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
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
~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
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
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
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.