[PPT] - Using this on the Z -process we get: 2 ( g ( T i ) g ( T )) 2 T t h PowerPoint Presentation

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Options on coupon-bearing bonds Non-trivial extension of ZCB options (from Bj¨

Uses change of numeraire. Idea developed independently by Jamshidian (JoF, ’89) and Geman (unpublished, same time). Important because

bonds with exotic embedded options.

Arbitrage-free economy where ZCB prices are driven 1-D BM, i.e. dP(t, T) = r(t)P(t, T)dt + σP (t, T)P(t, T)dW Q(t). Look at a coupon bond with payments α1, . . . , αN occurring at dates T1, . . . , TN. The price of this coupon bond is πB(t) =

αiP(t, Ti).

Last ingredient: A strike-K, expiry-T European call-option on the coupon bond. Let πC(t) denote its price, (and as usual β be the bank-account). Then πC(t) = β(t)EQ

β(T)

=

αiβ(t)EQ

β(T)

β(T)

αiP(t, Ti)EQTi

P(T, Ti)

αiP(t, Ti)QTi

(1)

Problem: Even with deterministic ZCB-price volatility, πB isn’t log- normal ( lognormals = lognormal), so it seems we won’t get a Black-Scholes type-expression (Qt·’s are Φ(a suitable point).) “It ain’t necessarily so.” Assume ZCB-volatility is of the form σP(t, T) = (g(T) − g(t))h(t). for some deterministic functions g (that is increasing & differentiable) and h (that is positive).

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Note

σf(t, T) = − ∂ ∂T σP(t, T) = −g′(T)h(t), i.e. it’s deterministic & multiplicatively separable.

is) the Hull/White (ex- tended Vasicek) model.

Look at the term QT

process Z(·, T, Ti) Z(t, T, Ti) = P(t, Ti) P(t, T) By direct application of Theorem 19.8 from Bj¨

dZ(t, T, Ti) = (g(Ti) − g(T))h(t)Z(t, T, Ti)dW T (t), where W T is a BM under the T-forward measure.

Remember that if a process Y solves the stochastic differential equa- tion dY (t) = µ(t; ω)Y (t)dt + σ(t; ω)Y (t)dW(t), then for s ≤ t we have Y (t) = Y (s) exp

2σ2(u; ω))du +

(Proof: Ito on “what’s inside the exp-function”.) Remember that is σ is deterministic then

σ(u)dW(u)∼N(0,

σ2(u)du), and independent of Ft.

Using this on the Z-process we get: P(T, Ti) = P(t, Ti) P(t, T) e−1

The first term in the “exp” is deterministic. The stochastic part of the second term, is the same for all Ti and putting H(t, T) =

where X is a N(0, 1) under QT and independent of Ft.

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πB(T) = πB(T; x)x=X =

”a pos. fct”(t, T, Ti) × e(g(Ti)−g(T))√

. Note that g is increasing and the sum is over i’s such that Ti > T so g(Ti) − g(T) > 0 and the x → πB(T; x) is a monotonely increasing (with R+ as domain). So it has an inverse function, formally π−B, and this function increasing, too. We then have QT

).

All in all we have found that the last term in (1) is KP(t, T)Φ(d(t, T)). How do we find d(t, T)? We have −d(t, T) = π−B(K) ⇔ πB(. . . , −d(t, T)) = πB(. . . , π−B(K)) = K, so we must find the solution to πB(. . . , −d∗) = K. But πB is a function we know explicitly, that’s easy to do numerically (bisecting or “goal seek”’ing in Excel).

