Introduction to Stochastic Calculus & Application in Finance

良心建議: take多啲math/stat, take少啲fina
學呢堆嘢最緊要有數底 fina自學都得

Msc 學過唒

連登真係好多勁人 獻醜啦

3.) Black-Scholes-Merton Model

好啦大家我地又繼續stoc cal嘅旅程

上回提要:
上一個post我地已經講到Black-Scholes equation
但係為咗等懶得追post嘅人都知道我地做緊乜
我就好快同大家review一次我地究竟做過啲乜啦

首先我地假設stock price(股票價錢)係follow Geometric Brownian Motion (GBM)
就好似下面呢幅圖咁:

咁S就係stock price啦 S_t就係呢隻stock at time t 嘅price
而T就係我地concern嘅Maturity time point
乜嘢嘅Maturity? 當然就係我地最終想知道嘅嗰隻derivative到期嘅日子啦

但係再講落去之前 我知道未接觸過stoc cal嘅大家齋睇條式應該一頭霧水
你地心裏面正常係會問以下嘅問題
:follow GBM嘅stock price有乜咁特別? 同現實世界嘅有乜唔同?
: 現實啲股票嘅圖表乜樣都有 你個model係咪真係咁把砲一條式就包曬?

咁我都可以老實答你 其實呢個model係錯嘅
點解錯? 佢錯嘅原因有三 三個都好簡單
1.)大家睇返我上個cm講嗰啲assumption 現實世界有冇可能做到?
2.)上面我地假設咗σ係constant 即係volatility係constant 事實係咪咁呢?
3.) 大家仲記唔記得wiener process係continuous everywhere but no where differentiable? 咁S_t講到尾都係wiener process砌出黎 佢同樣有呢個property 但係現實世界嘅stock price係咪真係continuous everywhere?
(p.s. Black-Scholes-Merton model嘅問題我係呢個section差唔多尾聲就會詳細講一次)

雖則係錯嘅model 但係每一個model都總有佢嘅用處
如果唔係我都唔洗花咁多筆墨介紹Black Scholes
Follow GBM嘅stock price其實已經非常非常似現實世界嘅stock price movement
大家如果仲記得 Follow GBM嘅stock price係會有Close form (上面幅圖嘅S_T)

咁我依家就假設T = 1 (year) 換言之即係Maturity係一年之後
然後因為close form裏面嘅T同t其實係求其揀 (arbitrarily chosen)
只要揀嘅兩個time point a,b 都係 0<=a<b<=T就可以
咁我就可以利用呢樣嘢 將time = 0 (now) 到 time = T =1 (maturity) 之間嘅interval割開m份
然後不斷用close form搵中間每一個step嘅price
最尾就plot到一幅follow GBM嘅stock price出黎啦
(p.s. 下圖係用vba plot sample path係兩條)

首先m = 100 (即係割開100份 由t=0出發 S_0經歷100個step先到S_T)


然後係m = 500


大家可以見到其實真係好似好似現實世界嘅stock price
只係大家要bare in mind我啱先喺上面講嗰兩個問題
記住呢個model唔係100%啱 (雖則根本冇一個model係100%啱 )

咁好啦 assume咗stock price嘅process 就到我地真正想搵嘅derivative出場
如果我地有一個derivative f 而佢嘅underlying係S
(e.g. For simplicity 大家可以當係european call/put)
咁我地就可以連埋隻stock砌以下呢個portfolio出黎


跟住靠self-financing portfolio嘅property (唔會有額外嘅資金流入or流出呢個portfolio)
呢個portfolio嘅differential form就可以寫成下圖咁樣 (唔depends on dh_1 and dh_2)


之後再靠我地assume咗嘅stock price process同ito's lemma
經過一輪運算 我地就砌到Black-Scholes equation出黎啦
(當然我地亦都要specify埋f係maturity嘅payoff)


我地剩低嘅任務其實話難唔算難 但係亦都講唔上係容易
就係要搵一個(可能唔止一個)可以satisfy上面嗰條BS equation (pde)嘅 f

點解我話唔算難? 因為solve pde嘅精髓其實只係得一個字: Try! (猜)
只要你specify到一個f出黎 而你代f入去LHS = RHS 咁呢個f就係我地想要嘅嘢
無論你用乜solve PDE嘅technique 背後嘅原理都只係猜!

