Chaper 4: Continuous-time interest rate models Lin Heng-Li December 5, 2011

4.3 The PDE Approach to Pricing The general principles in this development are that 𝑟(𝑡) is Markov Prices 𝑃(𝑡,𝑇) depend upon an assessment at time t of how 𝑟(𝑠) will vary between t and T The market is efficient, without transaction costs and all investors are rational.

4.3 The PDE Approach to Pricing Suppose that 𝑑𝑟 𝑡 =𝑎 𝑟 𝑡 𝑑𝑡+𝑏 𝑟 𝑡 𝑑𝑊(𝑡) 𝑑𝑃 𝑡,𝑇 =𝑃 𝑡,𝑇 [𝑚 𝑡,𝑇 𝑑𝑡+𝑆 𝑡,𝑇 𝑑𝑊 𝑡 ] Where W(t) is a Brownian motion under P The first two principles ensure that 𝑎 𝑡 =𝑎(𝑡,𝑟 𝑡 ) 𝑏 𝑡 =𝑏 𝑡,𝑟 𝑡 𝑃 𝑡,𝑇 =𝑃(𝑡,𝑇,𝑟(𝑡)) Thus, under a one-factor model, price changes for all bonds with different maturity dates are perfectly (but none-linearly) correlated. 課本page 60

4.3 The PDE Approach to Pricing By Itô’s lemma 𝑑𝑃= 𝜕𝑃 𝜕𝑡 𝑑𝑡+ 𝜕𝑃 𝜕𝑟 𝑑𝑟+ 1 2 𝜕 2 𝑃 𝜕𝑟 2 𝑑𝑟 2 = 𝜕𝑃 𝜕𝑡 𝑑𝑡+ 𝜕𝑃 𝜕𝑟 𝑎𝑑𝑡+𝑏𝑑𝑊 + 1 2 𝜕 2 𝑃 𝜕𝑟 2 𝑏 2 𝑑𝑡 = 𝜕𝑃 𝜕𝑡 +𝑎 𝜕𝑃 𝜕𝑟 + 1 2 𝑏 2 𝜕 2 𝑃 𝜕𝑟 2 𝑑𝑡+𝑏 𝜕𝑃 𝜕𝑟 𝑑𝑊 Where 𝑚 𝑡,𝑇,𝑟 = 1 𝑃 𝜕𝑃 𝜕𝑡 +𝑎 𝜕𝑃 𝜕𝑟 + 1 2 𝑏 2 𝜕 2 𝑃 𝜕𝑟 2 𝑆 𝑡,𝑇,𝑟 = 1 𝑃 𝑏 𝜕𝑃 𝜕𝑟 m(t,T,r)與S(t,T,r)之所以會有1/P可對照前一張投影片 (a.1) (a.2)

4.3 The PDE Approach to Pricing Consider two bonds with different maturity dates T1and T2 ( 𝑇 1 < 𝑇 2 ) At time t, suppose that we hold amounts − 𝑉 1 (𝑡) in the 𝑇 1 -bond and 𝑉 2 (𝑡) in the 𝑇 2 - bond Total wealth 𝑉 𝑡 = 𝑉 2 𝑡 − 𝑉 1 𝑡 Begin to consider the market price of risk (1)

4.3 The PDE Approach to Pricing The instantaneous investment gain from t to t+dt is 𝑑𝑉 𝑡 =− 𝑉 1 𝑡 𝑃 𝑡, 𝑇 1 𝑑𝑃 𝑡, 𝑇 1 + 𝑉 2 𝑡 𝑃 𝑡, 𝑇 2 𝑑𝑃 𝑡, 𝑇 2 =− 𝑉 1 𝑡 𝑚 1 𝑑𝑡+ 𝑆 1 𝑑𝑊 + 𝑉 2 𝑡 𝑚 2 𝑑𝑡+ 𝑆 2 𝑑𝑊 = 𝑉 2 𝑚 2 − 𝑉 1 𝑚 1 𝑑𝑡+ 𝑉 2 𝑆 2 − 𝑉 1 𝑆 1 𝑑𝑊 for notational compactness, we write mi=m(t,Ti,r(t))

4.3 The PDE Approach to Pricing We will vary 𝑉 1 (𝑡) and 𝑉 2 (𝑡) in such a way that the portfolio is risk-free. Suppose that, for all t, 𝑉 1 (𝑡) 𝑉 2 (𝑡) = 𝑆(𝑡, 𝑇 2 ,𝑟 𝑡 ) 𝑆(𝑡, 𝑇 1 ,𝑟 𝑡 ) = 𝑆 2 𝑆 1 then 𝑉 2 𝑆 2 − 𝑉 1 𝑉 1 𝑆 1 =0 (2)

4.3 The PDE Approach to Pricing Hence, the instantaneous investment gain 𝑑𝑉 𝑡 = 𝑉 2 𝑚 2 − 𝑉 1 𝑚 1 𝑑𝑡+ 𝑉 2 𝑆 2 − 𝑉 1 𝑆 1 𝑑𝑊 =𝑉 𝑚 2 𝑆 1 − 𝑚 1 𝑆 2 𝑆 1 − 𝑆 2 𝑑𝑡

