錦衣衞 2018-12-13 18:21:36 I can email you my WhatsApp if you have a dummy email account. Otherwise I can create one.

吉吉吉 2018-12-13 22:43:12 may be better for both to use a new email for chatting

吉吉吉 2018-12-13 22:46:14 Allows differentiation on stochastic process. Alternative proof to the existence and smoothness of a density function. Asymptotic expansion of stochastic process, hence closed form of a density function and option pricing models

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But I didn't go in depth as well.

宇智波月巴 2018-12-14 18:02:56 好似幾過癮 我sem break有時間嘅話望下先

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宇智波月巴 2018-12-14 18:27:15 小弟尋日終於完sem

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應該呢一兩日會有post㗎啦
對唔住要大家等咁耐

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我沒有放棄 2018-12-15 01:51:15 要學functional analysis先會易明啲

宇智波月巴 2018-12-17 16:35:21 3.) Black-Scholes-Merton Model
(d) Application of risk-neutral pricing formula
隔咗成六頁我地終於可以繼續講bond 同bond option

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等我好快review一次之前講到邊啦 我諗大家都已經唔係幾記得

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----上回提要----
我地依家只會講zero-coupon bond 即係好似下圖咁

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為咗方便起見 下面全部嘅"bond"都係指zero-coupon bond (default-free)
而如果我地用year 1白痴方法計嘅話

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咁zero-coupon bond嘅價錢就應該係


但係呢個post話明係stochastic calculus 點會接受呢啲咁兒戲嘅計法

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所以我地就要首先介紹一個用黎描述 (instantaneous) interest rate嘅model
就係上次講到嘅Vasicek model 佢嘅樣就好似下圖咁


而上次我地亦都講到 其實我地可以將bond睇做一隻underlying係interest rate嘅derivative

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所以我地之前講過嘅risk-neutral valuation全部可以照用
所以我地就可以將bond price寫成下面幅圖咁

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-----------------------

好啦開始進入今次嘅正文

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嗰個discount factor究竟會係咩樣

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其實答案就係下面幅圖咁 我地會變咗用integration去揾discount factor
咁點解要用integration? 其實下圖都已經比埋個解答大家
(p.s.呢個諗法唔係完全啱 不過會令大家易啲理解)
因為依家interest rate已經係一個 (continuous) stochastic process
即係代表由 t 到 T 呢段時間入面 每一秒嘅interest rate都 (可能) 唔同
所以一個可行啲嘅做法係揾一個average嘅interest rate
而如果我地要揾average 咁就變相要知道點樣sum曬由 t 到 T 咁多點唔同嘅interest rate
而呢樣正正就係integration可以處理嘅嘢 所以in general其實discount factor都可以寫做下圖呢個樣

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咁依家我地重新寫多次bond price條式出黎啦

大家有冇發覺最尾嗰舊expectation好熟面口

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冇錯啦

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呢個樣嘅expectation咪就係用完Feynman-Kac Formula之後我地會得到嘅嘢

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如果大家已經唔記得乜嘢係Feynman-Kac又懶得追post就睇下圖啦

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係咪真係完全一樣呢

換言之 我地依家手上面呢個expectation (bond price) 其實就係一條pde嘅solution黎
咁即係話透過Feynman-Kac 我地其實係可以揾得返原本條pde係點樣嘅

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如果我地對返位 同埋當interest rate係bond嘅underlying嘅話 咁條pde就應該係下圖咁樣


相信有睇開嘅讀者應該都知道下一步係點做
有條pde但係唔知solution? 咁咪try到知為止囉

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但係今次大家夠運啦唔洗自己try 因為Vasicek已經試咗個form比我地

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如果我地假設bond price solve完出黎就會好似上圖呢個form咁嘅話
我地就可以將呢個form塞返入條pde然後繼續做落去

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大家睇下幾神奇

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塞咗入去掉下位就將一條pde變咗做兩條ode
而更加緊要係呢兩條ode其實都非常容易solve

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咁我地就快快手solve埋佢地啦

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雖然啲步驟都幾長下

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但係其實solve嘅過程係唔難 只需要每步慢慢寫就寫到
依家我地有齊A同B 咁不如我地將全部嘢塞返入去bond price條式到 睇下我地依家手上面有啲乜

原來不知不覺我地已經solve到interest rate follows Vasicek Model嘅時候嘅Bond price條式啦

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今次講住咁多先 下個cm開始講Bond option

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心急嘅讀者其實可以自己試下用 ito's lemma 揾咗 dP(t,T) 係咩先
當然interest rate都係follows vasicek model
然後望下P嘅dynamics有啲咩特別嘅地方

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旮旯生物 2018-12-17 21:38:14 支持

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穿越牛熊 2018-12-17 22:22:32 留名學嘢
利申：computational finance grad

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Angular 2018-12-18 13:55:29 睇完我決定下個sem唔take....

無窮幻變天氣 2018-12-18 17:44:04 想學好stochastic好耐
我ee undergrad都只係好粗略學過
機緣巧合下去左一間bank做risk
重新學過所有野
發覺fin engineering都幾有趣
呢個post真心學到好多
之前都係睇paul wilmott學quan fin
但學得麻麻 始終自學無人教
呢個post易入手好多 build up番我個底

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宇智波月巴 2018-12-18 18:05:35 data analysis其實學inference, regression, time series會比較有用 而且一定要識programming
如果識python or R 是但一隻就更加好

我呢個post講嘅嘢其實比較偏向derivative pricing, financial engineering, risk management 多啲
所以會比較數學啲

Angular 2018-12-18 18:18:18 下個sem take那科叫Computational Finance似乎同 financial engineering差不多

有無巴打知會讀D咩

宇智波月巴 2018-12-18 18:29:04 應該都會關option pricing事 但係computational finance涉獵嘅嘢更加廣
應該time series, technical analysis, execution所有可以寫programme嘅嘢都會掂

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插_班女學生 2018-12-19 03:35:12 未計下forward, future, option price
跟住就用bull shit (BS)formula 去計option price
再複雜d就計下合成option price
仆街d就要你去記得delta gamma theta rho vega 條式

錦衣衞 2018-12-21 03:14:56 我都喺自學緊quant finance，宜家搵緊工。有冇工可以refer？

宇智波月巴 2018-12-21 18:33:34 巴打仲係澳洲？

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錦衣衞 2018-12-21 18:37:52 喺呀，啱啱開始搵工。搵到工之前都唯有留喺澳洲，始終澳洲好住D。

宇智波月巴 2018-12-21 19:50:01 有得揀都儘量唔好返黎香港

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呢度愈黎愈似地獄

錦衣衞 2018-12-21 20:29:18 所以我都喺prefer London, 始終同Melbourne嘅lifestyle似啲，而且又多啲靚女

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就算返香港都喺諗住儲幾年工作經驗然後去London

宇智波月巴 2018-12-22 01:21:22 唉我都想可以係香港出面揾到工

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依家報咗cuhk mphil 唔知會唔會收

錦衣衞 2018-12-22 02:05:40 可以嘅話最好讀PhD

宇智波月巴 2018-12-22 02:18:05 cu risk man應該剩係得MPhil可以報

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PhD嘅話就要讀PhD in Stat

同埋我都驚一黎就讀PhD自己會頂唔順

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所以想讀完MPhil大概知道research係咩一回事先再讀PhD

錦衣衞 2018-12-22 02:20:29 可以MPhil轉PhD

宇智波月巴 2018-12-22 02:43:31 即係可以讀到一半轉PhD degree?

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不過MPhil其實都係兩年 真係讀到嘅話就到時同prof傾下