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Chaper 4: Continuous-time
interest rate modelsLin Heng-Li
December 5, 2011
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The general principles in this development are that
◦ is Markov
◦ Prices depend upon an assessment at time t of
how 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
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Suppose that
Where W(t) is a Brownian motion under PThe first two principles ensure that
Thus, under a one-factor model, price changes for all bonds with different maturity dates are perfectly (but none-linearly) correlated.
4.3 The PDE Approach to Pricing
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By Itô’s lemma
Where
4.3 The PDE Approach to Pricing
(a.1)
(a.2)
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Consider two bonds with different maturity dates T1and T2 (
At time t, suppose that we hold amounts in the -bond and in the -bond
Total wealth
4.3 The PDE Approach to Pricing
(1)
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The instantaneous investment gain from t to t+dt is
4.3 The PDE Approach to Pricing
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We will vary and in such a way that the portfolio is risk-free.◦ Suppose that, for all t,
then
◦ By (1) and (2)
4.3 The PDE Approach to Pricing
(2)
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Hence, the instantaneous investment gain
Since the portfolio is risk-free, by the principle of no arbitrage
and
4.3 The PDE Approach to Pricing
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This must be true for all maturities. Thus, for all T>t
is the market price of risk.◦ Cannot depend on the maturity date◦ Can often be negative.
(Since is usually negative, suppose the volatility be positive, we have Thus, must be negative to ensure that expected returns are greater than the risk-free rate.)
4.3 The PDE Approach to Pricing
(b)
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From (b), we have And from (a.1) Equate the two expressions,
4.3 The PDE Approach to Pricing
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This is a suitable form to apply the Feynman-Kac formula
◦ The boundary condition for this PDE
4.3 The PDE Approach to Pricing
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By the Feynman-Kac formula there exists a suitable probability triple with filtration under which
◦ (s) () is a Markov diffusion process with ◦ Under the measure Q, satisfies the SDE
◦ is a standard Brownian motion under Q
4.3 The PDE Approach to Pricing
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Suppose that◦ satisfies the Novikov condition
◦ We define
By Girsanov Theorem, there exists an equivalent measure Q under which (for ) is a Brownian motion and with Radon-Nikodym derivative
4.3 The PDE Approach to Pricing
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Note that we have
)
4.3 The PDE Approach to Pricing
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The Feynman-Kac formula can be applied to interest rate derivative contracts.
Let be the price at time t of a derivative which will have only a payoff to the holder of at time T
4.3 The PDE Approach to Pricing
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Suppose that
By Itô’s lemma
From market price of risk
4.3 The PDE Approach to Pricing
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From above, we will have
Apply to Feynman-Kac formula
◦ subject to
4.3 The PDE Approach to Pricing
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By the Feynman-Kac formula , we have
where
4.3 The PDE Approach to Pricing