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Extra info for Chris Marker and Documentary Filmmaking: 1962-1982 (Phd Thesis)
The Cheyette model is an instantaneous short rate model embedded in the HJM-framework that models the instantaneous forward rates. 1) where f mkt represents the market instantaneous short rate at time t = 0, for maturity T . In order for this model to be arbitrage free, the drift term must be of the form: y σ(x, s)ds. 1), and integrating both sides from 0 to t, yields 31 32 CHAPTER 4. THE CHEYETTE MODEL WITHOUT STOCHASTIC VOLATILITY t T t σ(u, s)ds du + σ(u, T ) f (t, T ) = f (0, T ) + 0 u 0 0 σ(u, T )dW Q (u).
3bp in the implied volatility. Hence, if we get these parameters from the calibration process, we have to include an uncertainty region of approximately 70bp, around the implied volatility at strike ATM − 100bp. This uncertainty is clearly greater than the 50bp difference observed at strike levels at ATM+100bp. 46 CHAPTER 4. 0266 Again, like in the 10Y-10Y case, we have for a small γ and small σ quite good pricing results for the 20Y-10Y swaption. 82%. 9bp in the implied volatility. This means, if we calibrate to a strike level of ATM − 100bp, we have to include an uncertainty region of approximately 20bp.
To calculate the expectation and variance note that t S0,m (t) = S0,m (0) + σ(s)λ(s)R0 dW Q 1,m (s). 0 Simple calculations show that µ ¯ = S0,m (0), and 2 σ ¯ = R02 t 2 2 R02 σ(s) λ(s) ds = 0 2 m t σ(s) 0 2 qiS (0)B(s, Ti ) ds. i=0 Under the assumption of a piecewise constant σ(t) this integral has an analytic solution. Suppose σ(t) is piecewise constant on intervals 0 = s0 < s1 < s2 < . . < sp = t, then t p 2 2 σ(s) λ(s) ds = 0 = σi2 si m qiS (0)qkS (0)B(s, Tj )B(s, Tk )ds s i−1 i=1 j,k=0 p m si qiS (0)qkS (0) σi2 B(s, Tj )B(s, Tk )ds.