term structure driven by general lévy processes bonaventure ho hkust, math dept. oct 6, 2005
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Outline
Why do we need to model term structure? Going from Brownian motion to Lévy process Model assumption and bond price dynamics Markovian short rate and stationary volatility structure Comparison between Gaussian and Lévy setting Conclusion and Discussion
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Why do we need to model term structure? Term structure is the information contained in the forward rate curve, short
rate curve, and yield curve observed from market data. There are deterministic relationships among forward rates (f), short rates
(instantaneous interest rate)(r) and zero-coupon bond prices (P).
Usually, a model on fixed income assumes certain dynamics on the zero-coupon bond (eg. HJM) or on forward rates (eg. LIBOR). One can calculate the model price for f, r, and P under this model.
Arbitrage opportunities arise when the predicted values for f, r, or P are different from the currently observed market data.
Therefore, an arbitrage-free model should incorporate the term structure into itself.
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4
History of the Gaussian model
No-arbitrage Existence of risk neutral measure What is risk neutral measure?
An equivalent probability measure under which all tradable securities, when discounted, are martingales.
Discount factor (numeraire) = Log-normal model
Initiated by Bachelier (1900), modeling the French government bonds using Brownian motion.
Samuelson (1965) gave the price process in exponential form Black and Scholes (1973) framework of log-normality of stock prices
Eg.) Heath-Jarrow-Morton model under the risk neutral measure
where W is a standard Brownian motion Major shortcomings
Cannot generate volatility smile/skew as shown in the market data Distribution itself does not fit the market data very well – fatter tail and high peak, jump!
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tdWTtdttrTtPTtdP ),()(),(),(
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Lévy Process Lévy process Lt – a generalization to the Brownian motion.
stochastic process with stationary and independent increments continuous in probability (P(|Lu -Lv|≥ε) 0, as u v for all v)
L0=0 a.s. viewed as Brownian motion with jump associated Lévy measure F characterized by (,,F) for drift, variance, and the Lévy measure for jump
Examples of Lévy processes Brownian motion W BM with jump
(Merton) Merton Jump Part (Compound Poisson jump) (Carr) Variance Gamma (Eberlein) Hyperbolic (used as an example in the paper)
6
Examples of Lévy ProcessesDriver’s name Lévy Measure
Brownian motion
Merton Jump Part
Variance Gamma
Hyperbolic
)exp(log)( 1vLEv
22v
2
2
2 2
)(exp
2
1
x
1
2
1exp 22vv
x
x
exp
2
2
0
22
2
11log vv
)()(exp
2
22
221
22
xx
K
))((
)(log
)(log2
1
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221
222
22
uK
K
vv
Remarks: K1 is the modified Bessel function of the third kind with index 1
2
2
2
2
224,
7
Motivation to use Lévy processes
To incorporate jump into the price dynamics. Merton (1976) added an independent Poisson jump process with normal jump size. Eberlein (author of the presenting paper) and Keller
“It is in certain way opposite to the Brownian world, since its (Lévy process) paths are purely discontinuous. If one looks at real stock price movements on the intraday scale it is exactly this discontinuous behavior what one observes.”
– Eberlein and Keller (1997), promoting the hyperbolic Lévy model.
In 1995, they performed empirical studies and revealed a much better fit of return distributions on stock prices if the Brownian motion is replaced by a Lévy process. However, evidence for non-Gaussian behavior on bond prices is not as complete.
In 2005, Eberlein & Özkan explore the LIBOR model using Lévy processes.
Volatility smile/skew cannot be generated by the log-normal model Jump and/or stochastic volatility can generate such smile/skew. I will discuss the “jump” part as a component of Lévy processes in this presentation, based mostly
on the paper by Eberlein and Raible. Stochastic volatility can be generated by the “time-change” method, which will be my research
topic. Idea: change the calendar time to a random “business-activity” clock.
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Model assumptions
1. (initial bond prices)P(0,T) is deterministic, positive, and twice continuously differentiable function in T for all T[0,T*] for a fixed time horizon T*
2. (boundary condition)P(T,T)=1 for all T[0,T*]
3. (volatility bounds)Define :={(s,T) : 0sT T*}(s,T)>0 for all (s,T), sT, and (T,T)=0 for all T[0,T*]
4. (integrability)There exists constants M, >0 such that
5. (volatility smoothness)(s,T), defined on , is twice continuously differentiable in both variables and is bounded by M from #4
MvdxFvx
x)1(,)()exp(
1
9
Bond price dynamics
In HJM model, the solution to the SDE is simply
Note that the above solution of the SDE is evaluated under the risk neutral measure.
Our generalization: Replace W by L, thus source of randomness = Replace the discount factor by a more general numéraire :(t)
Under this numéraire, P(t,T) is a martingale under this measure when expressed in terms of units of (t).
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10
Lemma 1.1If f is left-continuous with limits from the right and bounded by M, then
where denotes the log MGF of L1
In particular, take f to be , we have
Here we relate the expected value of the source of randomness to the log MGF of the Lévy process, which is known when the process is specified.
Basic Lemmas(1)
)exp(log)( 1vLEv
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st dsTsdLTsEXE00
)),((exp),(exp:
tt
s dssfdLsfE00
))((exp)(exp
Proof
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Lemma 1.2Forward rate process f(,T) has the form
where 2 denotes the partial derivative of in its second variable (T)
Lemma 1.3The numéraire (t) is given by , where (t) is the usual money market account process
Remark: Substituting the result back to our initial assumption, we obtain
For Gaussian model, (u)=u2/2, we obtain the usual case
Basic Lemmas(2)
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)(exp)(
t
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),())),(()((exp),0(),(
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Proof
Proof
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Term structure of the volatility
We want to explore the class of volatility structure so that the short rate process is Markovian.
