Introduction to Stochastic Calculus & Application in Finance

樓主係咪姓曾

LM

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【民陣6.9反惡法遊行】
日期：6月9日（日）
集合地點：銅鑼灣東角道
時間：下午2時30分集合，3時正起步
遊行路線：銅鑼灣東角道遊行至金鐘添美道

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4.) Girsanov theorem & its application

e.g. Quanto derivatives (Part II cont.)

緊接返上次完嘅位 下圖就係我地依家嘅完整setting

如果幅圖入面有咩唔明唔記得 請追返上面嘅post
沒問題嘅話我地就可以開始解答上個post最尾嘅兩條問題

(iv) "Dividend" of foreign stock

明明我地assume隻stock冇dividend (in foreign market)
點解最尾會有個類似dividend嘅物體喺條式裡面?

因為我地只係assume咗隻foreign stock喺foreign market入面冇dividend
但係依家我地用緊DOMESTIC risk-neutral measure Q_d
而S_f嘅price dynamics應該係in terms of foreign currency ($)
所以currency嘅唔同變相導致咗呢舊類似“dividend”嘅物體出現
(更加準確咁講應該係類似continuous dividend先啱)

而呢個類似dividend嘅嘢又有乜咁特別?

特別之處就係佢encode咗exchange rate dynamics X(t)嘅資訊
咩意思? 你再認真啲睇下舊“dividend”
σ_X咪就係X(t)嘅volatility
而ρ其實係描述緊S_f同X(t)嘅correlation
(因為ρ本身就係呢兩條dynamics各自嘅wiener processes嘅correlation)
然後最奇妙嘅地方就係 如果你諗真啲咁樣encode法其實都幾合理
上面提到係因為domestic同foreign market嘅分別 (currency difference)先會產生呢舊“dividend”
咁呢個difference可以用啲咩嚟表達? Exchange rate咪就係最好嘅candidate
所以X(t)嘅“資訊”出現喺呢舊“dividend”入面係make sense嘅

咁實際上呢舊“dividend”究竟有咩意義?
我下面寫一個非常informal嘅argument就會大概睇到

係啦其實大概就係 (% chg of) S_f 同 (% chg of) X(t) 嘅covariance
由呢個argument你都會見到(% chg of) S_f 同 (% chg of) X(t) 嘅correlation其實就係ρ

其實除咗呢種由econ角度出發嘅解釋
我地可以用更加數學嘅方法去理解 (下面會提到)
不過我地首先要處理咗跟住嘅問題先

(v) Relationship between Q_d and Q_f

其實Q_d有乜咁特別? 如果我全部嘢都變曬under Q_f會有乜唔同?

首先define清楚咩叫Q_d同Q_f先
Q_d = DOMESTIC risk-neutral measure
Q_f = FOREIGN risk-neutral measure
我地之前一路都係將所有dynamics由P轉去Q_d
咁點解要揀Q_d? 佢係咪真係有特別過Q_f嘅地方?
如果好formal咁講其實係冇嘅
我將全部dynamics由P轉去Q_f其實都係一樣valid
只係咁轉嘅話 我地用嘅main argument就會變成下面呢句
All discounted FOREIGN asset price processes under Q_f is a Q_f - martingale

「由P轉去Q_d」同「由P轉去Q_f」實際上邊個簡單啲就好主觀嘅
不過轉去Q_f有個幾明顯嘅優點
就係我地會即刻知道S_f under Q_f嘅dynamics係點樣
S_f (foreign stock) 本身就係一隻foreign asset
其實個情形咪同普通Black-Scholes入面嗰隻stock一樣
所以根據我地玩咗咁耐Black-Scholes嘅經驗 我地已經可以即刻conclude到下面呢條式


咁跟住揾埋X(t) under Q_f嘅dynamics就搞掂 但係問題就嚟啦
本身我地define咗X(t)係in terms of domestic currency (£)

如果我地用foreign嘅viewpoint去睇 咁就首先要將X(t)整到係in terms of foreign currency ($)先
咁又唔係話太難解決 我地直接consider 1/X(t)就搞掂

然後模仿返我地之前用過嘅observation => Notice that B_d(t)/X(t) is a psuedo-foreign asset
不過接踵而來嘅又係一大堆又長又悶嘅ito's lemma
到最尾我地就會得到X(t) under Q_f嘅dynamics
(p.s. 喺呢度我就剩係show最尾嘅result 我諗大家睇ito's lemma都已經睇到厭曬)

Q_d同Q_f之間有乜關係?