咁點解我又話講唔上容易? 你地都應該估到啦
點撚樣諗個f出黎try先得㗎 我冇方向咁try試撚到死都未solve到啦
但係大家唔洗驚 因為大家連try嘅時間都可以慳返
下一個cm介紹嘅Feynman-Kac formula會為大家即時解決所有煩惱
唔洗1秒即刻知道個f係乜
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剩返1200幾字應該講唔曬 一樣下個cm再戰

你係咪馬料水大學既學生 ？

你問得出都知答案啦

呢個 course 簡直係成個 major 既精綷，加上超 charm 既 presentation，無讀過簡直係浪費哂成個 program

Btw 我補充多一點
feynman-Kac Formula當然唔係無中生有老屈出黎
每一條theorem背後都有proof
正如我之前都係逐個step做比你地睇

但係Feynman-Kac嘅問題係佢真正嘅proof實在太難
要理解呢個proof所required嘅數學程度
我諗呢度90%嘅人都唔會有

同埋呢個post都算係半個科普post啦
雖然本身入場門檻已經好高
但係我都唔想連肯留到依家嘅讀者都趕走埋
所以真正嗰個proof我唔會係度寫出黎啦
大家有興趣就自己search下 feynman path integral

不過講完risk neutral pricing之後
其實係可以寫到一個比較簡單嘅proof (其實都唔可以叫做proof )
我應該會係section最尾做一次比大家睇

佢話依家啲學生廢
見到啲數就掉頭走
讀rmsc都唔敢take呢個course 渣㗎

咁似logistic regression 嘅做法嘅?

邊一part似?
regression只係學過皮毛 同埋依家已經近乎唔記得曬

其實佢超撚好 grade，又學到野
take take 埋埋 d 廢鳩 course 都係想 gpa 靚姐，但最終真係浪費時間

全個department教得最好係佢

見到BSM 即刻refresh哂d memory

想問咩係"Feynman-Kac嘅問題係佢真正嘅proof"??

一般FK的變種或做研究上得出的推廣都係定義好相關的gain process, 然後Ito's formula + f solves a PDE 去show the gain process is a martingale, 再加上PDE的boundary conditions 去得出representation.

係咪有textbook以外的proof??

邊個course

https://en.wikipedia.org/wiki/Path_integral_formulation#Wick_rotation_and_the_Feynman%E2%80%93Kac_formula

你講嗰一個approach應該就係我上堂學嘅”proof”
但係原本feynman-Kac formula係develop from quantum mechanics
Feynman同Kac係研究path integral嘅wick rotation嗰時夾份整咗出黎
但係quantum mechanics啲數實在太難
好老實講我完全睇唔明

真正嘅proof應該係好似下面條link咁
(Section 8.4&8.5)
https://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/lecture-notes/sec8.pdf

佢undergrad剩係教兩個course
唔洗講出腸啦

3.) Black-Scholes-Merton Model

(c) Black-Scholes-Merton Formula (Risk Neutral pricing formula)
我都知上一part有好多嘢要消化 好快再同大家review一次

首先我地本來已經有Black-Scholes equation係手
呢條equation話咗比我地聽 只要你assume咗Stock price follows GBM
咁任何靠呢隻stock整出黎嘅derivatives嘅price都要satisfy下圖嘅pde

而f(T,S)就係呢隻derivative係maturity嘅payoff (Boundary condition for pde)

咁f(T,S)點解會等於一個剩係depends on S嘅φ?
因為at maturity呢隻derivative已經唔會再depends on time
(Maturity個下已經係呢隻derivative最後嘅time point 無可能再比時間影響)
所以無論個payoff有幾複雜佢都只會depends on stock price
簡單嘅例子有以下兩個 (S_T = Stock price at maturity):
European Call: φ(S) = Max(S_T - K , 0)
European Put: φ(S) = Max(K - S_T , 0)

