arbitraging parallel shifts of yield...

Arbitraging Parallel Shifts of Yield Curves

Peter Carr

NYU/MS

ICBI Global Derivatives USA, Nov. 16, 2012

Kevin Atteson, Peter Carr, Jian Sun (NYU/MS)Arbitraging Parallel Shifts of Yield Curves 11/16/2012 1 / 24

Disclaimer

The views represented herein are the authors’ own views and do not necessarilyrepresent the views of Morgan Stanley or its affiliates and are not a product ofMorgan Stanley Research.


Introduction

Practitioners in the interest rate space like to think in terms of yield curves.

However, quants prefer to model forward rate curves rather than yield curves.

Can quants model the yield curve instead? A potential stumbling block isthat not all yield curve dynamics are arbitrage-free.

In particular, some textbooks argue that yield curves can’t move by parallelshifts.


Parallel Shifts of Yield Curves and Butterfly Spreads

A parallel shift of a yield curve occurs when all yields change by the sameamount.

If yield curves move only by parallel shifts, one can try to exploit such movesby forming a butterfly spread.

A butterfly spread of three zero coupon bonds involves going long thenearest and furthest bond; the zero coupon bond with intermediate maturityis shorted. One can ensure that the portfolio has zero cost, zero duration,and strictly positive convexity.

If yield curves move only by parallel shifts, is such a butterfly spread anarbitrage?


The Setup: Bond Prices

Consider an economy with three zero coupon bonds paying one dollar atmaturity dates T1, T2, and T3.

Let t = 0 be the current time and assume that 0 < T1 < T2 < T3 <∞.

For t ∈ [0,Ti ] and i = 1, 2, 3, let Pt(Ti ) > 0 denote the price at time t ofthe bond paying one dollar at its maturity date Ti .

We don’t require that the 3 bond prices be below one, but they are assumedpositive.


The Setup: Yields

Recall that for t ∈ [0,Ti ] and i = 1, 2, 3, Pt(Ti ) > 0 denotes the price attime t of the bond paying one dollar at its maturity date Ti .

These three bond prices give rise to three well-defined yield processes:

yt(Ti ) ≡ −ln Pt(Ti )

Ti − t, for t ∈ [0,Ti ) and i = 1, 2, 3.

Suppose that these three yields move only by parallel shifts until the nearestbond matures. Mathematically, this means that there exists a real-valuedstochastic process {at , t ∈ [0,T1]} independent of the 3 maturities such thatfor i = 1, 2, 3:

yt(Ti ) = y0(Ti ) + at , for t ∈ [0,T1].

For any t ∈ [0,T1], we interpret at as the amount that the three yields havemoved in common over the time period [0, t].

Do such dynamics necessarily imply arbitrage?


a is for Arbitrage?

Well, if at = 0 and the initial yield curve is flat, then it stays flat, and thereis no arbitrage.

So what if there is positive probability of positive volatility in yields. We canform a butterfly spread with no cost, no duration, and positive convexity.

Now is there arbitrage?


B is for Butterfly

Recall we are assuming that the 3 yields move only by parallel shifts i.e.

yt(Ti ) = y0(Ti ) + at , for t ∈ [0,T1],

for some non-trivial stochastic process a independent of maturity.

To try to arbitrage these dynamics, let:

λ ≡ T3 − T2

T3 − T1

be a number strictly between 0 and 1 which is uniquely defined by the threematurity dates.

Define a Butterfly Spread as a static portfolio of the three bonds with λdollars initially invested in the T1 bond, negative one dollar initially investedin the T2 bond, and 1− λ dollars initially invested in the T3 bond.


Valuing the Butterfly

For a ∈ R, let V (a, t) denote the value at time t ∈ [0,T1] of the butterflyspread, given that at = a. This value function is obtained by the sum

product of initial investment with gross return Pt(Ti )P0(Ti )

:

V (a, t) = λPt(T1)

P0(T1)− Pt(T2)

P0(T2)+ (1− λ)

Pt(T3)

P0(T3), a ∈ R, t ∈ [0,T1].