Furthermore, we want to restrict our volatility structure to be stationary. That is, .
It will be proven that the volatility has either Vasiček volatility structure or Ho-Lee volatility structure!
or
for real constants
)(~),( tTTt
)(1ˆ
),( tTaea
Tt
)(ˆ),( tTTt
0,0ˆ a
13
Proof steps(1)
Lemma 2.1Suppose 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
Lemma 2.2The 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.
Corollary:
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t
sdLsfX0
)(: t
sdLsgY0
)(:
T
TtTtTtTtUt
),(
),(],,0[),,(),( 222
),()()(),(2 TtTtTt
Proof
Proof
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Proof steps(2) & Hull-White revisit
Theorem 2.3Further assume that the volatility structure is stationary, then it must be either of Vasiček or Ho-Lee structure.
CorollaryUnder 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)
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tatatat
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Proof
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Comparing forward rates
Using similar steps, we can derive the forward rate process.
Comparing it to the Gaussian case, we obtain:
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Examples using Lévy processes (hyperbolic)
Eberlein and Keller (1995) used hyperbolic Lévy motion to model stock price dynamics. They claimed that the hyperbolic model allows an almost perfect statistical fit of stock return data.
In order to compare with the Gaussian case, we restrict L1 to be centered, symmetric, and with unit variance. That is, we pick
K1 and K2 denotes the modified Bessel function of the third kind with index 1 and 2 respectively.
We investigate the case =10 (density close to normal) and =0.01 (considerably heavier weight in the tails and in the center than normal).
Vasiček volatility structure, Initial term structure: flat at f(0,t)=0.05 for all t
)(
)(0
2
1
K
K
5.0015.0ˆ a
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Comparing forward rates
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.
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Comparing bond optionsBond 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
01 )),()(:);,,0( KTtPtEKTtC
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20
Comparing bond options result
Strike price Gaussian
Hyperbolic
(=10) (=0.01)
0.90 0.048731 0.048731 0.048731
0.91 0.039219 0.039219 0.039222
0.92 0.029707 0.029708 0.029725
0.93 0.020217 0.020227 0.020290
0.94 0.011095 0.011105 0.011170
0.95 0.004002 0.003961 0.003615
0.96 0.000741 0.000741 0.000757
0.97 0.000058 0.000074 0.000162
0.98 0.000002 0.000005 0.000035
0.99 0.000000 0.000000 0.000008
1.00 0.000000 0.000000 0.000002
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.
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Conclusion
Lévy process – a generalization to Brownian motion that allows jumps, which is more realistic to model bond/stock price movements.
Under the assumption of Markovian short rate and stationary volatility structure, the only possible volatility structures are Ho-Lee and Vasiček.
We can re-derive the mean-reverting short rate process (Hull-White) when we utilize Vasiček volatility even under Lévy process.
When we use hyperbolic model, forward rates predicted by Lévy model is always slightly higher than the Gaussian model.
For bond option, the price differences form a W-shaped against the forward/strike price ratio, due to heavier weight in the center and in the tail for the hyperbolic distribution.
22
Discussion
Future research on time-changed Lévy process, which can capture both jump and stochastic volatility.
Apply time-changed Lévy process to the LIBOR model and explore the term structure under the model.
Apply time-changed Lévy process to model stock/bond price movement for option pricing with correlation, or to model firm value movement for credit derivative pricing (for structural model evaluation)
23
References
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.
25
Appendix
Appendix 1.1
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LtfELtfELtfEiii
))((exp))((exp
)(exp)(exp)(exp
t
iii
t
si
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dLsfELtfEi
0
0
))((exp))((exp
)(exp)(exp
BACK
26
Appendix
Appendix 1.2
From (1), and lemma 1.1,
Take –log, we get,
Differentiate w.r.t. T, we have,
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s dsTsdLTstTPTtP00
),(exp),(exp)(),0(),(
tt
s dsTsdLTstTPTtP00
),(),()(log),0(log),(log
t
s
t
dLTsdsTsTsTfTtf0
2
0
2 ),(),(),(`),0(),(
BACK
27
Appendix
Appendix 1.3
From (1), and lemma 1.1,
From (2), setting tT
Integrate r(T),
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s
t
dLtsdststP
dLtsdststP
ttPt
0000
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1),(exp)),((exp
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),()(
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000
),()),((),0(log
),()),((lim),0(log
),()),((
lim),0()(
BACK
28
Appendix
Appendix 2.2From (2),
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.
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sdLUs ),(2 )(, TZT
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29
Appendix
Appendix 2.2 (con’t)
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,
T
sdLUs0
2 ),(T
)(),(),(
0
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2 TZGdLUsT
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),(),( 22 TsUs
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30
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)(),()(),(
)(),()(),(
),(),(
),(),(|)(
2
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s
T
s
U
T
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T
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sT
T
s
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T
sT
Appendix
Appendix 2.2 (con’t)
() Assume , then ),(),( 22 TsUs
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31
Appendix
Appendix 2.2 (con’t)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
1
),(
),()(
),()(
*2
2
*2
Ts
TsT
Tst
)()(),(2 TtTt
BACK
32
Appendix
Appendix 2.3
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
(Ho-Lee)
If a0, then
(Vasiček)
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