跟住我地就靠Feynman-Kac formula (see below):


得到咗Black-Scholes equation入面嘅 f 嘅analytical solution

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而其實Black-Scholes-Merton formula (Risk neutral pricing formula)
就係指satisfies Black-Scholes equation嘅呢個f
f(t,S)就等於at time t 呢隻derivative嘅price(價錢)

~This section is for interested readers~
大家應該會發覺我上面幅圖個expectation同再上面嗰幅嘅有啲唔同
我上面嗰幅其實係condition on F_t 而再上面嗰幅係冇任何condition
咁邊一個expectation先啱? 嚴格黎講就應該係conditional expectation先啱
因為F_t如果大家仲記得 其實係指information up to time t (類近嘅意思係咁啦)
而的確我地係time t嗰一個位 只會知道up to time t嘅資訊 (完全係講緊廢話)
所以我地嘅expectation係應該condition on information up to time t

大家如果有少少基礎嘅finance底 其實可能已經覺得有啲奇怪
: 點解條BSM formula好似係將at maturity嘅payoff discount返去time t咁?
: 等陣先 咁其實咪即係搵緊present value?

的確係相當似 不過我想大家再思考多少少嘢先
大家如果睇返我上一個section最尾 應該會見到有一個令人好困惑嘅位其實係未解決

(我下面會用(1)同(2)去代表呢兩條SDE)
一開頭Black-Scholes model係assume咗Stock price follows GBM
不過應該係assume咗上圖嘅(1)
然後我地整下整下去到最尾 搵到一個satisfies BS equation嘅f
而呢個 f 嘅underlying stock嘅price無錯都係follows GBM
不過竟然係follow上圖嘅(2)
大家放心我地冇計錯數 我地只要consider埋跟住落黎嘅嘢就會水落石出

唔知大家仲記唔記得上圖嘅GBM都有"Close form" solution?
跟住我地就好似下圖咁搵 (2) 嘅stock price嘅distribution出黎


我係度就唔詳細講咩係log-normal distribution 大家自己上wiki睇都會好快睇得明
依家個重點就係我想搵stock price S_T嘅expectation出黎
其實上圖都寫低埋expectation嘅樣 我地就塞返a同b入去睇下真正嘅樣係點


然後因為我地其實有曬所有information up to time t 所以S_t係一個constant
我地將個S_t由expectation裏面抽返出黎 搬去右手邊 就會有下圖嘅結果


大家放心我地冇痴線 呢個咁古怪嘅結果係啱嘅
依家呢個結果話咗比我地聽
未來嘅expected stock price其實就等於我地將依家嘅stock price乘一個forward rate
換句話說就係我地依家隻股票in expectation只係賺緊risk-free rate
(記住依家呢個S係follows (2) 而呢個S係satisfies BS equation嘅 f 嘅underlying stock price)

如果有個人無端端係財經台同大家講 今日0005嘅close price係60.5
咁聽日收市佢expected嘅close price就係60.5*exp(r(T-t))
大家肯定覺得呢條友有病
但係當我地做緊derivative pricing嘅時候 呢個講法就絕對無錯
甚至可以話係你唔咁樣講就會price錯呢隻derivative
而呢一個concept其實就係risk-neutral pricing/valuation/measure

我地係price一隻derivative嘅時候
其實就係要搵一種expectation E* (or probability measure)
令到我地上面講嘅野係無錯
i.e. 未來嘅expected stock price其實就等於我地將依家嘅stock price乘一個forward rate [E(S_T) = S_t * exp(r(T-t))]
然後就用呢一種measure 將at maturity嘅payoff discount返去依家嘅time point
咁就會等於呢隻derivative係依家嘅fair price
(其實就完全等同我地用完feynman-kac得出黎嘅嘢)

而大家可以見到 當S follows上圖嘅(2)嘅時候
的而且確E(S_T) = S_t * exp(r(T-t))
但係原本BS-model明明assume個stock price follows (1)
咁中間究竟發生咩事?

