(i) Background
首先我地要回憶返應該用啲乜嘢黎 model interest rate
其實我都剩係講過一個model
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就係Vasicek Model
下圖就係Vasicek Model嘅樣 同埋 under Vasicek Model 嘅 (non-defaultable) zero-coupon bond price
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詳細嘅derivation就暫時唔再寫 因為都係solve ODE冇乜咁特別
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遲啲一次過再執
咁我地依家有咗 zero-coupon bond price 嘅 "Close form" 之後
我地就可以再考慮一樣嘢
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點解我上面要用引號括住Close form? 原因就係interest rate r_t 依家已經係stochastic (random)
所以就算我地寫得出呢個Close form 其實zero-coupon bond price都唔會係deterministic
咁如果bond price都係stochastic嘅話 咁佢都應該會有一條屬於自己嘅dynamics
如果大家心水清都會知道我地可以靠Ito's lemma去揾返bond price嘅dynamics
[ Reasoning: the Bond price P we derived above was just like applying a function g to r_t, and Ito's lemma could help us in finding the dynamics of g(r_t) ]
成個mean term即刻變咗做rP
咁我地就可以寫低 bond price P 嘅dynamics啦
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(ii) Structure
咁依家就真係萬事俱備 可以開始講bond option
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首先顧名思義 Bond option就係option on (non-defaultable zero-coupon) bond啦
同普通option一樣都可以分Bond call option同Bond put option
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同樣亦都有European同American之分
但係for simplicity我依家只會討論European bond call
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( Note1: European bond put price可以用put-call parity揾到)
( Note2: 而American style嘅option由於牽涉到hitting time 所以要容後講完Laplace transform再一次過處理)
而下圖分別就係European bond call嘅structure同payoff
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簡單咁講就係 at time T你有權用 $ K 買underlying bond
咁當at time T 嘅 market bond price P( T, T_1; r_T) > K
好自然你就會用 $K 買bond 然後 $P 賣返出去
所以呢個case 嘅 Payoff at time T 就會係 $P-K
但係當 P <= K 嘅時候 Payoff就會係 0
點解唔係 $K-P ? 大家要記住 Call/Put 係比你買賣underlying嘅權利
future/forward 先係一定要買underlying
所以當你發現 at time T 嘅 market bond price P 仲低過 K
咁好自然你就唔會exercise張call 咁Payoff at time T 就會 = 0
如果將兩個case合埋就可以好似上圖咁用maximum function寫低
至於點解我地要咁寫?
因為Black-scholes call price可以話係最basic嘅option pricing formula
將其他exotic option嘅price寫返做Black-Scholes call price咁嘅款就會極度方便我地計數
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同埋大家都可以思考下當中嘅強大之處
大家要記得呢隻係exotic option 我地本身suppose佢嘅pricing formula可能會好撚複雜
但係最尾我地都可以用返最普通european call price嘅樣寫低條pricing formula
This is the beauty of derivative pricing