Repeated use of the relation dW τ(t) = dW Q(t) − σP (t, τ)dt. allows us to perform the same analysis on the QTi

is DIY) and arrive at: Result The time-t price of a call-option on a coupon bearing bond is given by πC(t) =

αiP(t, Ti)Φ(d∗(t, Ti)) − KP(t, T)Φ(d∗(t, T)), where for any τ ∈ {T, T1, . . . , Tn}, d∗(t, τ) is defined implicitly as the solution to the equation

αi P(t, Ti) P(t, T) e

Remarks The technique will work for other 1-factor models than the Gaussian

CIR), except you don’t get Φ’s but a more complicated distribution function. The technique will not work (without possibly crude approximations) in a multidimensional setting. Why? Well, first ax + by = K defines a line, not a single point. Could try rewriting

a

b

= K.

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But then z would be different for different Ti’s. Then what: You can do various “rank 1”-approximations. Claus Munk has done nice work. PhD-course participants : How’s that for continuity! “Topics ...”-course participants : Final project: Implement the for-

dan prices (from the embedded options in “realer”).

Simple forward rates; LIBOR A simple forward rate L(t; S, T) specifies the cash-flow for a loan agreement where

Note that this rate is quoted on a discretely compounded basis. If L(0; 1, 1.25) = 0.04 then you have to pay back 1.01; if the 0.04 were taken as continuously compounded you’d have to pay back exp(0.25∗ 0.04) = 1.010050. The usual simple no-arbitrage argument (DIY) shows that 1 + (T − S)L(t; S, T) = P(t, S) P(t, T) ⇒ L(t; S, T) = 1 T − S

P(t, T) − 1

Such simple rates are called LIBOR.

With T = S + δ we may write Lδ(t; S), Lδ(t; t) is called (δ-) spot LIBOR. Immediate (Technical) Observation Lδ(t; T) is a QT+δ martingale.

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Floating rate bonds & Swaps Look at a tenor-structure; a set of dates where something interesting happens

A floating rate bullet bond has the cash-flows δLδ(Ti−1; Ti−1)

at Ti for i ≤ N−1 , and 1+Lδ(TN−1; TN−1) at date TN. The cash-flows are stochastic so finding the arbitrage-free price seems to require a dynamic model.

It doesn’t. We have ci = 1 P(Ti−1, Ti) − 1 for i ≤ N − 1 and the time-t value of the “-1” is of course −P(t, Ti). Now consider the following trading strategy:

units & a net-cash-flow of 0.

At a cost of P(t, Ti−1), this perfectly replicates the 1/P(Ti−1, Ti)-cash- flow from the floating rate bullet. Hence the arbitrage-free price of cash-flow ci is P(t, Ti−1) − P(t, Ti). The arbitrage-free price of the floating rate bullet is FlBull(t) =

(P(t, Ti−1) − P(t, Ti)) + P(TN−1) = P(t; T0), as the sum telescopes. In particular, if t = T0 (“vi st˚ ar p˚ a en ter- minsdato”) then the floating rate bullet has value 1.The floating rate bond has par value.

Note that this result is easily extended to any type of floating rate bond (eg. serial or annuity) with deterministic instalment plan. If H(Ti) denotes remaining principal and A(Ti) is the principal repaid at time Ti then H(Ti−1) =

A(Tj). The Ti-cash-flow from the bond is ci = A(Ti)+δLδ(Ti−1; Ti−1)H(Ti−1) = A(Ti)+δLδ(Ti−1; Ti−1)

A(Tj). A portfolio with A(Ti) units of the Ti-bullet has exactly the same cash-flows, and its price (assuming t = T0) is

the new bond has par value too.

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A plain vanilla interest rate swap is contract that consists of

want as notional principal)

You can think of this as contract that swaps floating rate interest payments for fixed rate payments (or vice versa). The value of the swap contract is Vswap(t) = P(t, T0) − P(t; TN) −

δκP(t; Ti)

In practice this equation is used backwards (at the time of initiation

κ∗(t) = P(t, T0) − P(t; TN)

. This is called the (par) swap rate. Note that it is specific to the swap considered; you get different swap rates if you move T0, δ or N around. The message is then:

dependence)

A couple of disclaimers/warnings: It is very important for the “volatility independence” that you swap the exact right rate at the exact right time. Swapping the 6M LIBOR every 3rd month induces volatility dependence. So does moving pay- ments to where they are first known. So-called convexity adjustment try to remedy that. Swaps can be made a lot more exotic with all kinds of embedded

Famous disaster: Proctor and Gamble vs. (literally, later) Bankers

the fact that P&G took a huge gamble on rates staying low.)

