但係依家呢個簡單版嘅FX market仲係非常無聊
因為我地仲剩係得兩隻risk-free嘅assets
咁話明係Foreign Exchange Market又點可以冇Spot exchange rate
所以我地 define Spot exchange rate = X(t) = units of £ / 1 unit of $
同時assume埋 X(t) 喺現實世界(P-measure)嘅dynamics係 GBM with drift term α_X and diffusion term σ_X
(ii) The problem with currency
大家可能會奇怪點解我無端端要specify X(t) under P-measure嘅dynamics
乜唔係under Q-measure個drift term一律變做 r_d 嘅咩?
之前無論stock定bond嘅Q-dynamics都係咁樣
咁我地點解唔直接specify under Q-measure嘅dynamics?
如果你咁諗就中伏
:^(
中伏嗰原因好簡單 就係我之前算係呃咗大家
:^(
我之前講stock其實只係講咗個simplified case
如果大家有認真睇我講Black-Scholes之前嘅assumption
就會見到我其實係assume咗stock pays no dividend
而個問題正正就喺呢度
stock with no dividend 同 exchange rate本質上係唔同嘅
如果你買咗一隻冇dividend嘅stock
理論上你嘅gain/loss全部都係嚟自stock price嘅fluctuation
即係 由你買咗呢隻stock返嚟 再到你賣返佢出去 嘅呢段時間 你係唔會有任何income落袋
但係currency嘅情況就好唔同
假設你依家有$1喺手
就算你之後做啲乜都好 under我地上面define嘅market structure你都一定有sure income
啲sure income係喺邊度嚟嘅? 咪就係Time value of money
你有$1喺手 咁你就可以擺落(foreign) bank account continuously 咁賺 r_f
所以e.g.一年之後 其實你已經有$exp(r_f)咁多嘅美金喺手 而唔係剩係得初頭嘅$1
(p.s. 有(continuous) dividend嘅stock就會同currency嘅情況非常相似)
(iii) The problem with "Q"
因為咁 我地就要靠另一啲方法去揾X(t) under Q-measure嘅dynamics
而眼利嘅朋友可能仲會睇到第二個問題 就係個"Q"字
依家我地有兩個market --- Domestic 同 Foreign
咁我依家講嘅Q-measure究竟係講緊邊一個market嘅risk-neutral measure?
為咗區分返呢兩個measures 我會用:
Q_d 代表 domestic 嘅 risk-neutral measure
Q_f 代表 foreign 嘅 risk-neutral measure
希望大家仲記得risk-neutral pricing嘅精髓係乜
如果我地想domestic market冇arbitrage 就一定要至少搵到一個Q_d-measure
such that 下面呢句一定要成立 (因為呢個正正係Q_d-measure應該有嘅property) The DISCOUNTED domestic asset price process under Q_d is a Q_d-martingale
而呢句就係我地用嚟揾X(t) under Q_d-measure嘅dynamics嘅關鍵
(iii) Dynamics of X(t) under Q_d-measure
跟住我地只要recognise到下面呢句key step其實就已經成功咗一半 At time t, B_f(t) units of $ are worth B_f(t)*X(t) units of £
所以買foreign currency然後擺落銀行賺r_f
其實就同買一隻 price process 係 B_f(t)*X(t) 嘅domestic asset係完全一樣 (只係currency唔同咗)
因為又就嚟爆字數 所以照舊喺圖入面解釋曬剩低嘅嘢
而最簡單嘅例子就係European Currency Call
Payoff at Maturity (Time T) = Max(X(T) - K , 0) , where K is in £
顧名思義你買咗呢隻call 咁at time T你就可以用 £K 去買 $1
而喺at time T嘅FX market裡面可以用 $1 去買 £X(T)
如果X(T) > K你就會exercise張call 所以個payoff就會係咁樣
但係大家唔好唔記得我地之前揾過啲乜
我地其實已經知道X(t) under domestic risk-neutral measure [Q_d - measure]條式係點
加多隻foreign stock係唔會改變到X(t)條式嘅任何嘢
因為本身揾X(t) under Q_d-measure條式都唔需要呢隻stock嘅dynamics
所以for X(t)我地可以直接用返上面嘅result
:^(
不過foreign stock under Q_d-measure條式我地就真係唔知
:^(
而呢個亦都係我地嘅main task --- 揾foreign stock under Q_d-measure條式出黎
下圖就係setting + 所有我地現時已知嘅嘢
:^(
(iii) Dynamics of S_f(t) under Q_d-measure
點樣揾S_f(t) under Q_d條式出黎?
