

Priceone side=exp⁡(−2r(u+d))\mathrm{Price}_{\mathrm{one~side}} = \exp (-\sqrt{2r} (u + d))

and the price of the bouncing option is

Price=exp⁡(−2r(u+d))E(ΨDexp⁡(−rτ))\mathrm{Price} = \exp (-\sqrt{2r} (u + d))\mathbb{E}\left(\Psi_{D}\exp (-r\tau)\right)

where DD be the subset where −d- d is hit first. The expectation term was calculated in the question 3.6 and we get

Price=exp⁡(−2r(u+d))sinh⁡(2rd)sinh⁡(2r(u+d))\mathrm{Price} = \exp (-\sqrt{2r} (u + d))\frac{\sinh(\sqrt{2r}d)}{\sinh(\sqrt{2r}(u + d))}

To generalize we add one bounce, we consider an option paying 1 if the process touches consecutively u,−du, - d and uu . In this case when uu is touched we have a new type of option. We will receive 1 after the down barrier and the up barrier are touched consecutively but there is no knockout feature. We price this option first. We consider a different Brownian Motion WtW_{t} starting at 0 and paying 1 if a down barrier at (−u−d)(- u - d) is touched and WtW_{t} returns to 0. We denote τ~\tilde{\tau} the first hitting time of (−u−d)(- u - d) .

Priceno knockout=exp⁡(−2r(u+d))E(exp⁡(−rτ~))\mathrm{Price}_{\mathrm{no~knockout}} = \exp (-\sqrt{2r} (u + d))\mathbb{E}(\exp (-r\tilde{\tau})) Priceno knockout=exp⁡(−22r(u+d))\mathrm{Price}_{\mathrm{no~knockout}} = \exp (-2\sqrt{2r} (u + d))

and the price of the option with 2 bounces is

Price2 bounces=exp⁡(−22r(u+d))E(ΨUexp⁡(−rτ))\mathrm{Price}_{2\mathrm{~bounces}} = \exp (-2\sqrt{2r} (u + d))\mathbb{E}\left(\Psi_{U}\exp (-r\tau)\right) Price2bounces=exp⁡(−22r(u+d))sinh⁡(2ru)sinh⁡(2r(u+d))\mathrm{Price}_{2\mathrm{bounces}} = \exp (-2\sqrt{2r} (u + d))\frac{\sinh(\sqrt{2r}u)}{\sinh(\sqrt{2r}(u + d))}

and generalizing to nn bounces, if the first triggering barrier is up

Pricenbounces=exp⁡(−n2r(u+d))sinh⁡(2ru)sinh⁡(2r(u+d))\mathrm{Price}_{n\mathrm{bounces}} = \exp (-n\sqrt{2r} (u + d))\frac{\sinh(\sqrt{2r}u)}{\sinh(\sqrt{2r}(u + d))}

and when the first triggering barrier is down

Pricenbounces=exp⁡(−n2r(u+d))sinh⁡(2rd)sinh⁡(2r(u+d))\mathrm{Price}_{\mathrm{n}\mathrm{bounces}} = \exp (-n\sqrt{2r} (u + d))\frac{\sinh(\sqrt{2r}d)}{\sinh(\sqrt{2r}(u + d))}