宇智波月巴 2019-2-24 19:57:00 此回覆已被刪除

錦衣衞 2019-2-24 23:09:38 It doesn’t work

宇智波月巴 2019-2-24 23:14:04 此回覆已被刪除

S.Lev 2019-2-24 23:21:29 first generation exotic:
european barrier, digital, continuous barrier, kiko

second generation exotic:
chooser, asian, lookback, fwd start, tarf, vol or var swap

but normally we dont call european barrier or digital "exotic" now as you can replicate it as vanilla combination

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宇智波月巴 2019-2-25 02:02:58 So basically it’s an ever changing term in the market

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Sooner or later some of the exotic products will not be as “exotic” as before due to its replicability and liquidity/trading vol

宇智波月巴 2019-3-6 08:35:21 推下先

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連續兩日兩個midterm

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應該今個weekend有新post

潛龍 2019-3-6 11:14:44 此回覆已被刪除

宇智波月巴 2019-3-10 06:28:40 4.) Girsanov Thoerem (Change of measure) & its application

e.g. Exchange option

(i) Background review
考曬midterm之後又可以繼續出post

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首先review一次 Cholesky decomposition 先 跟住就可以講 exchange option
上次最尾我地講到可以用下面幅圖嘅 transformation 去 generate 兩粒 correlated 嘅 bivariate normal R.V.s


而去到 wiener processes 身上
其實個 transformation 都係一模一樣 而呢個做法就係 (2D版) Cholesky Decomposition
(p.s. n dimension版我會留返講 simulation 嗰時先處理 大家唔洗擔心我肯定會講)
咁再將呢個 technique 放落去兩條 GBM 入面就會變咗下圖咁
最尾個結果就係我地將自己心目中嘅 correlation structure 加咗入去呢兩條 SDE 入面


你地應該會問 最尾嗰舊 E(...) = ρdt 係想講乜?

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唔緊要 大家可以用一個 好heuristic 嘅方法去諗

所以大家可以當呢句係講緊兩條 wiener processes 嘅 covariance 係 ρdt
亦即係 imply 緊佢地嘅 correlation 就係 ρ
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(ii) Structure
Exchange option 最簡單嘅版本應該剩係牽涉兩隻stock 1 同 2
咁同之前一樣我剩係會討論 european exchange call option
呢隻 deriv 喺 maturity (Time T) 嘅 payoff 就係：


如果照字面上理解就係喺maturity嗰時
我地會賣咗 stock 2 同時買返 stock 1
咁話明係call option 如果上面咁樣做我地賺唔到錢自然就唔會exercise呢隻option

而買邊隻賣邊隻其實都係 issuer 同 investor 自己定
甚或乎個payoff可以係賣1手 stock 2 同時買返3手 stock 1 都ok
總之 issuer 同 investor 兩邊一開始 agree 咗個 terminal payoff 就得
咁依家 for simplicity 我地剩係討論上面幅圖嘅payoff嘅exchange call option
其餘情況其實都好類似 好容易就generalize到上去 我就唔特別分開再做
(p.s. 其實係因為我懶

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(iii) Purpose
買呢啲 option 嘅目的其實黎黎去去都係嗰啲

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多數唔係 hedging 就係 speculation
而因為 exchange call option 嘅 payoff 其實算係食緊兩隻 stock 嘅 price difference
所以一個好合理嘅做法就係如果你speculate緊/backtest到 兩隻stock有 -ve correlation
咁好自然under你嘅assumption/backtest result 兩個stock price 通常都會反向咁走
exchange call option就可以幫你將呢個關係轉化做 $$$

又或者你本身已經 hold 住咗 stock 2
因為某啲原因你唔想再 hold 譬如你覺得佢已經升到冇得再升
而同時你又覺得stock 1買得過 咁exchange call option就幫到你
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(iv) Pricing
其實上面都講到出腸 大家都應該估到我想做啲乜

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拎得嚟做 underlyings 嘅兩隻 stock 冇乜可能冇correlation
就算真係冇 學識咗 Cholesky decomposition 之後
我地依家都已經有能力consider一個有correlation嘅general case
所以點都當咗有correlation先

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你好肯定佢地冇嗰時再set ρ = 0
下面就係我地依家嘅setting


希望大家仲記得 GBM solve 完之後係咩樣
唔記得唔緊要 我寫多一次當係reference 之後睇得舒服啲


跟住其實都係 risk-neutral valuation 同 Girsanov Theorem

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去圖

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做到呢個位 有舊exp(...)喺expectation入面
個form又咁正路 所以又係時候用Girsanov Theorem