其實原理同上次非常相似 我地argue嘅嘢都係一樣 ALL DISCOUNTED domestic asset price processes under Q_d is a Q_d-martingale
就算我地加多一隻foreign stock落個model 呢一句都一定要啱
如果呢一句唔啱 咁我地個model就會有arbitrage
:^(
根據Martingale representation theorem
(Assume filtration F_t係純粹由wiener processes砌成 依家我地嘅case係)
一條process under Q_d要冇drift term (dt term)先會係Q_d-martingale
所以我地argue嘅嘢其實就等同下面呢句 ALL DISCOUNTED domestic asset price processes under Q_d have no drift term
而上次我地就係用no drift term + Girsanov Theorem揾到X(t) under Q_d嘅dynamics
今次其實都係類似
而第一個key step亦都同上次差唔多
我地需要realize到下面呢句 咁就可以正式開波
:^(
At time t, 1 unit of foreign stock, worth $S_f(t) in foreign currency, is worth £X(t)*S_f(t) in domestic currency
所以買foreign stock S_f(t)就同買一隻(psuedo-) domestic stock S_f(t)*X(t)係完全一樣 (只係currency唔同)
其實只要再細心啲睇多次個setting 呢個問題就迎刃而解
上面已經提過我地其實係知道X(t) under Q_d嘅dynamics
因為derive嘅過程根本唔需要用到S_f(t)條式 i.e. 唔牽涉第二個wiener process
所以換句話說 其實我地已經有第一條wiener process under P 同 under Q_d 嘅關係
e.g. Quanto derivatives (Part I)
(i) Background and setting
我地終於嚟到Girsanov Theorem呢一個section嘅尾聲
喺呢個section我地分別講過Bond option, Equity exchange option 同埋 Standard equity option with stochastic interest rate (Black-Scholes x Vasicek)
連埋我地之前討論過嘅Standard equity option, Forward start equity option, Zero coupon bond
大家可以見到其實全部都係圍繞住Stock同埋Fixed income呢兩類assets
咁financial market入面當然仲有其他唔同種類嘅assets
當中我地未提及過而又比較重要嘅有兩個classes
1. FX (Foreign Exchange)
2. Commodity
但係我喺呢個post就唔會講Commodity住 所以focus咗喺FX market先
FX market 包括Foreign Exchange同埋一啲Currency related嘅derivatives
而Quanto derivatives就係泛指一啲牽涉domestic同foreign currency嘅deriv
一開始我地唔洗諗得太複雜住 首先assume咗一個簡單啲嘅FX market先
假設 Domestic = 英國 UK , Foreign = 美國 US
咁即係 Domestic currency = Pounds sterling (£) , Foreign currency = US dollars ($)
然後再assume埋 Domestic short rate = r_d , Foreign short rate = r_f
咁我地就可以各自 construct 一個 Domestic 同 Foreign 嘅 risk-free asset
Domestic risk-free asset = B_d(t) , Foreign risk-free asset = B_f(t)
簡單啲咁諗大家可以當呢兩個risk-free assets做 Domestic 同 Foreign嘅Bank accounts
(p.s. 大家可以暫時ignore曬credit risk嘅問題 假設擺錢落銀行係完全risk-free先)
放錢落 Domestic (Foreign) Bank account 就一定會continuously咁賺short rate r_d (r_f)
但係依家呢個簡單版嘅FX market仲係非常無聊
因為我地仲剩係得兩隻risk-free嘅assets
咁話明係Foreign Exchange Market又點可以冇Spot exchange rate
所以我地 define Spot exchange rate = X(t) = units of £ / 1 unit of $
同時assume埋 X(t) 喺現實世界(P-measure)嘅dynamics係 GBM with drift term α_X and diffusion term σ_X
下圖係我地呢個簡單版FX market嘅summary
大家可能會奇怪 點解risk-free assets嗰兩條equation會係咁寫
唔緊要 solve一次比大家睇就會明
大家見到其實B_d(t)就好似domestic compounding factor咁
而的確擺錢入domestic bank account其實就係用 r_d 做緊continuous compounding
所以呢條equation的確就係bank account嘅dynamics 而同B_f(t)嗰條都係同樣道理
如果掉返轉咁睇 我地將一隻domestic asset嘅price process除以B_d(t)其實就係做緊(domestic) discounting
而呢一點對我地後面嘅reasoning非常重要
(ii) The problem with currency
大家可能會奇怪點解我無端端要specify X(t) under P-measure嘅dynamics
乜唔係under Q-measure個drift term一律變做 r_d 嘅咩?