Prices are related to yields by: Pt(Ti ) = e−yt(Ti )(Ti−t), t ∈ [0,Ti ), i = 1, 2, 3.

Substituting into the top equation implies that for a ∈ R, t ∈ [0,T1]:

V (a, t) = λe−yt(T1)(T1−t)

e−y0(T1)T1− e−yt(T2)(T2−t)

e−y0(T2)T2+ (1− λ)

e−yt(T3)(T3−t)

e−y0(T3)T3.

Since the 3 yields move only by parallel shifts, i.e.yt(Ti ) = y0(Ti ) + at , for t ∈ [0,T1], substitution implies that fora ∈ R, t ∈ [0,T1], the value function for the butterfly spread is:

V (a, t) = λe−a(T1−t)+y0(T1)t − e−a(T2−t)+y0(T2)t + (1− λ)e−a(T3−t)+y0(T3)t .


Zero Initial Cost and Duration

Setting t = 0 in the value function for the butterfly spread shows how theinitial value of the butterfly depends on the shift amount a:

V (a, 0) = λe−aT1 − e−aT2 + (1− λ)e−aT3 , a ∈ R.

At time 0, the total amount that each yield has moved is zero, i.e. a0 = 0.Setting a = 0 implies that the initial cost of the butterfly spread is also zero:

V (0, 0) = λ− 1 + (1− λ) = 0.

The initial duration of the butterfly spread is defined as the negative of itsinitial rate sensitivity:

D = −∂V (a, 0)

∂a

∣∣∣∣a=0

= λT1 − T2 + (1− λ)T3.

Since λ ≡ T3−T2

T3−T1,D = T3−T2

T3−T1T1 − T2 + T2−T1

T3−T1T3 = 0. Hence, the initial

value and the initial duration of the butterfly spread are both zero.


Non-negative Value Profile

We now show that the initial value profile of the but. spread is non-negative:

V (a, 0) ≥ 0, a ∈ R.

First, recall the standard def’n of a convex real function of a real variable:Definition: Let f (x) be a function mapping R to R. Let x1 and x3 be anytwo distinct points and suppose x2 lies between them, i.e. there existsα ∈ (0, 1) such that:

x2 = αx1 + (1− α)x3.

Then the function f (x) is convex if:

f (x2) ≤ αf (x1) + (1− α)f (x3).

To apply this to our problem, suppose we take x1 = −aT1 and x3 = −aT3.If we take x2 = −aT2, then α = λ. Since the exponential functionf (x) = ex , x ∈ R is convex, the convexity definition implies that:

e−aT2 ≤ λe−aT1 + (1− λ)e−aT3 , a ∈ R.

It follows that V (a, 0) ≥ 0 for a ∈ R.


Butterfly as Arbitrage?

Recall we have shown that the initial value function V (a, 0) of the butterflyspread is nonnegative everywhere with zero height and slope at a = 0.

So if yields can move only by non-trivial parallel shifts over [0,T1), then isthis butterfly spread an arbitrage?


Time Decay

The answer is no.

The passage of time forward from t = 0 can possibly lower the initial valueprofile.

If the forward movement of calendar time at t = 0 causes the value profileto drop, and if yields do not change right at t = 0, then a mark to marketresults in a loss.

Yield curves can move only by parallel shifts and there need not be anarbitrage.


Convex Yield Curves

Now suppose the three initial yields are convex in maturity date, i.e.:

y0(T2) ≤ λy0(T1) + (1− λ)y0(T3),

where recall λ ≡ T3−T2

T3−T1.

Note that yield curves which are either flat in maturity or linear in maturityare also considered to be convex in maturity.

Suppose we assume both that the initial yield curve is convex in maturity,and that the three yields move only by non-trivial parallel shifts. Thetheorem on the next slide shows that now the butterfly spread is anarbitrage opportunity.


Theorem

Assume that for t ∈ [0,T1], the three yields yt(T1), yt(T2), yt(T3) move only bynon-trivial parallel shifts, i.e.:

yt(Ti ) = y0(Ti ) + at , for t ∈ [0,T1].