4.3 The PDE Approach to Pricing This must be true for all maturities. Thus, for all T>t 𝑚(𝑡,𝑇,𝑟 𝑡 )−𝑟(𝑡) 𝑆(𝑡,𝑇,𝑟 𝑡 ) =𝛾(𝑡,𝑟 𝑡 ) 𝛾(𝑡,𝑟 𝑡 ) is the market price of risk. Cannot depend on the maturity date Can often be negative. (Since 𝜕𝑃 𝜕𝑟 is usually negative, suppose the volatility 𝑏 𝑡,𝑟 𝑡 be positive, we have 𝑆 𝑡,𝑇,𝑟 𝑡 <0. Thus, 𝛾(𝑡,𝑟 𝑡 ) must be negative to ensure that expected returns are greater than the risk-free rate.) (b)

4.3 The PDE Approach to Pricing From (b), we have 𝑚 𝑡,𝑇,𝑟 =𝑟 𝑡 +𝛾 𝑡,𝑟 𝑆(𝑡,𝑇,𝑟) And from (a.1) 𝑚 𝑡,𝑇,𝑟 = 1 𝑃 𝜕𝑃 𝜕𝑡 +𝑎 𝜕𝑃 𝜕𝑟 + 1 2 𝑏 2 𝜕 2 𝑃 𝜕𝑟 2 Equate the two expressions, 𝜕𝑃 𝜕𝑡 + 𝑎−𝑏𝛾 𝜕𝑃 𝜕𝑟 + 1 2 𝑏 2 𝜕 2 𝑃 𝜕𝑟 2 −𝑟𝑃=0

4.3 The PDE Approach to Pricing This is a suitable form to apply the Feynman- Kac formula 𝜕𝑃 𝜕𝑡 +𝑓 𝑡,𝑟 𝜕𝑃 𝜕𝑟 + 1 2 𝜌 2 𝑡,𝑟 𝜕 2 𝑃 𝜕𝑟 2 −𝑅 𝑟 𝑃+ℎ(𝑡,𝑟)=0 𝑓 𝑡,𝑟 =a−γ𝑏 𝜌 𝑡,𝑟 =𝑏 𝑡,𝑟 𝑅(𝑟)=𝑟 ℎ(𝑡,𝑟)=0 The boundary condition for this PDE 𝑃 𝑇,𝑇,𝑟 =𝛹 𝑟 =1

4.3 The PDE Approach to Pricing By the Feynman-Kac formula there exists a suitable probability triple (Ω,ℱ,𝑄) with filtration ℱ 𝑡 :0≤𝑡<∞ under which 𝑃 𝑡,𝑇,𝑟 𝑡 = 𝐸 𝑄 [exp⁡(− 𝑡 𝑇 𝑟 𝑠 𝑑𝑠 )| ℱ 𝑡 ] 𝑟 (s) (𝑡≤𝑠≤𝑇) is a Markov diffusion process with 𝑟 𝑡 =𝑟 𝑡 Under the measure Q, 𝑟 (𝑢) satisfies the SDE 𝑑 𝑟 𝑢 =𝑓 𝑢, 𝑟 𝑢 𝑑𝑢+𝜌 𝑢, 𝑟 𝑢 𝑑 𝑊 𝑢 = a−γ𝑏 𝑑𝑢+𝑏𝑑 𝑊 𝑢 𝑊 𝑢 is a standard Brownian motion under Q 課本246~248頁

4.3 The PDE Approach to Pricing Suppose that 𝛾 𝑠,𝑟(𝑠) satisfies the Novikov condition 𝐸 𝑄 exp 1 2 0 𝑇 𝛾 𝑠,𝑟 𝑠 2 𝑑𝑠 <∞ We define 𝑊 𝑡 =𝑊 𝑡 + 0 𝑡 𝛾 𝑠,𝑟(𝑠) 𝑑𝑠 By Girsanov Theorem, there exists an equivalent measure Q under which 𝑊 𝑡 (for 0≤𝑡≤𝑇) is a Brownian motion and with Radon-Nikodym derivative 𝑑𝑄 𝑑𝑃 =exp⁡[− 0 𝑇 𝛾 𝑡,𝑟 𝑡 𝑑𝑊(𝑡) − 1 2 0 𝑇 𝛾 𝑡,𝑟 𝑡 2 𝑑𝑡 ] 原本的模型 r= a dt + b dW 是在P測度底下的 我們必須證明P Q 兩種測度約當(等價) 才能進行套用

