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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Term Structure Driven by general Lévy processes
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.
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 3Differences 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.
Thank you for participating.
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.
Howeveris 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