MIT OpenCourseWare
http://ocw.mit.edu
15.997 Practice of Finance: Advanced Corporate Risk Management
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Valuation
MIT Sloan School of Management
15. 997 Advanced Corporate Risk Management
John E. Parsons
Review
„
„
„
„
„
„
The risk-neutral valuation framework starts by going one step
deeper than the traditional risk-adjusted discount rate.
The risk-adjusted discount rate (1) assumes a linear exposure to the
underlying factor risk, and (2) applies a single discount factor to the
expected cash flow.
This is equivalent to applying different discount factors to the “up”
and “down” states. These are the forward state prices.
The risk-neutral valuation framework recovers these forward state
prices.
With these in hand, we are no longer restricted to valuing only linear
exposures. We can value any non-linear pattern of exposure.
The risk-adjusted discount rate method cannot do this because it
stopped before recovering the two different discount factors, and
only applied the “average” discount factor to the “average” or
expected cash flow.
2
1
Turbo-Charged Valuation
„
„
„
Once we have recovered the risk-neutral probabilities (i.e. the
forward state prices) we can value any non-linear pattern of
exposure.
Whole body of “option pricing” or “derivative pricing” theory is just
working out a compendium of formulas for different problems.
Behind them is the general principle.
Illustrate this with a case study…
3
Valuing Option-Like Investments
„
„
„
„
Use risk-neutral method…
Model the stochastic process for the underlying risk variable.
Specify the risk-neutral process, i.e., calculate the risk neutral probabilities.
Value any derivative claim using the risk-neutral method, i.e., using
the risk-neutral probabilities to determine the certainty equivalent,
and discounting at the risk-free rate.
4
2
An Aluminum Supply Contract with a Structured
Price Clause
„
„
„
„
aluminum producer and major consuming firm
20-year term
Fixed annual quantity
Price clause
¾
¾
¾
¾
¾
¾
¾
¾
¾
Floor price equal to $1,644/Ton
Between $1,644 and $1,808/Ton, the world market price
From $1,808 to $2,308/Ton, price rises by the world market
price,
From $2,308 to $3,008/Ton, price rises by 25% of the world
market price,
From $3,008/Ton, price rises by 50% of the world market
Price
All trigger values grown at the price of inflation.
5
Contract Payoff Diagram
4000
3500
contract price
3000
2500
2000
1500
1000
500
0
0
1000
2000
3000
4000
world market price
6
3
Option Analogy
„
Payoff diagram can be converted into a batch of options on the
aluminum price
¾
¾
„
„
Find the options!
The 20 year contract means a series of options, one set for each year
Options can be priced using a model for the price evolution and a
model for pricing risk
The value of the contract is the value of the bundle of options
reshaping the payoff, summed across the full series of maturities.
7
Floor Price as a Put Option
„
„
Receive the world market price,
Plus a put option with the right to surrender the world market price in
exchange for the floor price, i.e., a put option on aluminum with a
strike price equal to the contract floor price.
8
4
Effect of a Put on the Payoff
4000
3500
contract price
3000
combined
2500
2000
1500
world market price
1000
500
put
0
0
1000
2000
world market price
3000
4000
9
A put option reshapes the payoff at the low end…
4000
3500
contract price
3000
2500
2000
1500
put value gained
1000
500
0
0
1000
2000
3000
4000
world market price
10
5
How to Capture the Remainder?
4000
3500
3000
contract price
Value surrendered
2500
2000
1500
put value gained
1000
500
0
0
1000
2000
3000
4000
world market price
11
Full Contract Equals…
„
„
„
„
„
Fixed price of $1,644/Ton
Plus a Call with the exercise price of $1,644/Ton
Minus a Call with the exercise price of $1,808/Ton
Plus 25% of a Call with the exercise price of $2,308/Ton
Plus 25% of a Call with the exercise price of $3,008/Ton.
12
6
Valuing the Options
„
„
„
„
Suppose the world market price of aluminum follows a random
walk…i.e., GBM
Choose a convenience yield, …i.e., a dividend rate and expected
rate of growth of aluminum prices
Estimate the volatility
Apply Merton’s variation on Black-Scholes, the value of an option on
dividend paying stock…
13
Spot Price Model Matters
Valuation with the Estimated Process:
ATM Calls of Varying Maturity
$350
Option value ($/Ton)
$300
$250
$200
$150
$100
O-U
RW
$50
$0
10/31/95
10/31/99
10/31/2003
10/31/2007
10/31/2011
10/31/2015
Date
Figure by MIT OpenCourseWare.
14
7
Spot Price Model Matters
Valuation with the Estimated Process:
ATM Calls of Varying Maturity
$350
Option value ($/Ton)
$300
$250
$200
$150
$100
O-U
RW
$50
$0
10/31/95
10/31/99
10/31/2003
10/31/2007
10/31/2011
10/31/2015
Date
Figure by MIT OpenCourseWare.
15
Spot Price Model Matters
Valuation with the Estimated Process:
ATM Puts of Varying Maturity
Option value ($/Ton)
$400
$300
$200
O-U
$100
RW
$0
10/31/95
10/31/99
10/31/2003
10/31/2007
10/31/2011
10/31/2015
Date
Figure by MIT OpenCourseWare.
16
8
Spot Price Model Matters
Valuing the Contract for Alternative Processes
Cumulative contract present
value ($Millions)
$300
$245
Million
RW
O-U
BV
$250
$200
$206
Million
$150
$140
Million
$100
$50
$0
10/31/95
10/31/99
10/31/2003
10/31/2007
10/31/2011
10/31/2015
Date
Figure by MIT OpenCourseWare.
17
Risk & Valuation: When Does It Matter
„
„
„
Underlying state variable has asymmetric risk – extreme downside.
Underlying state variable risk does not grow linearly with time –
don’t compound discount rate.
Project payoff is non-linear – e.g., a call option on the underlying
state variable.
18
9
Main Myth of Risk-Neutral Valuation
„
„
Do you need traded securities?
No.
¾
¾
„
RN valuation is just about consistent valuation of all non-linear payoffs.
Any equilibrium model like the CAPM implies a risk-neutral valuation
framework.
Traded securities make the starting point of the valuation more
reliable. But this has nothing to do with the RN framework.
19
Second Myth of Risk-Neutral Valuation
„
„
Real-option pricing is a competitor to the DCF framework.
RN is just an extension of the DCF framework.
20
10