Term Structure Driven by general Lvy processes

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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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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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Lévy Process

• Lévy process Lt a generalization to the
  Brownian motion.
• stochastic process with stationary and
  independent increments
• continuous in probability (P(Lu -Lve) ? 0, as
  u ? v for all v)
• L00 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)

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Examples of Lévy Processes
Remarks K1 is the modified Bessel function of
the third kind with index 1
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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

• (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
• (boundary condition)P(T,T)1 for all T?0,T
• (volatility bounds)Define ??(s,T) 0?s?T
  ?T?(s,T)gt0 for all (s,T)???, s??T, and
  ?(T,T)0 for all T??0,T
• (integrability)There exists constants M, ?gt0
  such that
• (volatility smoothness)?(s,T), defined on ?, is
  twice continuously differentiable in both
  variables and is bounded by M from 4

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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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Basic Lemmas(1)

• Lemma 1.1
• If 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.

Proof
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Basic Lemmas(2)

• Lemma 1.2
• Forward rate process f(?,T) has the form
• where ?2 denotes the partial derivative of ? in
  its second variable (T)
• Lemma 1.3
• The numéraire ?(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

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
  Vasicek volatility structure or Ho-Lee volatility
  structure!
• or
• for real constants

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Proof steps(1)

• Lemma 2.1
• Suppose the CF of L1 is bounded, with real
  constants C, ?, ?gt0, such that
• If f, g are continuous functions such that
  f(s)??kg(s) for all s, then the joint
  distribution of X and Y is continuous w.r.t.
  Lebesgue measure ?2 on ?2, where
• Lemma 2.2
• The short rate process r is Markovian if and only
  if
• where 0ltTltUltT, and note that ? may depend on T
  and U, but not on t.
• Corollary

Proof
Proof
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Proof steps(2) Hull-White revisit

• Theorem 2.3
• Further assume that the volatility structure is
  stationary, then it must be either of Vasicek or
  Ho-Lee structure.
• Corollary
• Under the above stationarity and Markovian
  assumptions, we can take the volatility to have
  the Vasicek 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 Vasicek model)

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).
• Vasicek volatility structure,
• Initial term structure flat at f(0,t)0.05 for
  all t

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Hyperbolic vs normal
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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 options
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
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Comparing bond options result
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
  Vasicek.
• We can re-derive the mean-reverting short rate
  process (Hull-White) when we utilize Vasicek
  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.

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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)

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References

• Carr, P. Geman, 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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THE END

• Thank you for participating.

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Appendix

• Appendix 1.1

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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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Appendix

• Appendix 1.3
• From (1), and lemma 1.1,
• From (2), setting t?T
• Integrate r(T),

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Appendix

• Appendix 2.2
• From (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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Appendix

• Appendix 2.2 (cont)
• 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 cant be
  continuous w.r.t. ?2 on ?2. By lemma 2.1,

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Appendix

• Appendix 2.2 (cont)
• (?) Assume , then

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Appendix

• Appendix 2.2 (cont)
• For the corollary, simply take UT, ?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)gt0 for t?T and
  ?(t,t)0. Therefore, ? ? 0.
• Then we can define
• and obtain our desired result, where

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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 a0, then
• (Ho-Lee)
• If a??0, then
• (Vasicek)

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