之前無論stock定bond嘅Q-dynamics都係咁樣
咁我地點解唔直接specify under Q-measure嘅dynamics?
如果你咁諗就中伏
中伏嗰原因好簡單 就係我之前算係呃咗大家
我之前講stock其實只係講咗個simplified case
如果大家有認真睇我講Black-Scholes之前嘅assumption
就會見到我其實係assume咗stock pays no dividend
而個問題正正就喺呢度
stock with no dividend 同 exchange rate本質上係唔同嘅
如果你買咗一隻冇dividend嘅stock
理論上你嘅gain/loss全部都係嚟自stock price嘅fluctuation
即係 由你買咗呢隻stock返嚟 再到你賣返佢出去 嘅呢段時間 你係唔會有任何income落袋
但係currency嘅情況就好唔同
假設你依家有$1喺手
就算你之後做啲乜都好 under我地上面define嘅market structure你都一定有sure income
啲sure income係喺邊度嚟嘅? 咪就係Time value of money
你有$1喺手 咁你就可以擺落(foreign) bank account continuously 咁賺 r_f
所以e.g.一年之後 其實你已經有$exp(r_f)咁多嘅美金喺手 而唔係剩係得初頭嘅$1
(p.s. 有(continuous) dividend嘅stock就會同currency嘅情況非常相似)
(iii) The problem with "Q"
因為咁 我地就要靠另一啲方法去揾X(t) under Q-measure嘅dynamics
而眼利嘅朋友可能仲會睇到第二個問題 就係個"Q"字
依家我地有兩個market --- Domestic 同 Foreign
咁我依家講嘅Q-measure究竟係講緊邊一個market嘅risk-neutral measure?
為咗區分返呢兩個measures 我會用:
Q_d 代表 domestic 嘅 risk-neutral measure
Q_f 代表 foreign 嘅 risk-neutral measure
希望大家仲記得risk-neutral pricing嘅精髓係乜
如果我地想domestic market冇arbitrage 就一定要至少搵到一個Q_d-measure
such that 下面呢句一定要成立 (因為呢個正正係Q_d-measure應該有嘅property)
The DISCOUNTED domestic asset price process under Q_d is a Q_d-martingale
而呢句就係我地用嚟揾X(t) under Q_d-measure嘅dynamics嘅關鍵
(iii) Dynamics of X(t) under Q_d-measure
跟住我地只要recognise到下面呢句key step其實就已經成功咗一半
At time t, B_f(t) units of $ are worth B_f(t)*X(t) units of £
所以買foreign currency然後擺落銀行賺r_f
其實就同買一隻 price process 係 B_f(t)*X(t) 嘅domestic asset係完全一樣 (只係currency唔同咗)
因為又就嚟爆字數 所以照舊喺圖入面解釋曬剩低嘅嘢
結果一個cm都係講唔曬