依家我地其實已經知道曬under Q_d同埋under Q_f嘅所有dynamics
下圖係一個簡短嘅summary


上面都提過Q_d其實冇乜咁特別
兩個risk-neutral measure都係完全valid嘅 (而且依家Q_d同Q_f各自都係unique嘅)
咁我諗一個natural question就係Q_d同Q_f其實有冇關係?
如果有嘅話 我地又可以點表達佢地之間嘅關係?

細心嘅巴打可能已經睇到 佢地一定有關係
而兩個measure之間嘅關係其實咪即係Girsanov Theorem
的確我地可以靠上面嘅dynamics去推敲返Q_d同Q_f之間嘅關係
更準確咁講 應該係兩條wiener processes各自under Q_d同under Q_f嘅關係


仲有一個位係值得留意嘅
當我地知道第一條wiener processes under Q_d同under Q_f嘅關係
用Cholesky decomposition同樣可以揾到第二條嘅關係

咁變相證明轉咗咁多次measure之後 個correlation structure都依然係preserve嘅

其實除咗呢種由econ角度出發嘅解釋
我地可以用更加數學嘅方法去理解 (下面會提到)

我地依家可以去返呢個位啦
希望大家仲記得 Girsanov Theorem做緊嘅嘢(好簡略咁講)其實只係shift緊個drift term
咁數學上嚟講 嗰舊"dividend"嘅出現其實只係因為我地由Q_f轉咗去Q_d (or vice versa)
變相個drift term都要跟住郁 而佢嘅financial meaning就係好似上面講咁樣
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4.) Girsanov theorem & its application

e.g. Quanto derivatives (Part II cont.)

(vi) Pricing quanto derivatives (General case)
搞咗咁耐 我地終於可以入真正嘅戲肉
依家個market豐富咗 自然可以price一啲更加有趣嘅quanto
但係我地依家只會討論domestically traded嘅quanto deriv
亦即係話呢啲deriv全部都會係in terms of domestic currency (£)
(原因: traded in foreign market嘅deriv根本唔洗理X(t) 咁成件事就變到好無聊)

In general我地跟住講嘅deriv都會好似之前簡單版market咁 係一個simple T-claim
換言之呢啲deriv嘅payoff就會係下圖咁款
(p.s. 總之個payoff淨係depends on value(s) at time T我都會一律叫佢地做simple T-claim)


跟住又係同之前簡單版market一樣
我地assume一個functional form俾呢啲simple T-claim嘅price
簡單咁講只要呢個function夠"Smooth"可以俾我地d起碼兩次就ok

然後又係用返嗰堆argument同ito's lemma
為咗唔好悶親大家 呢part我就唔show steps 做法同之前完全一樣只係更加tedious
而最尾我地得到嘅結論都係一樣
就係我地可以照用(domestic) risk-neutral valuation去price呢啲simple T-claim

(p.s. 我地亦都可以用Feynman-Kac揾返條pde出嚟 不過礙於字數所限就唔寫出嚟)

(vii) Pricing quanto derivatives (1st example)
第一個例子就係一隻struck in domestic currency嘅call on foreign stock
又foreign又domestic又struck唔知想點?
唔緊要大家一睇payoff就會知佢做緊乜


上面提過依家全部payoff都係要quoted in domestic currency (£)
所以你想一隻call on foreign stock可以擺落domestic market入面trade
我地就唯有諗辦法令佢嘅payoff由foreign變返做domestic
而呢個case嘅辦法就係將X(T)乘落S_f(T)
咁我地就會得到at maturity quoted in £ 嘅 foreign stock price
因為成個payoff最尾要係quoted in £
所以個strike亦都會係quoted in £
(p.s. hence the name "struck in domestic currency")

咁要price呢隻call都唔算太難
我地首先要consider X(t)*S_f(t)嘅dynamics先 (under Q_d)

依家有兩條 (correlated嘅) wiener processes喺入面 咁點算?
其實唔難解決 唔知大家仲記唔記唔得我喺bivariate normal嗰個section提過
如果A同B係jointly normal distributed嘅話
咁 A+B 都係照follows normal distribution (sum of normal R.V.s is still normal)
依家正正係呢個情況 兩條wiener processes乘埋各自嘅coefficient分別就係A同B
所以我地只要揾到A+B嘅mean同variance就得

咁我地就知道in distribution sense可以得到下圖嘅結論

然後我地就可以重寫X(t)*S_f(t) under Q_d嘅dynamics (in distribution sense)