If the three initial yields y0(T1), y0(T2), y0(T3) are convex in maturity:

y0(T2) ≤ λy0(T1) + (1− λ)y0(T3), λ ≡ T3 − T2

T3 − T1,

then the butterfly spread is an arbitrage opportunity.


Nontrivial Parallel Shifts of Convex Yield Curves Imply Arb.

Proof: Let L(a, t) be the value of just the long part of the but. spread, i.e.:

L(a, t) = λe−a(T1−t)+y0(T1)t +(1−λ)e−a(T3−t)+y0(T3)t , a ∈ R, t ∈ [0,T1].

Again applying the convexity definition, suppose we takex1 = −a(T1 − t) + y0(T1)t and x3 = −a(T3 − t) + y0(T3)t. Fix α atλ ≡ T3−T2

T3−T1∈ (0, 1) and using the convexity of f (x) = ex , x ∈ R:

L(a, t) ≥ e−λa(T1−t)+λy0(T1)t−(1−λ)a(T3−t)+(1−λ)y0(T3)t

= e−a(T2−t)+λy0(T1)t+(1−λ)y0(T3)t , a ∈ R, t ∈ [0,T1].

Note that the inequality is strict for a point (a, t) with a 6= 0 and t > 0,since T3 > T1. Since the exponential function is increasing, the assumedconvexity of yields in maturity implies that:

e−a(T2−t)+λy0(T1)t+(1−λ)y0(T3)t ≥ e−a(T2−t)+y0(T2)t , a ∈ R, t ∈ [0,T1].

Finally, the above pair of inequalities implies that L(a, t) ≥ e−a(T2−t)+y0(T2)t

and hence V (a, t) ≥ 0, a ∈ R, t ∈ [0,T1]. Furthermore, the inequality isstrict for a point (a, t) with a 6= 0 and t > 0 QED.


Necessity of Parallel Shifts of Convex Yield Curves for Arb.

It can be shown that under non-trivial parallel shifts, the convexity of theinitial yields in maturity is also a necessary condition for the butterfly spreadto be an arbitrage opportunity.

In other words, if the three initial yields are strictly concave in maturity, theneven though yield curves move only by non-trivial parallel shifts, a butterflyspread will not be an arbitrage opportunity. In particular, the initial theta isnegative and hence money will be lost on the butterfly spread if yields donot move initially.


Intuition Based on Pull to Par

Under positive interest rates, zero coupon bonds are priced below par. Along position in a bond is convex in the rate shift and is pulled upward topar as time moves forward.

Since convexity is quadratic in term while duration is linear, a but. spreadcan be formed which has zero cost, zero duration, and positive convexity.

However, this butterfly spread is not necessarily an arbitrage, just as beinglong a zero delta straddle is not necessarily an arbitrage. The time decay ofa portfolio can kill the profit.

For long bond positions, the passage of time helps value. In a butterflyspread, money can only be lost if the short bond position is pulled down tonegative par too fast.

If yields are initially convex, then the time decay on the short bond is lowerthan the time gain on the two long bonds. This is because the calendar timederivative is just yield times price. As a result, the portfolio can’t lose.


Extensions

One can clearly extend this work to a single factor model when the volatilityof each yield is known. The HJM drift restriction on forward rates is just aspecial case of such a framework.

While no one believes that yields move only by parallel shifts, it is interestingto note that the dominant component in a PCA is parallel shifts and thathistorically, the average shape of the yield curve has been concave.

This work has been extended to swap rates. If 3 swap rates are initiallyconvex in duration and if these 3 swap rates can move only by non-trivialparallel shifts, then a butterfly spread of the 3 swaps is an arbitrage.

One can also extend this kind of thinking to other contracts, e.g. CDS andoptions. In all of these cases, one has a monotonic function of price e.g.yield, CDS spread, or volatility that is constant only in a toy model. If onewishes to realistically randomize, the convexity of derivative security prices inthe construct prevents an arbitrary evolution of the construct.


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