Applications of the Return Form of Arbitrage Pricing: Equity Derivatives

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Applications of the Return Form of Arbitrage
PricingEquity Derivatives
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Deriving Equations for Derivative Assets
Three step algorithm

• Derive factor models for returns of tradable
  assets.
• (often involves Itos lemma.)
• (2) Apply absence of arbitrage condition.
• (m1 sl)
• (3) Apply appropriate boundary conditions and
  solve.

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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Black-Scholes (Again)
Step 1 Derive factor models for returns of
tradable assets.
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Black-Scholes (Again)
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Black-Scholes (Again)
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Black-Scholes (Again)
Third Equation
The Black-Scholes Equation
This is for an option on a non-dividend paying
asset which follows a geometric Brownian motion.
Step 3 Apply appropriate boundary condition and
solve!
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Step 3
These formulas are basic...know them!!!
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Step 3
European Calls and Puts
We derived the equation for geometric Brownian
motion.
But, the equation doesnt depend on the mean
return.
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Step 3
European Calls and Puts
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Step 3
American Calls and Puts
We have the option to exercise early.
It is not difficult to show that early exercise
of an American call on a non-dividend paying
stock is never optimal, hence it has the same
value as a European call. Early exercise for a
put, however, can be optimal.
In general, we need to use numerical techniques
to solve for American options. The boundary
condition given above is nasty!
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Terminology
European and American call and put options are
often referred to as plain vanilla options.
Other derivatives are then called exotics.
There are (too) many Binary or digital
options Barrier options Compound
options Chooser options...
Just because they are called exotic doesnt
mean they are difficult. Often they are just a
different boundary condition for the
Black-Scholes equation.
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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A stock paying a continuous dividend (Merton).
Step 1 Derive factor models for returns of
tradable assets.
Underlying Stock
Lets think about the proper dynamics for this
tradable asset...
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A stock paying a continuous dividend (Merton).
Dividend
Over time dt, the dividend is qSdt.
You purchase a portfolio of 1 share value is vt
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A stock paying a continuous dividend (Merton).
Stock price
Dividend
Over time dt, the dividend is qSdt.
StdtqStdt
vtdt
Dividend
Price
St
vt
dt
You purchase a portfolio of 1 share value is vt
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An important principle
You cannot just buy the price of the stock!!!!
You must buy a portfolio which consists of a
single share!
Since this is what we will purchase, it must
satisfy our absence of arbitrage conditions.
The price of the stock does not have to satisfy
the conditions because it cannot be purchased!!!!
This is a very important point.
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A stock paying a continuous dividend (Merton).
Note, value of the derivative depends on price of
the stock S, not v!!!!
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A stock paying a continuous dividend (Merton).
Now we have models of returns for our tradable
assets.
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(No Transcript)
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Third Equation
This is for a derivative on a stock paying a
continuous dividend.
Step 3 Apply appropriate boundary condition and
solve!
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Step 3
European Calls and Puts
These formulas are basic...know them!!!
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Step 3
Which assets pay continuous dividends?
How about A stock index Foreign
currencies Commodities with a convenience yield
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Step 3
Do dividends make an option more or less valuable?
Dividends bleed away the price...
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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A stock paying a cash dividend.
Step 1 Derive factor models for returns of
tradable assets.
Underlying Stock
Pays a cash dividend of Dt at time t.
On the other hand, the price drops when the stock
goes ex-dividend.
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A stock paying a cash dividend.
Risk Free Asset
Underlying Asset Price
Value dynamics
Note, value of the derivative depends on price of
stock S, not v!!!!!!
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A stock paying a cash dividend.
Risk Free Asset
Value dynamics
Derivative
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A stock paying a cash dividend.
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A stock paying a cash dividend.
Third Equation
This is for an option on a stock paying a lump
(cash) dividend.
Step 3 Apply appropriate boundary condition and
solve!
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Step 3
For European options, only the value of S at time
T matters
D
x
S
x
t
T
So, you can just act like the stock started at a
lower price and there were no dividends!
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Step 3
For European options, only the value of S at time
T matters
x
S
x
De-rT
t
T
So, you can just act like the stock started at a
lower price and there were no dividends!
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Options on Futures (Black,1976)
Step 1 Derive factor models for returns of
tradable assets.
