Extending the Linear Functional Form

1
Extending the Linear Functional Form
2
Change of numeraire
Martingales and equivalent martingale measures
Random interest rates
Risk neutral worlds, rates of return, and market
price of risk
Where is the pde hiding now?
Time to think...
3
Return to our linear pricing functional
Assume interest rates are constant and start with
1 in the bond
Our old pricing formula!
4
Normalize by the stock
We have derived a different pricing formula under
a different equivalent measure!
5
We saw this trick earlier in our solution
of Margrabes exchange one asset for another.
When we divide by the bond, we can think of this
as measuring everything in units of the bond.
Alternatively, we can measure in units of the
stock. This is what happens when we divide by the
stock.
This is known as a change of numeraire.
A numeraire asset is the asset that we measure
everything else in terms of.
Be careful, it must be positive. Dividing by
zero doesnt make sense!
6
The general approach is as follows
Step 1 Select a numeraire asset (say S)
7
Change of numeraire
Martingales and equivalent martingale measures
Random interest rates
Risk neutral worlds, rates of return, and market
price of risk
Where is the pde hiding now?
Time to think...
8
What do we do with early cash flows Reinvest in
the numeraire!
Assume Bt is the numeraire
t 0
t 1
t 2
t 3
t 4
t 5
t 6
9
(No Transcript)
10
A path dependent derivative
Option is written
For a lookback option, for instance, if I buy at
time t, then I need to know then maximum price
that was achieved in 0,t in order to be able to
price it.
In general, I need to know all the available
information up to time t.
11
Martingales
A stochastic process Xt is a martingale if
Notes
Martingales have the property that your
expectation of them in the future is whatever
they are today.
They model a fair gamble your expected money
after playing is what you start with.
In particular, martingales have zero mean return
12
Equivalent Martingale Measures
In general, all assets normalized by the
numeraire are martingales under the forward risk
neutral probabilities with respect to the
numeraire.
Hence, forward risk neutral probabilities are
also known as equivalent martingale measures.
13
Change of numeraire
Martingales and equivalent martingale measures
Random interest rates
Risk neutral worlds, rates of return, and market
price of risk
Where is the pde hiding now?
Time to think...
14
A more general pricing formula
Assume the interest rates are random.
As my numeraire, I take a security that starts
with 1 and is just continuously rolled over at
the (random) short rate. (this is often referred
to as the money market account)
15
A more general pricing formula
This is the generalization of our original risk
neutral pricing formula.
If interest rates were constant, or
deterministic, I could pull them out of the
expectation. But here they are random, so I
cant.
When people refer to risk neutral pricing, they
usually mean this formula.
16
Change of numeraire
Martingales and equivalent martingale measures
Random interest rates
Risk neutral worlds, rates of return, and market
price of risk
Where is the pde hiding now?
Time to think...
17
What is the expected return on securities in this
risk free world?
18
has zero mean return under Q.
Hence
All securities earn the (random) short rate!!!
19
Rates of returns in other risk neutral worlds
20
Rates of returns in risk neutral worlds (Case 1
correlation)
21
Rates of returns in risk neutral worlds (Case 2
uncorrelated)
The numeraire follows
Take another security
with
22
Rates of returns in risk neutral worlds (Case 2
uncorrelated)
The market prices of risk are the volatilities of
the numeraire asset.
23
Change of numeraire
Martingales and equivalent martingale measures
Random interest rates
Risk neutral worlds, rates of return, and market
price of risk
Where is the pde hiding now?
Time to think...
24
Lets use the money market account as the
numeraire
25
In general, the pde can be derived as follows
Choose a numeraire. Then the market price of
risk in the forward risk neutral world is given
by the volatilities of the numeraire.
In the forward risk neutral world all assets
normalized by the numeraire are martingales.
Hence, the normalized value must have mean drift
0.
Use Itos lemma to compute the mean drift in the
forward risk neutral world, and set it equal to
zero. This is your Black-Scholes pde.
Using the money market account as the numeraire
is a convenient choice, since then you know all
assets earn the short rate. So, you can use
Itos lemma to compute the mean return of a
derivative and set it equal to the short rate.
This is what I did in the previous slide, and it,
of course, will give the same answer.
26
Change of numeraire
Martingales and equivalent martingale measures
Random interest rates
Risk neutral worlds, rates of return, and market
price of risk
Where is the pde hiding now?
Time to think...
27
Here is a silly question...
If you know the mean return of an asset, m, and
its payoff function, what is its price?
Let f be a geometric Brownian motion
This is a pricing formula...
Why dont we use this as our pricing formula????
28
I can think of at least two good reasons...

• Mean returns are hard to determine.
• This doesnt say anything about arbitrage.

Actually, we essentially do use this formula to
value equity and many other securities. In that
case, we use theories such as CAPM to try to help
us determine m.
But maybe there is still hope for this
approach...
29
Lets make this approach respect absence of
arbitrage.
So, all mean returns must satisfy this
relationship for some market price of risk, l.
Great! So then I have...
Now it respects absence of arbitrage!
But I still have to determine l, which is
equivalent to finding m.
30
Okay, so I still have to estimate m, or
equivalently, l.
Lets say I am really bad at estimating m, and I
think that l 0!
Now I use this formula to price!
Even though I was wrong about the mean return, I
still price all assets with no arbitrage, and I
price the bond and underlying stock correctly!
This is a linear pricing functional that fits the
data!
31
Assume that instead I had estimated mean returns
such that l1. Then I would have
Again, this wouldnt allow arbitrage, and it
would fit the data!
In fact, I can choose l to be whatever I want.
As long as I set mean returns equal to m r
ls, then I wont have arbitrage and I will fit
the data.
Any choice of l will provide a valid absence of
arbitrage pricing formula!
32
When I choose different numeraires, I am
selecting different market prices of risk!
Sometimes, this makes the computation of the
expectation easier! Hence, we get nice formulas
by choosing the correct numeraire for our
calculations.
If you like, you can think that you are just
estimating the mean returns to be something
different. Your estimate may not be correct, but
that doesnt matter, because the prices that you
determine will not allow anyone to arbitrage off
of your mistake!
In complete markets, it doesnt matter how bad
your estimate is, everyone will arrive at the
same AOA price. In an incomplete market,
different estimates will lead to different
prices. You can argue over whose is correct, but
you cant arbitrage off it!
33
Appendix
34
Lets look at this reasoning in a couple of cases
Black-Scholes
35
Lets look at defaultable bonds
If you recall we were getting formulas that
looked like those for a normal bond, but the
interest rate was increased by the intensity of
default. Here is one way to argue for that
formula...
payoff with default, discount at r.
payoff without default, discount at rl.
If I throw away the possibility of default, I
would have to discount at a higher rate in order
to obtain the same price.