Term Structure Driven by general L é vy processes Bonaventure Ho HKUST, Math Dept. Oct 6, 2005 Outline Why do we need to model term structure? Going from Brownian motion to L évy process Model assumption and bond price dynamics Markovian short rate and stationary volatility structure
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HKUST, Math Dept.
Oct 6, 2005
where W is a standard Brownian motion
Remarks: K1 is the modified Bessel function of the third kind with index 1
Under this numéraire, P(t,T) is a martingale under this measure when expressed in terms of units of (t).
If f is left-continuous with limits from the right and bounded by M, then
where denotes the log MGF of L1
Forward rate process f(,T) has the form
where 2 denotes the partial derivative of in its second variable (T)
Thenuméraire (t) is given by , where (t) is the usual money market account process
Remark: Substituting the result back to our initial assumption, we obtainFor Gaussian model, (u)=u2/2, we obtain the usual case
for real constants
Suppose the CF of L1 is bounded, with real constants C, , >0, such that
If f, g are continuous functions such that f(s)k·g(s) for all s, then the joint distribution of X and Y is continuous w.r.t. Lebesgue measure 2 on 2, where
The short rate process r is Markovian if and only if
where 0<T<U<T*, and note that may depend on T and U, but not on t.
Further assume that the volatility structure is stationary, then it must be either of Vasiček or Ho-Lee structure.
Under the above stationarity and Markovian assumptions, we can take the volatility to have the Vasiček volatility structure. Then the short rate process follows:
If we take L to be W, we revert back to the Hull-White model (or the extended Vasiček model)
K1 and K2 denotes the modified Bessel function of the third kind with index 1 and 2 respectively.
Standard normalHyperbolic vs normal
Figure 1: Forward rate predicted by hyperbolic Lévy (=0.01)
Figure 2: fhyper(t,T)-fGauss(t,T)
Forward rates predicted by hyperbolic Lévy motion are marginally higher than that predicted by Brownian motion.
Bond call option:
current time = 0, option maturity = t, bond maturity = T, strike = K
In the Gaussian case, there is an analytic solution:
However, in the Lévy setting, the expectation becomes:
Fortunately, a numerical solution is available because the joint density function for the last two stochastic terms can be found. (Very complicated)
Comparison method: We compare the pricing difference against the various forward price/strike price ratio.
Option maturity = 1yr, bond maturity = 2yr
Note that at-the-money strike 0.951
Figure 3 Differences in option pricing vs forward/strike price ratio
As one can see, for =10, the difference is minimal, but for =0.01, At-the-money option is lower for the hyperbolic model (~10%) while the in-the-money and out-of-the-money prices are slightly higher, forming the W-shaped pattern as show in Figure 3.
Carr, P. G´eman, H., Madan, D., Wu, L., Yor, M. (2003) Option Pricing using Integral Transforms. Stanford Financial Mathematics Seminar (Winter 2003).
Carr, P., Wu, L. (2003) Time-Changed Lévy Processes and Option Pricing. Journal of Financial Economics, Elsevier, Vol 71(1), 113-141.
Eberlein, E., Baible, S. (1999). Term structure models driven by general Lévy processes. Mathematical Finance, Vol 9(1), 31-53.
Eberlein, E., Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli, 1, 281-299.
Eberlein, E., Keller, U., Prause, K. (1998)New insights into smile, mispricing and value at risk: the hyperbolic model. Journal of Business 71.
Eberlein, E., Özkan, F. (2005) The Lévy Libor Model. Finance and Stochastics 9, 327-348.
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From (1), and lemma 1.1,
Take –log, we get,
Differentiate w.r.t. T, we have,
From (1), and lemma 1.1,
From (2), setting tT
we can see that r is Markov iff Z(T) is Markov, where
() Assume r is Markov. Then
is independent of because of the independent increments
of L. Thus the last two terms are equal, implying the equality of the first two terms.
However is measurable w.r.t. . Therefore,
is some function of Z(T), say
. Then the joint distribution of X and Y, where
is only defined on (x,G(x)), thus can’t be continuous w.r.t. 2 on 2. By lemma 2.1,
() Assume , then
For the corollary, simply take U=T*, then we have 2(t,T*)= 2(t,T), where is independent of t (yet it may depend on T and T*). If =0, then 2(t,T*)=0 for all t. However, this implies that (t,T)=constant for all T, which violates the assumption that (t,T)>0 for tT and (t,t)=0. Therefore, 0.
Then we can define
and obtain our desired result, where
Write . Then . Writing
we have Rearranging terms, we have
Since both sides cannot depend on t or T, it must equal to some constant a.
If a=0, then
If a0, then