MATH3090/7039: Financial mathematics Lecture 4
Yield curves
Yield curves
Copyright By PowCoder代写 加微信 powcoder
Arbitrage relationships Implied forward rates Stripping yield curves
Bonds, duration and convexity
Bond pricing revisited
Interest rate risk: duration and convexity
Stochastic interest rate models
Binomial model of yield curves
Real markets do not have a single interest rate. Instead, they have bonds of different maturities, some paying coupons and others not paying coupons . . . In summary, instead of having a single interest rate, real markets have a yield curve, which one can regard as a function of finitely many yields plotted versus their corresponding maturities.
. Calculus for Finance II: Continuous Time Models
Yield curves
Yield curves
Arbitrage relationships Implied forward rates Stripping yield curves
Bonds, duration and convexity
Bond pricing revisited
Interest rate risk: duration and convexity
Stochastic interest rate models
Binomial model of yield curves
Valuation by arbitrage and replication
Recall the law of one price or the valuation by arbitrage principal: • Assets with equal future cashflows/payoff must have the same
• The value of an asset is the value of any portfolio which replicates the asset’s future cashflow structure.
Today we start to utilise this approach: • Arbitrage-free yield curve.
• Stochastic interest rate models.
Yield curves
Yield curves
Arbitrage relationships Implied forward rates Stripping yield curves
Bonds, duration and convexity
Bond pricing revisited
Interest rate risk: duration and convexity
Stochastic interest rate models
Binomial model of yield curves
Yield curves
• Consider this Bloomberg Markets link.
The yield curve or the term structure of interest rates expresses the relationship between the interest rate or yield or YTM on an issue of fixed interest securities with respect to time to maturity.
• For example, 5-year Govt bonds typically trade at different yields from, say, 30-year Govt bonds.
• Bonds of the same type issued from the same issuer will have their own yield curves.
In other words we plot YTM as a function of time to maturity.
The yield curve: spot and forward rates
• The term spot trading refers to trading asset and securities in the usual manner: delivery or settlement takes place today.
• The forward market refers to trading in assets and financial securities for delivery or settlement at a future date.
◦ You make a contract with another participant to buy or sell an asset at a future date for an agreed price.
• Trading in spot markets is at spot prices or rates or yields.
• Trading in forward markets is at forward prices or rates or yields.
Zero yield curves: arbitrage relationships
Suppose a 10-year zero-coupon bond (zero) is trading spot at 5% and a 20-year zero is trading spot at 7%. Suppose there is a forward market to trade 10-year zeros with delivery in 10 years time. What should the forward yield y be?
Solution: For there to be no arbitrage in the yield curve, we require
1 20 10 10 1.0720 10
1.07 =1.05 (1+y) so y= 1.0510 −1=0.0904, or
e0.07(20) = e0.05(10)ey10 so y = 0.07(20) − 0.05(10) = 0.09. 10
What happens if this is not true?
Zero spot yield curve
Some notation
{t0 =0,t1,…,tN−1,tN =T}isasetofN+1datesmeasuredas
years from the current date t0.
yi,j is the forward rate or yield for a zero-coupon bond being delivered in ti years and maturing in tj years from now.
We are assuming i < j, and hence ti < tj.
Note that y0,j is a spot rate or spot yield for any j = 1,...,N.
Zero yield curves: accumulation processes
The (zero-coupon) accumulation process between points ti and tj is A(i, j ) = (1 + yi,j )tj −ti and A(i, j ) = eyi,j (tj −ti )
for discrete and continuous compounding with constant yields, and
tj y(s)ds
A(i, j) = exp
for continuous compounding with nonconstant yields, assuming i < j.
Zero yield curves: arbitrage relationships
If there is no-arbitrage in the zero yield curve then
A(0, N ) = A(0, 1) . . . A(N − 1, N ) = A(n − 1, n). n=1
Under discrete compounding, this reads as
(1+y0,N )T = (1+y0,1 )t1 . . . (1+yN −1,N )T −tN −1 = (1+yn−1,n )tn −tn−1
and under continuous compounding we get
ey0,NT = ey0,1t1 ...eyN−1,N(T−tN−1) = eyn−1,n(tn−tn−1).
Zero yield curves: no arbitrage
More generally
A(i, k) = A(i, j)A(j, k)
in other words
(1+yi,k)tk−ti =(1+yi,j)tj−ti(1+yj,k)tk−tj or
eyi,k(tk−ti) = eyi,j(tj−ti)eyj,k(tk−tj)
forall 0≤i
• Of course, instead of investing $1 from ti to tk, we can partition the investment over [ti,tk] as follows. First we invest $1 from ti to tj, for some ti < tj < tk, to get $A(i,j) at time tj, and then invest $A(i, j) from tj to tk to get $A(i, j)A(j, k) at time tk.
