大家諗下都知 現實啲deriv咁撚複雜 邊有可能剩係model一隻underlying嘅dynamics就夠
隻deriv隨時可能係base on 3-4隻underlyings 仲未計啲痴線payoff
所以如果我地想繼續講落去 就一定要知道點樣同時model幾隻有correlation嘅underlyings (e.g. Apple and Google, S&P500 and SX5E, EUR and CHF)
咁我地就可以price到啲比較複雜嘅deriv
:^(
(ii) Background
咁唔知大家仲記唔記得
我地暫時consider過嘅model入面 其實所有嘅randomness都係嚟自Wiener Process
(Black-Scholes同Vasicek入面嘅randomness都係嚟自Wiener Process)
假設我地想model 2隻有correlation嘅underlyings dynamics
咁好自然我地就需要兩條correlated嘅Wiener Processes
:^(
然後再各自用自己嘅 r 同 σ 砌到變做兩條 in Q measure 嘅 SDE
而大家仲記唔記得Wiener Process嘅definition係乜?
只要符合曬以下三點就可以叫做wiener process
:^(
1.) Start at 0 , i.e. W(t_0) = 0
2.) Stationary Increment, i.e. W(t+h) - W(t) ~ N(0,h)
3.) Independent Increment, i.e. W(t_4) - W(t-3) independent with W(t_2) - W(t_1) [where t_1 < t_2 < t_3 < t_4]
如果我地將 2.) 入面嘅 t = 0 再加埋 1.)
咁我地就會得到 W(h) ~ N(0,h)
所以好大概咁講 整兩條correlated嘅wiener processes其實就好似整兩粒correlated嘅Normal R.V.s 咁
(iii) Multivariate Normal Distribution
如果依家我話 X follows Normal Distribution, i.e. X ~ N( μ , σ^2) 咁我諗大家都好易理解
Multivariate Normal顧名思義就係多過一粒 R.V. (e.g. X_1 , X_2) 而佢地jointly咁follow (multivariate) normal distribution
:^(
如果真係有兩粒 R.V.s , i.e. X_1 and X_2 , 咁呢個case我地就會話 X_1 and X_2 follows a Bivariate Normal Distribution (with certain correlation or covariance)
:^(
當然 X_1 同 X_2 各自都係follow Normal Distribution 但係in general佢地嘅mean同variance都會唔同
咁我地應該點用鬼畫符去表達 "X_1 and X_2 follows a Bivariate Normal Distribution"
:^(
:^(
其實唔難
:^(
Linear algebra is our friend
:^(
只要將 X_1 同 X_2 寫做一支 (random) vector 就可以表達到個意思
而 in general 支random vector 就會好似下圖咁
每一粒 X_i 都有各自嘅mean同variance 而佢地互相都有correlation (或者應該講有covariance)
而X_1, X_2 , ... , X_n 夾埋一齊就係follow Multivariate Normal Distribution
Do you guys remember me? Had been very very busy in the last few weeks!
宇智波月巴
2019-2-22 20:21:41
Yes of course
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:^(
How’s your job search going on?
錦衣衞
2019-2-22 20:25:42
Submitted my thesis just before Christmas.
Got an internship underway (they said they’re preparing the contract so fingers crossed). Also interviewing for some full time positions elsewhere.
宇智波月巴
2019-2-22 21:21:33
May I know what’s your thesis about?
Good to hear that you got an internship offer, is it a quant internship?
Btw wish you good luck for the upcoming interview
:^(
錦衣衞
2019-2-22 21:28:54
Thanks!
My thesis is on Wishart processes, which can be considered as a matrix version of Square root processes (CIR processes). I studied the integrated Wishart bridge processes.
The internship is quant research at an investment bank. (you probably know which bank
:^(
)
宇智波月巴
2019-2-23 00:03:23
Oh I have heard of Wishart process before, but never study it in a serious manner
:^(
, maybe I could do that in this summer
Quant research so strong
:^(
:^(
Are you doing the internship in London?
潛龍
2019-2-23 00:13:43
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宇智波月巴
2019-2-23 00:30:49
:^(
錦衣衞
2019-2-23 00:36:14
It’s just a research project in Sydney
:^(
潛龍
2019-2-23 00:48:33
此回覆已被刪除
宇智波月巴
2019-2-23 00:50:59
Oh I thought you have found one in London
:^(
Given the job market environment now, I think quant research internship in Sydney is already a really great opportunity. At least you get a chance to ask for return offer and it should be much easier than finding a full-time position directly.
想學但係數底弱 希望睇note 學