More Applications of Linear Pricing

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More Applications of Linear Pricing
2
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
3
Exchange one asset for another
4
Exchange one asset for another
where
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Exchange one asset for another
where
Substituting in terms of S1 and S2 gives the
final answer
6
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
7
Futures contracts and the risk neutral measure
t (start)
time 0
time T
tdt
time T value
mark to market (at time tdt)
time/position
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Futures contracts and the risk neutral measure
t (start)
time 0
time T
tdt
t
Total Cost 0
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Plug into our risk neutral pricing formula
The futures price is the expected price of the
stock at time T in a risk neutral world.
10
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
11
Forward contracts and the bond forward risk
neutral world
payoff occurs
T
t
0
Forward prices are a tradable (St) divided by
B(tT)
Hence, under B(tT) as the numeraire, forward
prices are martingales.
The forward price is the expected price of the
stock at time T in a bond (B(tT)) forward risk
neutral world.
12
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
13
Spot and Forward Rates
Rates are quoted as yearly rates. Hence, the
actual rate applied over time t1 is t1R(0t1).
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Forward interest rates and the bond forward risk
neutral world
payoff occurs
Tt
T
t
0
The forward rate is tradables divided by B(tTt).
Hence, forward rates are martingales under the
numeraire B(tTt).
15
Forward interest rates and the bond forward risk
neutral world
payoff occurs
Tt
T
t
0
16
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
17
Swaps and Swap rates
An interest rate swap is an agreement to exchange
a fixed rate of interest S for a floating rate of
interest on the same notional principal P.
The swap rate S is the fixed rate that makes a
swap have zero value.
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Forward swap rates and the annuity forward risk
neutral world
Tt
T2t
Tnt
T
t
0
To be a bit more precise, let S(tT) be a forward
swap rate where the swap begins at time T.
where the Floating side is tradable (for instance
as coupons on a floating rate bond).
Therefore, forward swap rates are martingales
under the numeraire A(tT).
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Forward swap rates and the annuity forward risk
neutral world
Tt
T2t
Tnt
T
t
0
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Summary
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Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
22
The set-up
T
t
0
The option allows me to purchase the asset at
time T for the strike price K
If the forward price at time t is greater than K,
then this option is worth FtT-K at time T.
Otherwise, the option is worthless because the
forward price is less than the strike.
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The set-up
(FtT-K)
T
t
0
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The set-up
(FtT-K)
T
t
0
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The set-up
(FtT-K)
T
t
0
26
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
27
A simple generalization of the Black-Scholes
formula
We can use Blacks model to generalize
Black-Scholes by thinking of Black-Scholes as an
option on a forward contract where the delivery
of the forward, and expiration of the option are
at the same time.
Hence, we can use Blacks formula with tT and
STFTT.
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A simple generalization of the Black-Scholes
formula
But, I want the solution in terms of S0
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A simple generalization of the Black-Scholes
formula
where
distribution function for a standard Normal (i.e.
N(0,1))
It looks like standard Black-Scholes, just use
the interest rate corresponding to the expiration
date!
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Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
31
A Bond Option
(Bc(T)-K)
T
0
Exercise decision
Payoff occurs

Then Blacks model says
32
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
33
Picture of a cap
Interest Rate
RK
A cap is an upper limit on the interest payments
corresponding to the life of a loan.
A caplet is an upper limit on a single interest
payment in a loan.
Hence a cap corresponds to a caplet for every
interest payment.
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An interest rate caplet
Tt
T
0
Interest rate set R(TTt)
Use B(tTt) as the numeraire
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An interest rate caplet (a second approach)
Tt
T
0
Interest rate set R(TTt)
A put option on a bond. We can use Blacks model
to value it (In practice, the first approach is
used more often.)
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Blacks Model is used to price a number of
interest rate derivatives
We showed how to price a caplet, which puts an
upper limit on interest rates at a specific time
in the future.
When we place an upper limit on all the interest
payments for a loan, this is called a cap.
Hence, a cap is just a portfolio of caplets. We
price it by linearity. Price each caplet and add
them together.
A floor is a lower limit on interest rates over
the life of a loan. A floorlet is for a single
interest payment.
A collar is a cap and a floor.
37
Exchange one asset for another
Futures, forwards, forward rates, and swap rates
Blacks model with stochastic interest rates
A generalization of Black-Scholes
Interest rate derivatives
Bond options
Caplets, etc.
Swaptions
38
A swap option (swaption)
Tt
T2t
Tnt
T
t
0
We have an option to enter into a swap at time T,
where we pay a fixed swap rate SK.
Assume the swap rate at time T is S(TT). If
this is greater than SK, then we exercise the
option. This is worth Pt(S(TT)-SK) at each swap
date.
Since the swap rate is known at time T, we can
discount the payoff back to time T.
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A swap option (swaption)
Tt
T2t
Tnt
T
t
0
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A swap option (swaption) (a second approach)
Tt
T2t
Tnt
T
t
0
(fixed bond-P)
Since a swap can be thought of as exchanging a
bond with fixed interest for a bond with floating
interest, a swaption can be thought of as an
option on a bond.
The strike price is the value of the floating
rate bond, which is always its principal P at
every reset date.
Blacks model for a bond option can be used to
price this. However, this approach is not used
often in practice.
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Appendix Alternate proofs of martingale
property for forwards, forward rates, and forward
swap rates.
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Forwards
Forward Rates
Swap Rates
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Forward contracts and the bond forward risk
neutral world
payoff occurs
T
t
0
Consider the bond forward risk neutral world with
B(tT) as the numeraire
The forward price is the expected price of the
stock at time T in a bond (B(tT)) forward risk
neutral world.
44
Forwards
Forward Rates
Swap Rates
45
Spot and Forward Rates
Rates are quoted as yearly rates. Hence, the
actual rate applied over time t1 is t1R(0t1).
46
Forward interest rates and the bond forward risk
neutral world
payoff occurs
Tt
T
t
0
Use B(tTt) as the numeraire
47
Forward interest rates and the bond forward risk
neutral world
payoff occurs
Tt
T
t
0
48
Forwards
Forward Rates
Swap Rates
49
Swaps and Swap rates
An interest rate swap is an agreement to exchange
a fixed rate of interest S for a floating rate of
interest on the same notional principal P.
The swap rate S is the fixed rate that makes a
swap have zero value.
50
Forward swap rates and the annuity forward risk
neutral world
Tt
T2t
Tnt
T
t
0
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Forward swap rates and the annuity forward risk
neutral world
Tt
T2t
Tnt
T
t
0
total payoff
at time T
Use the annuity A(t) as the numeraire