As you may guess, if the two strategies do not produce the same value at time tk, i.e. A(i, j)A(j, k) ̸= A(i, k), then arbitrage exists. Do you know why?
The no-arbitrage yield curve conditions on L4.12/13 intuitively say the same thing: a yield curve has no-arbitrage if, based on that curve, an investment of $1 at time ti will have the same value at time tk > ti, regardless of how you partition the investment over the period [ti,tk].
This can be expressed as A(i,k) = A(i,j)A(j,k) for all i < j < k.
Zero yield curves: zero-coupon forward rates
yj,k for 0
As the yield curve changes shape, bond prices fluctuate. This is called interest rate risk.
We can quantify this risk by a measure called duration.
Write the price of a bond as a function of y the YTM:
T −1 C C + F B(y) = +
(1+y)t (1+y)T In deriving Newton’s method, we calculated that
B′(y) = − t=1
Now note that
(1 + y)B′(y) = −
T (C + F ) (1 + y)T+1
(1 + y)t+1
T (C + F ) (1+y)t (1+y)T
Let ∆(1 + y) = ∆y denote a small change (1 + y) and thus in y, and ∆B(y) = B(y + ∆y) − B(y).
%∆(1+y)= ∆(1+y) = ∆y and %∆B(y)= ∆B(y).
%∆B(y) ∆y→0 %∆(1 + y)
= 1+y lim B(y+∆y)−B(y)
Duration measures the sensitivity of B wrt changes in y by
B(y) ∆y→0 ∆y
= 1+yB′(y). B(y)
Hence, we see that the duration of a bond is
D= (1+y)B(y) =− B(y)
t=1 (1+y)t B(y)
T−1 tC + T(C+F)
It is the price elasticity of the bond with respect to its yield to maturity.
The modified duration
D=B′(y)= 1 D B(y) (1 + y)
is also often seen.
Note that we will have to use C2 , y2 , 2t, and 2T if semiannual compounding.
Duration: example
DurationofaT =3yr,F =$1,000,8%couponbondaty=8%is:
Duration: Immunisation
The duration measure D < 0. However, the absolute value |D| years
tells us how long (in years) to hold a bond so that our position is immunised against a 1-off shift in the yield curve.
An example best illustrates the idea.
Immunisation: example
Consider a 3-year F = $100, 000 bond with coupons paid annually at c = 6% and with YTM of 5%. We find that |D| = 2.7007 years. The price of the bond in 2.7007 years time is
B = 106, 000 = 104, 463.49. (1.05)0.2993
If you reinvest the coupons, in 2.7007 years time they are worth Coupon 1 : 6, 000 × 1.051.7007 = 6, 519.11
Coupon 2 : 6, 000 × 1.050.7007 = 6, 208.67.
The final value of your bond position adds these up and is 117, 191.27.
Immunisation: example (cont.)
Suppose interest rates immmediately increased by 1%, to 6%, after you bought the bond.
The price of the bond in 2.7 years time is
B = 106, 000 = 104, 167.57. (1.06)0.2993
If you reinvest the coupons, in 2.7 years time they are worth Coupon 1 : 6, 000 × 1.061.7007 = 6, 625.06
Coupon 2 : 6, 000 × 1.060.7007 = 6, 250.04.
The final value of your bond position adds these up and is 117, 042.67. This is slightly different to the original value due to the duration measure being a linear approximation.
The convexity of a bond with respect to y is: C = 1 B′′(y)
2 B(y) T−1
(1+t)tC + (1+T)T(C+F)
(1+y)t+2 B(y)
T−1 (1+t)tC + (1+T)T(C+F)
t=1 (1+y)t (1+y)T . B(y)
Bond price movements
Question: If interest rates y change by ∆y, then how much will the bond price B(y) change?
Answer: Recall ∆B(y) = B(y + ∆y) − B(y) and truncate the Taylor series expansion
∆B(y) = B′(y)∆y + 1 B′′(y)(∆y)2 + . . . 2!
to ∆B(y) = B′(x)∆y + 12B′′(y)(∆y)2 = DB(y)∆y + CB(y)(∆y)2,
or write %∆B(y) = D∆y + C(∆y)2.
Yield curves
Yield curves
Arbitrage relationships Implied forward rates Stripping yield curves
Bonds, duration and convexity
Bond pricing revisited
Interest rate risk: duration and convexity
Stochastic interest rate models
Binomial model of yield curves
Stochastic interest rates
Recall y(t) is the instantaneous continuously compounded yield at t. We previously assumed y(t) was a deterministic function.
