3.) Black-Scholes-Merton Model
(d) Application of risk-neutral pricing formula
同之前講嘅一樣啦 喺再落下一隻derivative --- bond option之前 首先我地要知道點搵bond price先
希望大家本身都對bond有最基本嘅認知
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如果想好快咁重溫一下就睇下圖啦
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簡單咁講 每一張bond都會講定一個face value同埋coupon rate
(For simplicity呢個example我假設coupon rate係fixed嘅 當然現實就唔一定係fixed)
然後只要你比錢買(依家假設你係t=0買入) 呢張bond嘅issuer就會每一期都比返coupon你
而coupon最簡單嘅計法就係= Face value * coupon rate (當然可能仲有其他更複雜嘅計法)
當呢張bond到期嘅時候 你就會拎埋最後一期嘅coupon同埋face value
所以換句話說其實你買bond就等於借錢比個issuer
良心建議: 希望冇fina底嘅各位 起碼花少少時間搞明乜嘢係bond先 之前講嘅其他嘢唔明都唔緊要
因為除咗股票 一個正常人最有機會買到嘅instrument就係bond
就算你只係擺錢入銀行做定期其實都已經相當於買咗一張bond
而bond亦都算係比較安全嘅instrument (除非你係專揀啲high yield bond黎買嘅癲佬咁我冇嘢講
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因為bond issuer原則上係一定要比coupon/face value你 如果佢比唔到就係違約
所以如果你揀啲credit rating好嘅bond issuer去買 基本上係非常非常安全嘅
亦都因為咁所以每個人嘅portfolio都最好有一定portion嘅bond渣係手 就算經濟轉差都唔至於一鋪清袋
咁review完就正式開始
首先我地跟住落黎只會處理zero-coupon bond 即係完全冇coupon嘅bond
而for simplicity我會assume埋face value = 1
佢嘅strucutre請睇下圖
![](https://na.cx/i/v8JN3jo.jpg)
大家都見到真係非常簡單
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但係我地只要處理咗最簡單嘅case 咁之後有coupon同face value>1嘅情況其實都好易可以generalize上去
所以我地集中火力處理咗zero-coupon bond先
首先我地諗下year 1嘅finance course係點教我地price呢隻zero-coupon bond先
假設我地依家係continuous compounding啦
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咁個result其實就好似下圖咁簡單
![](https://na.cx/i/FevHSoz.png)
大家可以見到zero-coupon bond嘅價錢其實就係個discount factor (depends on interest rate r)
所以當我地問「點樣price一隻zero-coupon bond?」嘅時候
其實變相即係問緊「應該點樣model interest rate?」
而上圖呢個pricing method嘅
大前提係interest rate係T-t呢段時間係constant or deterministic
我之前都提過呢個assumption係唔realistic嘅 因為interest rate本質係random
We can do better than this
正如我地之前用GBM去model stock price
跟住我都會介紹一個stochastic嘅interest rate model 然後用佢黎model interest rate
而呢一個model嘅名就係Vasicek model
(CIR model我應該會係section 4尾聲再提 所以請大家等陣啦
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呢個model嘅SDE其實就好似下圖咁
![](https://na.cx/i/g1TB3VS.png)
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請注意 呢條SDE已經係risk-neutral world Q入面嘅dynamics
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我地可以見到呢個model嘅drift term同GBM都幾唔同 不過都唔算係複雜
首先乜嘢係b? 大家可以當b就係interest rate嘅long term mean
而a就係interest rate返翻去呢個mean嘅rate
最尾volatility σ我地for simplicity暫時都係當佢constant先
所以單靠呢條SDE我地其實已經implicit咁假設咗interest rate嘅movement係一個mean reverting process
何謂mean reverting? 就係當你比interest rate亂咁郁一段好長嘅時間 最尾佢一定會返翻去自己嘅long term mean
大家可能會問點解我地要整條唔同嘅SDE出黎玩自己 然後又imply interest rate係mean reverting
我地梗係有原因先咁做
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因為有好多empirical study都指出interest rate的確係會revert to mean 所以我地先會用一個咁嘅model去嘗試描述有呢個性質嘅interest rate
(其實呢種process真正嘅名係叫做Ornstein–Uhlenbeck process 有興趣就自己wiki
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咁當然我又唔係話Vasicek model已經係最準確嘅model (大家見到volatility又係constant就知準極有限
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不過係我地呢個程度 Vasicek model係比較簡單同容易去理解
兼且佢已經足夠幫我地去解釋分析好多嘢 所以大家暫時同住佢玩遊戲先
依家有咗個model就可以返翻去原先嘅問題
假設interest rate follows Vasicek model嘅dynamics
咁一隻zero-coupon bond at time t嘅fair price應該係乜?
我想大家再認真思考下我上面講過嘅呢條式
![](https://na.cx/i/FevHSoz.png)
上面我提過呢隻zero-coupon bond最後其實都係depends on r (當然仲會depends on t)
換個角度諗 其實我地大可以當bond嘅underlying就係interest rate 正如equity call嘅underlying係stock咁
所以我地亦可以大約將bond睇成係一種interest rate derivative
心水清嘅朋友睇我左講右講無論點都要講到"derivative"呢隻字出黎都應該估到我用意如何
我地一路學咁耐derivative pricing 最核心嘅concept係乜?
冇錯就係risk-neutral pricing (如果你唔知我講緊乜就代表你真係要追post
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如果我地將bond當成一隻underlying係interest rate嘅derivative咁睇 咁成個問題就變到非常簡單
我地首先define清楚啲notation先
我會用P(t,T)去表示一隻係time T 到期嘅zero-coupon bond, at time t嘅fair price
咁好自然P(T,T) = 1 因為我已經assume咗face value = 1 所以隻bond到期只會拎返$1
根據risk-neutral pricing嘅神髓 我地會知道P(t,T) = the fair price of zero-coupon bond at time t其實就等於下圖咁樣
大家又再諗下 我地之前assume stock price follows GBM嘅時候
我地係assume曬r同σ都係constant
所以exp[-r(T-t)] 係一個(up to information t) constant 我地其實係因為咁所以先可以將呢個discount rate寫係expectation出面
咁依家interest rate已經係stochastic我地就冇得偷雞再咁做
不過其實呢舊discount rate嘅form唔難搵到出黎亦都唔難去理解
但係個問題係我又寫到爆字數
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下個cm再戰
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(預告:一個好耐冇出現嘅F字頭朋友會再次登場
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