然後再透過 Cholesky decomposition
我地亦都可以知道埋 dW2(t) 喺 Q-world 同 Q*-world 之間嘅關係


然後貌似就好似卡死咗咁

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括號入面一個Q一個Q*又好似唔可以再化簡
其實我地並冇卡死 只係我需要大家accept咗一啲嘢先

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我寫得出呢個claim就肯定係啱嘅
至於點解啱我就留到最後先講 因為原因都比較technical 有興趣嘅讀者可以碌到最尾望下

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咁依家我地有齊 dW1(t) 同 dW2(t) 喺 Q-world 同 Q*-world 嘅關係
順便就可以揾埋 S1 同 S2 喺 Q*-world 嘅 dynamics
其實我地無論如何都喺要知佢地喺 Q*-world 嘅SDE係點樣
因為用完 Girsanov Theorem 之後 個expectation已經taken under Q*
所以如果我地仲繼續用 under Q-world 嘅 dynamics 就會計錯數 (因為個mean已經shift咗)
而下圖就係S1 同 S2 喺 Q*-world 嘅 dynamics


我地搞完一大輪終於可以繼續計粒 V(t,S1(t),S2(t))

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眼力好又心算好嘅朋友可能已經睇到神奇嘅地方喺邊
我地就再繼續做落去啦

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大家應該都發現 做到最尾其實同bond option嘅做法愈黎愈似

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寫埋尾二嗰一步大家就可以睇到其實真係非常之似


所以我地亦都可以好似做bond option嗰時咁
用Black Scholes Call price formula 嘅 form去寫低 V(t,S1(t),S2(t))
點解呢個寫法valid我喺#500已經解釋過 如果有唔明可以去睇下先


打咗差唔多4000字
最尾都係去返Black Scholes call price formula
This is the beauty of option pricing

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(v) 下回提要
European call price under Black Scholes model with stochastic interest rate

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小新小葵 2019-3-10 23:24:10 Stochastic interest rate range

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無敵星星 2019-3-10 23:30:05 推一推支持下樓主
希望有機會可以睇曬

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潛龍 2019-3-10 23:31:09 此回覆已被刪除

潛龍 2019-3-10 23:31:19 此回覆已被刪除

宇智波月巴 2019-3-10 23:34:25 多謝各位

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喺度同聽日考4001嘅人講聲加油

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插_班女學生 2019-3-11 06:35:06 stochastic interest rate 係咪姐係個堆short term model, HJM??
係的話等你救命

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宇智波月巴 2019-3-11 06:53:06 應該有排先講到short rate model

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依家stochastic interest rate我只係考慮緊最basic嘅Vasicek model
hull white model cir model嗰堆短期內唔會講住
因為我想由build yield curve開始講起 咁樣個flow會連貫啲
依家用Vasicek model你當係過下手癮

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其實好老實講 fixed income我都唔係話好有信心可以講得好
包埋credit risk要整dual/multi curve就更加複雜
同埋fixed income嘅content已經足夠額外開多個post講

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睇下呢度有冇高手得閒肯出招

旮旯生物 2019-3-11 08:46:34 留名

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潛龍 2019-3-12 12:21:07 此回覆已被刪除

宇智波月巴 2019-3-12 13:12:51 啱啱考完exam ifm又有時間寫

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我望返上一個cm發現原來打漏咗個proof

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應該係爆字數之後唔記得咗打
陣間打埋上黎先

潛龍 2019-3-12 14:25:54 此回覆已被刪除

插_班女學生 2019-3-12 14:47:45 我篇論文就係要寫short rate model

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仲未諗到題目

錦衣衞 2019-3-12 19:38:59 In my opinion, it is more common to write

dWt dBt = \rho dt

without the expectation as these deferential notations are understood in a L2 sense anyways.

宇智波月巴 2019-3-12 20:21:20 then I will drop the expectation from now on haha

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Thx for the reminder

宇智波月巴 2019-3-12 20:21:51 巴打讀緊mphil/phd？

插_班女學生 2019-3-12 20:26:20 master姐
個tutor話我可以諗個新model

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宇智波月巴 2019-3-12 20:45:46 邊間讀緊？
依家做short rate多數都要dual/multi curve setting先有意思
如果唔係你account唔到credit risk 啲price都係錯曬
但係個model就會變到好複雜

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諗新model我覺得更加痴線

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我諗最傳統嗰兩三個都已經比人explore曬