4.3 The PDE Approach to Pricing Note that we have 𝑑𝑃 𝑡,𝑇 =𝑃 𝑡,𝑇 𝑚 𝑡,𝑇,𝑟 𝑡 𝑑𝑡+𝑆 𝑡,𝑇,𝑟 𝑡 𝑑𝑊 =𝑃 𝑡,𝑇 [𝑚 𝑡,𝑇,𝑟 𝑡 𝑑𝑡+𝑆(𝑡,𝑇,𝑟 𝑡 ){𝑑 𝑊 −𝛾 𝑡,𝑟 𝑡 𝑑𝑡} =𝑃 𝑡,𝑇 𝑚 𝑡,𝑇,𝑟 𝑡 −𝛾 𝑡,𝑟 𝑡 𝑆 𝑡,𝑇,𝑟 𝑡 𝑑𝑡+𝑆 𝑡,𝑇,𝑟 𝑡 𝑑 𝑊 𝑚 𝑡,𝑇,𝑟 𝑡 −𝛾 𝑡,𝑟 𝑡 𝑆 𝑡,𝑇,𝑟 𝑡 𝑑𝑡+𝑆 𝑡,𝑇,𝑟 𝑡 𝑑 𝑊 =𝑃 𝑡,𝑇 (𝑟 𝑡 𝑑𝑡+ 𝑆 𝑡,𝑇,𝑟 𝑡 𝑑 𝑊 ) 因此 在側度Q底下 任何債券的期望報酬皆等於無風險利率 rf Q是一個風險中立測度

4.3 The PDE Approach to Pricing The Feynman-Kac formula can be applied to interest rate derivative contracts. Let 𝑉 𝑡 be the price at time t of a derivative which will have only a payoff to the holder of 𝛹 𝑟(𝑇) at time T

4.3 The PDE Approach to Pricing Suppose that 𝑑𝑟 𝑡 =𝑎 𝑟 𝑡 𝑑𝑡+𝑏 𝑟 𝑡 𝑑𝑊(𝑡) 𝑑𝑉 𝑡,𝑇,𝑟 =𝑉 𝑡,𝑇,𝑟 [𝑚′ 𝑡,𝑇,𝑟 𝑑𝑡+𝑆′ 𝑡,𝑇,𝑟 𝑑𝑊 𝑡 ] By Itô’s lemma dV= 𝜕𝑉 𝜕𝑡 +𝑎 𝜕𝑉 𝜕𝑟 + 1 2 𝑏 2 𝜕 2 𝑉 𝜕𝑟 2 𝑑𝑡+𝑏 𝜕𝑉 𝜕𝑟 𝑑𝑊 𝑚′ 𝑡,𝑇,𝑟 = 1 𝑉 𝜕𝑉 𝜕𝑡 +𝑎 𝜕𝑉 𝜕𝑟 + 1 2 𝑏 2 𝜕 2 𝑉 𝜕𝑟 2 𝑆′ 𝑡,𝑇,𝑟 = 1 𝑉 𝑏 𝜕𝑉 𝜕𝑟 From market price of risk 𝛾 𝑡,𝑟 𝑡 = 𝑚′(𝑡,𝑇,𝑟 𝑡 )−𝑟(𝑡) 𝑆′(𝑡,𝑇,𝑟 𝑡 ) 𝑚′ 𝑡,𝑇,𝑟 =𝑟 𝑡 +𝛾 𝑡,𝑟 𝑆′ 𝑡,𝑇,𝑟

4.3 The PDE Approach to Pricing From above, we will have 𝜕𝑉 𝜕𝑡 + a−γ𝑏 𝜕𝑉 𝜕𝑟 + 1 2 𝑏 2 𝑡,𝑟 𝜕 2 𝑉 𝜕𝑟 2 −𝑟 𝑡 𝑉=0 Apply to Feynman-Kac formula 𝜕𝑉 𝜕𝑡 +𝑓 𝑡,𝑟 𝜕𝑉 𝜕𝑟 + 1 2 𝜌 2 𝑡,𝑟 𝜕 2 𝑉 𝜕𝑟 2 +ℎ(𝑡,𝑟)−𝑅 𝑟 𝑉=0 subject to 𝑉 𝑇 =𝛹 𝑟(𝑇) 𝑓 𝑡,𝑟 =a−γ𝑏 𝜌 𝑡,𝑟 =𝑏 𝑡,𝑟 𝑅(𝑟)=𝑟 ℎ(𝑡,𝑟)=0

4.3 The PDE Approach to Pricing By the Feynman-Kac formula , we have V t = 𝐸 𝑄 exp − 𝑡 𝑇 𝑟 𝑠 𝑑𝑠 𝛹 𝑟 𝑇 ℱ 𝑡 where 𝑑 𝑟 𝑢 =𝑓 𝑢, 𝑟 𝑢 𝑑𝑢+𝜌 𝑢, 𝑟 𝑢 𝑑 𝑊 𝑢 = a−γ𝑏 𝑑𝑢+𝑏𝑑 𝑊 𝑢 =𝑎∗𝑑𝑢+𝑏∗ 𝑑 𝑊 𝑢 −γ𝑑𝑢 =𝑎∗𝑑𝑢+𝑏∗𝑑𝑊 =𝑑𝑟(𝑢)