Financial Engineering

1
Financial Engineering

• Term Structure Models
• Zvi Wiener
• mswiener_at_mscc.huji.ac.il
• tel [phone]

2
Interest Rates

• Dynamic of IR is more complicated.
• Power of central banks.
• Dynamic of a curve, not a point.
• Volatilities are different along the curve.
• IR are used for both discounting and defining
  the payoff.
• Source Hull and White seminar

3
Main Approaches

• Blacks Model (modification of BS).
• No-Arbitrage Model.

4
Notations

• D - face value (notional amount)
• C - coupon payments (as of par), yearly
• N - maturity

See Benninga, Wiener, MMA in Education, vol. 7,
No. 2, 1998
5
Continuous Version

• Denote by Ctdt the payment between
• t and tdt, then the bond price is given by

Principal should be written as Diracs delta.
6
Forward IR

• The Forward interest rate is a rate which
  investor can promise today, given the term
  structure.
• Suppose that the interest rate for a maturity of
  3 years is r310, and the interest rate for 5
  years is r511.
• No borrowing-lending restrictions.

7
Forward IR

• r310, r511.
• Lend 1,000 for 3 years at 10.
• Borrow 1,000 for 5 years at 11.
• Year 0 -1,0001,000 0
• Year 3 1,000(1.1)3 1331
• Year 5 -1,000(1.11)5 -1658
• Is identical to a 2-year loan starting at year 3.

8
Forward IR
Forward interest rate from t to tn.
9
Forward IR
Continuous compounding
10
Forward IR
11
Estimating TS from bond data

• Idea - to take a set of simple bonds and to
  derive the current TS.
• Too many bonds.
• Too few zero coupons.
• Non simultaneous pricing.
• Very unstable!

12
Estimating TS from bond data

• Assume that
• r15.5, r25.55, r35.6, r45.65, r55.7.
• Bond prices
• 1 year 3 979.766
• 2 years 5 982.56
• 3 years 3 918.164
• 4 years 7 1030.94
• 5 years 0 740.818

13
Estimating the TS

• We can easily extract the interest rates from the
  prices of bonds.
• However if the bond prices are rounded to a
  dollar, the resulting TS looks weird.
• Conclusion TS is very sensitive to small errors.
  Instead of solving the system of equations
  defining a unique TS it is recommended to fit the
  set of points by a reasonable curve representing
  TS.
• Another problem - time instability.

14
Is flat TS possible?

• Why can not IR be the same for different times to
  maturity?
• Arbitrage
• Zero investment.
• Zero probability of a loss.
• Positive probability of a gain.

15
Is flat TS possible?

• Form a portfolio consisting of 3 bonds maturing
  in one, two, and three years and without coupons.
• Choose a, b, c units of these bonds.
• Zero investment
• ae-r be-2r ce-3r 0
• Zero duration
• -ae-r - 2be-2r - 3ce-3r 0

16
Is flat TS possible?

• Two equations, three unknowns
• ae-r be-2r ce-3r 0
• -ae-r - 2be-2r - 3ce-3r 0
• Possible solution (r10)
• a 1, b -2.21034, c1.2214

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Arbitrage in a flat TS
18
Arbitrage in a flat TS

• However even a small costs destroy this
  arbitrage.
• In many cases the assumption that TS is flat can
  be used.

19
Yield

• Denote by P(r,t,tT) the price at time t of a
  pure discount bond maturing at time tT t.
• Define yield to maturity R(r, t,T) as the
  internal rate of return at time t on a bond
  maturing at tT.

20
Yield

• The relation between forward rates and yield

When interest are continuously compounded the
average of forward rates gives the yield.
21
TS model

• Assume that interest rates follow a diffusion
  process.

What is the price of a pure discount bond
P(r,t,T)?
Implicit one factor assumption!
22
TS model

• Substituting dr we obtain

Taking expectation and dividing by dt we get
23
TS model

• Using equilibrium pricing models assume that

Here ? is the risk premium. The basic bond
pricing equation is (Merton 1971,1973)
24
TS model

• Merton has shown that in a continuous-time CAPM
  framework, the ration of risk premium to the
  standard deviation is constant (over different
  assets) when the utility function is logarithmic.