事實就係當我地用完feynman-kac之後 其實我地已經轉變咗個probability measure
大家可以想像我地由現實世界P (Physical world)跳咗入去另一個世界Q (Risk-neutral world)

Note that:
為咗區分兩個世界各自嘅expectation
我會用"E"去表達現實世界嘅expectation
而"E^Q"就係講緊risk neutral嘅expectation

而係Risk-neutral world入面 stock price就係會follow (2)
兼且E^Q(S_T) = S_t * exp(r(T-t))
而只有係呢個world入面做derivative pricing
我地先可以得出係現實世界入面呢隻derivative嘅fair price
而呢一個就係derivative pricing嘅精髓 --- risk neutral pricing
(所以我之前先會話其實知道risk-free rate就夠 本身隻stock嘅mean係點我根本唔care)

所以其實上面嘅圖我係混淆咗大家
真正嘅寫法應該係咁

巴打有冇唸過以後點搵工？我喺澳洲都係就嚟畢業，我想做Quant，澳洲基本冇乜工可以搵。有冇巴打可以指點迷津，點條生路細佬行吓

3.) Black-Scholes-Merton Model

(c) Black-Scholes-Merton formula (Risk-neutral pricing formula)
痴線上個cm爆字數
我再補充多少少關於risk-neutral嘅concept
(因為呢個concept真係好重要 所以係值得花多啲筆墨去講)

我係上一個cm講過

我地係price一隻derivative嘅時候
其實就係要搵一種expectation E* (or probability measure)
令到我地上面講嘅野係無錯
i.e. 未來嘅expected stock price其實就等於我地將依家嘅stock price乘一個forward rate [E*(S_T) = S_t * exp(r(T-t))]
然後就用呢一種measure 將at maturity嘅payoff discount返去依家嘅time point
咁就會等於呢隻derivative係依家嘅fair price

然之後因為我地就用"E^Q"黎denote risk-neutral world入面嘅expectation
但係其實我地仲可以present得好少少

唔知大家仲記唔記得咩係martingale? (唔知我講緊乜嘅朋友就係時候追post)
如果用呢個concept嘅話 我地就可以將上面嘅野rephrase一次

我地係price一隻derivative嘅時候
其實就係要搵一種expectation E^Q (risk neutral probability measure)
令到下圖嘅equality成立

換言之 e^(-rt)S_t (discounted stock price)其實係martingale

咁假設呢隻derivative嘅price係f(t,S_t)
佢嘅discounted price [e^(-rt)f(t,S_t)] under 呢一個risk-neutral measure都要係martingale
i.e.好似下圖咁樣


然後我地就用呢一種measure 將at maturity嘅payoff discount返去依家嘅time point
咁就會等於呢隻derivative係依家嘅fair price
i.e.

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咁講完risk-neutral pricing 我地係時候拎呢條formula去實戰下
講咗咁耐我地都冇真正specify過個 f(T,S_T) = φ(S_T ) 究竟係乜樣
咁我地不如就拎普通嘅European call (認購證) 黎做例子啦

European call係maturity嘅payoff係好似下圖咁樣:


所以佢係依家呢個time point (For simplicity我assume依家係t = 0) 嘅fair price係:


跟住我又係會喺圖入面講曬所有解釋
因為分開打實在太麻煩 唔講咁多 去圖










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打到手殘終於打完
咁既然上面我搵到call price嘅close form formula under BS model
put price嘅formula就唔需要好似上面咁痛苦咁搵
我地靠put-call parity就得啦

下一個sub-section 亦都係section 3嘅最尾一part
我會同大家分析下幾隻複雜少少嘅derivative (但未係exotic)
然後再用上面嘅BS call /put price formula去示範比大家睇
點解BSM model咁強大

我都想有人話比我聽之後點搵工
畢業等失業

呢本入門唔錯
利申：睇過一半

最鍾意risk neutral probability 呢個手法，好elegant