Option structures & LIBOR market models A caplet contract pays off δ(Lδ(Ti−1, Ti−1) − ω)+ at time Ti Owning a caplet can be thought of as having an insurance against paying high interest. The time-t arbitrage-free price of a caplet is πcaplet(t) = δβ(t)EQ

β(Ti)

δP(t, Ti)ETi

Recall that Lδ(; Ti−1) is a QTi martingale.

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So one way to to specify an arbitrage-free model is as dLδ(t; Ti−1) = γ⊤(t; Ti−1)Lδ(t; Ti−1)dW Ti(t) (2) for some deterministic (possible vector-valued) function γ. This is called the (lognormal) LIBOR market model. Put v2(t, T) =

||γ(u; T)||2du. Then a standard B/S-like calculation (DIY) shows that πcaplet(t; Ti−1, δ, κ) = δP(t, Ti)

where d± = (ln(Lδ(t; Ti−1)/κ) ± 1

If γ is 1D & constant (in 1st argument) then v2 = γ2(Ti−1)Ti−1 and the formula is the so-called Black’s formula. Other assumption: γ(t, T) = γ(T − t) where γ is piecewise constant. Not clear what a reasonable volatility specification is. A cap contract is a series of caplets; its price is simply the sum of caplet prices. For many years, market practice was to price – or at least quote – caps with the Black formula. Here is a formal, arbitrage-free models that supports this.

Papers with the model by [Miltersen, Sandmann, Sondermann], [Brace, Gatarek, Musiela] and [Jamshidian] appeared virtually simultaneously in 1997. Immediate hit. Understandably so. Justifies what was being done & takes as input real observables. Quoting prices in terms of Black-volatility does not actually mean that you belive in the lognormal model. Cap prices are quoted as “flat volatility”, ie. the same constant γ that when plugged into caplets & summed gives the price.

The models are actually more complicated than they look:

σP(t, T) = −

δLδ(t, T − δk) 1 + δLδ(t, T − δk)γ(t, T − δk). Not a Markovian structure. So simulation requires a lot of book- keeping.

And if 3M LIBOR has lognormal volatility structure, then 6M LIBOR hasn’t.

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time-indices & integrations. There is an extensive literature on market models. Nice recent articles by Pelsser, Driessen, deJong.

Another option-type contract is the swaption

❄

❄

❄ ❄

❄

❄

The time-Tl value of the swaption (i.e. the swaption price at its expiry date) is πswopt(Tl; Tl, Tm, Tn, δ, ω) = δ(κ(Tl; Tm, Tn, δ) − ω)+

P(Tl, Tj).

So for t < Tl, the swaption price can be written as πswopt(t; ...) = P(t; Tl)EQTl

P(Tl, Tj)

If Tl = Tm then that we can rewrite the swaption pay-off as

αjP(Tm, Tj)

, with αj = δω for j ≤ n − 1 & αn = 1 + δω. So the swaption is really a put option on a coupon-bearing bond. The ideas from earlier in the day was used by BGM to derive an approximate swaption-price formula in a lognormal LIBOR market model.

Put X(t) = δ n

numeraire, so it induces an equivalent martingale measure QX. Then πswopt(t; Tl, Tm, Tn, δ, ω) = X(t)EQX

. and the process {κ(t; Tm, Tn, δ)}t is a QX-martingale. This, known as the swap-measure approach, can lead to Black-type formulas for swaptions. A few calculations show that lognormal volatility of swaprates is not consistent with lognormal LIBOR volatility.

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