Sharpe ratio
25
TS model

• For a pure discount bond we have

Thus by Itos lemma
26
TS model

• Hence for the risk premium we have

The basic equation becomes
27
Vasiceks model

• Ornstein-Uhlenbeck process

28
Vasiceks model

• Discrete modeling

Negative interest rates. Can be used for example
for real interest rates.
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Shapes of Vasiceks model

• All three standard shapes are possible in
  Vasiceks model.
• Disadvantages
• calibration, negative IR, one factor only.
• There is an analytical formula for pricing
  options, see Jamshidian 1989.

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Extension of Vasicek

• Hull, White

31
CIR model

• Precludes negative IR, but under some conditions
  zero can be reached.

32
CIR model
33
CIR model
34
CIR model

• Bond prices are lognormally distributed with
  parameters

35
CIR model

• As the time to maturity lengthens, the yield
  tends to the limit

Different types of possible shapes.
36
One Factor TS Models
37

• ?1 ?2 ?3 ?1 ?2 ?
• Cox-Ingersoll-Ross 0.5
• Pearson-Sun 0.5
• Dothan 1.0
• Brennan-Schwartz 1.0
• Merton (Ho-Lee) 1.0
• Vasicek 1.0
• Black-Karasinski 1.0
• Constantinides-Ingersoll 1.5

38
Black-Derman-Toy

• The BDT model is given by
• for some functions U and ?.
• Find conditions on ?2, ?3, and ?2 under which the
  Black-Karasinski model specializes to the BDT
  model.

39
The Gaussian One-Factor Models

• For ?3 ?2 0 we get a Gaussian model, in which
  the short rates r(t1), r(t2), ,r(tk) are jointly
  normally distributed (under the risk-neutral
  measure).
• Special cases Vasicek and Merton models.
• In this case a negative ?2 is mean reversion.

40
The Gaussian One-Factor Models

• For a Gaussian model the bond-price process is
  lognormal.
• An undesirable feature of the Gaussian model is
  that the short rate and yields on bonds are
  negative with positive probability at any future
  date.

41
The Affine One-Factor Models

• The Gaussian and CIR models are special cases of
  single factor models with the property that the
  solution has the form

42
The Affine One-Factor Models
The yield for all t is affine in r
Vasicek, CIR, Merton (Ho-Lee), Pearson-Sun.
43
TS Derivatives

• Suppose a derivative has a payoff
• h(r,t) prior to maturity, and
• a terminal payoff g(r,?) when exercised (?
• Then by the definition of the equivalent
  martingale measure, the price at time t is
  defined by

44
TS Derivatives
45
TS Derivatives

• By Feynman-Kac theorem it can be equivalently
  written as a solution of PDE

With boundary conditions
46
Bond Option

• A European option on a bond is described by
  setting
• h(x, t) 0,
• g(x, ?) Max( f(x, ?) - K, 0).

47
Interest Rate Swap

• Can be approximated as a contract paying the
  dividend rate
• h(r, t) rt - r, where r is the fixed leg
• g(r,?) 0.

48
Cap

• Is a loan at variable rate that is capped at some
  level r. Per unit of the principal amount of
  the loan, the value of the cap is defined when
• h(rt, t) Min(rt,r)
• g(r?,?) 1 (sometimes 0)

49
Floor

• Similar to a cap, but with maximal rate instead
  of minimal
• h(rt, t) Max(rt,r)
• g(r?,?) 1 (sometimes 0)

50
MBS

• Mortgage Backed Securities
• Sinking fund bond. At origination a sinking fund
  bond is defined in terms of a coupon rate, a
  scheduled maturity date, and an initial
  principle.
• At each time prior to maturity there is an
  associated scheduled principle.

51
MBS

• Assume that the coupon rate is ? and principal
  repayment is at a constant rate h.

For a given initial principal p0. The schedule
is chosen so that at time T the loan is repaid.
52
MBS

• Home mortgages can be prepaid. This is typically
  done when interest rates decline.
• Unscheduled amortization process should be
  defined.
• It has psychological and economical factors.
• Standard solution - Monte Carlo simulation.

53
Monte Carlo

• X(?) - random variable
• Let Y be a similar variable, which is correlated
  with X but for which we have an analytic formula.

54
Monte Carlo

• Introduce a new random variable
• (here Y is the analytic value of the mean of
  Y(?) and ? - is a free parameter which we fix
  later)

55
Monte Carlo

• Calculate the variance of the new variable

56
Monte Carlo
we can reduced variance! The optimal value of the
parameter ? is
57
Monte Carlo
This choice leads to the variance of the
estimator
where ? is the correlation coefficient between X
and Y.