Step 2 Apply AOA but for futures!
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Step 2 Apply
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Third Equation
The Black-Scholes Equation for an option on a
futures contract.
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Step 3
It looks like the equation for an option on an
asset that pays a continuous dividend, except
here the continuous dividend rate is the risk
free rate!
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Step 3
In some ways Blacks model is more fundamental
then the other models we have seen. (recall our
analysis of market price of risk.)
If you can price futures contracts in terms of
spot prices, then you can use Blacks formula to
derive all our previous formulas.
As an exercise, you should try this for assets
paying a continuous or cash dividends.
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Assets with Poisson Jumps (Cox and Ross, 1976)
Be careful. Should set the mean of dp to 0.
This doesnt matter in the current setting, but
it can matter in other settings.
Lets do it anyway...
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Assets with Poisson Jumps (Cox and Ross, 1976)
Bond
Asset price
Derivative
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Step 2 Apply
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Third Equation
This is a partial differential/difference
equation.
Step 3 Apply appropriate boundary condition and
solve!
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Step 3
Closed form solution for a European Call Option
Looks a lot like the Black-Scholes formula but
with Poisson instead of Gaussian.
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Mertons Jump Diffusion Model (1976)
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Mertons Jump Diffusion Model (1976)
Merton makes a big assumption
Put another way, the jump risk has zero beta.
We know that this also means l2 and l3 are zero!
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Mertons Jump Diffusion Model (1976)
Step 2
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Third Equation
Step 3 Apply appropriate boundary condition and
solve!
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When Y is lognormal, there is a closed form
solution for a call option
Messy...but at least we can calculate it.
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An easy special case
Question Does bankruptcy make an option worth
more or less?
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Option Prices versus the Risk Free Rate
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Stochastic Volatility (Hull and White)
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Stochastic Volatility (Hull and White)
I wont go into solutions of this equation...
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Option to exchange one asset for another
(Margrabe, 1978)
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Option to exchange one asset for another
(Margrabe, 1978)
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Option to exchange one asset for another
(Margrabe, 1978)
Step 3
Looks a lot like Black Scholes.
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Option to exchange one asset for another
(Margrabe, 1978)
Step 3
A touch of intuition
We are measuring S2 in units of S1.
This in known as a change of numeraire. We will
see this trick later...
By the way, can a standard call option be seen as
exchanging one asset for another?
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Black-Scholes
Poisson (Cox and Ross)
Dividends
Jump diffusion model (Merton)
Options on futures (Black)
Multiple factors
Exchange one asset for another (Margrabe)
Stochastic Volatility (Hull and White)
Path Dependent
Option on an average (Asian options)
Option on Max, Min
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Path Dependence
Some derivatives depend on the path of the
underlying asset.
We need to write an Ito equation for this. The
dependence on the entire path is a problem!
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Path Dependence
The general approach is to try to capture the
path dependence with another variable.
Lets see how this would work for an Asian option
Now we can apply Itos lemma and continue in
typical fashion...
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Path Dependence
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Path Dependence
Step 2 Apply
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Path Dependence
We can pull the same sort of trick for other path
dependence
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Path Dependence
So, and option on a max solves the Black-Scholes
equation, but the boundary conditions are
different.
I wont expand on this now since we will see
another approach to this later in the course...
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Summary of Return Form Approach
Three step algorithm

• Derive factor models for returns of tradable
  assets.
• (often involves Itos lemma.)
• (2) Apply absence of arbitrage condition.
• (m1 sl)
• (3) Apply appropriate boundary conditions and
  solve.
• (how to solve is your problem.)

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References
Merton, R. C., Option pricing when underlying
stock returns are discontinuous, Journal of
Financial Economics, 3 (1976) 125-144. Cox, J.
C. and S. A. Ross, The valuation of options for
alternative stochastic processes, Journal of
Financial Economics, 3 (1976) 145-166. Black,
F., The pricing of commodity contracts, Journal
of Financial Economics, 3 (1976)
167-179. Merton, R. C., Theory of rational
option pricing, Bell Journal of Economics and
Management Science, 4 (1973) 141-183. Black, F.
and M. Scholes, The pricing of options and
corporate liabilities, Journal of Political
Economy, 81 (1973) 637-659. Margrabe, W., The
value of an option to exchange one asset for
another, Journal of Finance, 33(1) (1978)
177-186. Hull, J. C., Options, Futures, and
Other Derivatives, 3rd Edition, Prentice Hall,
1997. Wilmott, P. Paul Wilmott on Quantitative
Finance, Volume 1 and 2, Wiley, 2000.