去到呢個位其實我地已經即刻知道最後答案係乜
依家條dynamics其實咪同普通black-scholes model嗰條一模一樣
只係我地要當X(t)*S_f(t)係一舊嘢咁睇同埋個volatility複雜咗咁解
所以最尾呢隻struck in domestic currency嘅call on foreign stock嘅price
都照樣可以用Black-scholes call price formula嘅方式寫出嚟


(viii) Pricing quanto derivatives (2nd example)
睇完上面個example 你可能會問
「咁點解X(T)一定要乘喺入面？乘喺max()出面唔得嘅咩？」
冇錯其實將X(T)乘喺max()出面一樣得 咁做嘅話我地就會得到第二個example:
Call on foreign stock (struck in foreign currency)
其實同上面最大嘅分別就係個strike
因為依家X(T)係最後先乘落去 所以個strike K_f應該仲係in foreign currency ($)
(p.s. hence the name "struck in foreign currency")
但係最尾成個payoff都一定係in terms of domestic currency (£)


跟住就開始同上面唔一樣 因為我地今次無需要再consider X(t)*S_f(t)
咁我地不如由(domestic) risk-neutral valuation開始做起
睇下中間會做到啲乜 然後再睇下點處理


哇舊exponent咁嘅樣 肯定係用Girsanov Theorem啦


等陣先 點解最尾條式咁熟口面?

仲記唔記得上個post我地討論過Q_d同Q_f嘅關係?
你再望下第一條wiener process under Q_d同under Q_f嘅關係條式

兩條式根本一模一樣
如果兩條式shift drift嘅幅度一樣 咁呢兩個measure亦都只能夠係一樣
所以用完Girsanov之後 我地其實係轉咗去Q_f-measure

當我地知道呢個新measure其實係Q_f嘅時候
其實就已經做完 因為淨低嘅嘢又係同普通black-scholes一樣


(ix) Pricing quanto derivatives (3rd example)
最尾呢個example其實係上面嗰個嘅小變種
到目前為止我地啲deriv都係冇lock死到exchange rate X(t)
咩意思? 你望下上面兩個example嘅payoff
我地係用X(T)去convert from $ back to £
但係咁就有uncertainty喺裡面 因為我地冇可能提前知道value at time T

假設依家我地喺time t
其中一個解決辦法就係fix死exchange rate做X(0)
亦即係話我地拎呢隻deriv一開始trade嗰刻嘅exchange rate
下圖就係依家嘅payoff


咁某程度上我地就好似limit咗exposure to exchange rate risk
不過事實係咪真係咁? 我地落手做一次就會知答案

大家都睇到實際上FX risk仲係存在
只係代表呢個risk嘅term走咗入pricing formula裡面以dividend嘅姿態出現

嚇撚死我以為又打唔曬 打到淨返最後一隻字
咁Part 4亦都正式喺度告一段落
打咗咁撚耐終於打完半個3 cred course嘅內容

跟住我會執一個完整版pdf
裡面會包括曬Part I - Part 4嘅所有嘢
而且裡面嘅內容如果當初講得唔好/唔詳細 我都會再執執佢
不過利申返先 因為小弟都有其他嘢做
所以應該都冇咁快會完成到
同埋再講多次 排版一定會好柒 請大家唔好介意

至於Part 5我依家偏向都係照原定計劃講simulation
唔講path-dependent option住嘅原因係如果我一開始講
就要連埋laplace transform或者fourier transform一齊講
然後連帶又要講埋好多其他關事嘅嘢
為咗快啲執好個pdf就唯有遲啲先講
同埋simulation多啲圖睇好似會吸引啲

跟住嘅更新速度我諗大概都係維持住兩個禮拜update一次
希望有興趣睇嘅可以耐心等下

反對《逃犯條例》修訂

白宮聯署
https://petitions.whitehouse.gov/petition/extradition-law-amendment-hong-kong-threat-personal-safety-and-freedom

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好重要所以要講三次

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問問，最後個個真係有fx risk?

如果用greeks去睇 dC/dX = 0, 個FX係q出現睇落只係令隻call平左

shit我其實係想打mitigate exposure 唔係limit exposure
深夜打文check極都係有錯

如果用greek睇的確係冇嘅
但係舊dividend會出現就係因為account咗FX risk
所以舊risk應該係limit咗 但係概念上仲存在

學Stochastic Calculus當然睇呢本書
https://www.springer.com/gp/book/9780387976556
識少少PG level Real analysis/Measure Theory + Probability應該睇得明

原來已經兩個月冇update
大家應該都知點解我冇辦法(亦都冇心情)update
不過我應承大家 場仗打完 我地贏咗之後我會繼續出post
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