A stochastic interest rate model is a model of the yield curve which allows y(t) to be a stochastic (random) process.
In this course we build discrete time stochastic interest rate models. In fact, we build a simple binomial model of the yield curve.
Binomial yield curve
We make the following assumptions:
• There are N time periods with dates
{t0 = 0,t1,...,tN−1,tN = T}.
• Thetimeperiodsareeach1year,sotn−tn−1 =1,n=1,...,N.
• The embedded 1-period forward rates/yields are random.
• The next period can have only 2 possible forward rates/yields.
• The 1-period forward rate is known at the start of its period.
• We are given a recombining forward rate lattice.
Objective: Calculate the yield curve from the forward rate lattice.
Binomial yield curve
Given y0,1 and the following tree or lattice of possible forward rates, we want to calculate the zero yield curve: y0,1, y0,2, y0,3, y0,4.
t0 Time period 1 t1 Time period 2 t2 Time period 3 t3 Time period 4 t4 y(uuu)
y(u) y(uud)
y0,1 y(ud)
y(d) y(ddu)
Binomial yield curve: notation
u is an up and d is a down movement in annual yields. y(uud . . . du) is the one period yield for time period n + 1.
uud . . . du is a sequence of up and down movements.
p is the probability of an up movement.
Since the forward rate lattice is recombining, y(uud . . . du) is the same
for any n-permutation of the up u and down d movements.
Binomial yield curve: two period model
The concept is best explained in a simple two-period setting.
• Let y0,1 be the period 1 yield.
• Three dates: {t0, t1, t2}.
• Two possible forward rates y(u) or y(d) with probabilities p & 1−p.
Question: What is the 2-period spot yield y0,2?.
Binomial yield curve: two period model
We have the 2-period forward rate lattice:
t0 Time period 1 t1 Time period 2 t2 y(u)
Binomial yield curve: two period model (cont.)
Either y(u) or y(d) is true on date t1 with probabilities p and 1 − p. For no arbitrage in the yield curve, we must have
(1 + y0,2)2 = (1 + y0,1)(1 + y(u)) or (1 + y0,2)2 = (1 + y0,1)(1 + y(d)). The binomial yield curve model sets:
11p1−p (1+y0,2)2 = 1+y0,1 1+y(u) + 1+y(d) .
Binomial yield curve: two period model
0,2 (1+y0,2 )2
isthepriceofazerowhichpaysF =$1att . 2
1 are the possible prices of zeros 1+y(d)
• P(u) = 1 1+y(u)
and P(d) =
starting at t1, maturing at t2 with a cashflow of $1.
• Hence we can write
P0,2 = 1 [pP(u) + (1 − p)P(d)].
• Rearranging, we calculate our two-period yield:
Binomial yield curve: two period model
We have the zero price lattice:
t0 Time period 1 t1 Time period 2 t2 P(u) 1 P(d) 1
Binomial yield curve: general N-period model
In this case:
• Start with a N-period orward rate lattice.
• Calculate y0,n for n ≤ N from the n-period sublattice.
• Work backwards in time through the lattice.
• Use the new calculated zero prices at each date.
• Construct the zero price lattice from the forward rate lattice. An example will best illustrate the idea.
Binomial yield curve: general N-period model Consider the following 5-period forward rate lattice.
Period 1 t1
t4 Period 5 t5 0.146
Binomial yield curve: general N-period model
To calculate e.g. the yield y0,4 we construct the 4-period zero price sublattice from the the 4-period forward rate/yield sublattice:
Period 1 t1
t3 Period 4 t4 0.117
Binomial yield curve: general N-period model
Some entries of the 4-period zero price sublattice are immediate, e.g.
0.895 = P (uuu) = 1 = 1 . 1 + y(uuu) 1.117
Period 1 t1
Period 2 t2 Period 3 P (uu )=?
Period 4 t4 1
Binomial yield curve: general N-period model 0.831 = P(uu) = 1 [pP(uuu) + (1 − p)P(uud)]
= 1 10.895 + 10.923 (setting p = 1). 1.094 2 2 2
Period 1 t1
Period 4 t4 1
Binomial yield curve: general N-period model Working backwards by repeating this process, we get P0,4 and then
y0,4 =0.067= 1 14 −1. P0,4
Hence, in this case we have y0,1 = 0.060 and y0,4 = 0.067.
To calculate y0,2, y0,3, and y0,5, we repeat the same process but for 2-period, 3-period, and 5-period lattices.
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com