38
FEC FINANCIAL ENGINEERING CLUB
FEC FINANCIAL ENGINEERING CLUB
PRICING EUROPEAN OPTIONS
AGENDA Stochastic Processes Stochastic Calculus Black-Scholes Equation
STOCHASTIC PROCESSES
A SIMPLE PROCESS Let with probability and with probability (for all t) and consider the
symmetric random walk, Assume that ’s are i.i.d.
Both and are random processes
A random/stochastic process is (vaguely) just a collection of random variables They could be i.i.d. They may be correlated—they may even have different distributions There is no general theory/application for random processes until more
context and structure is applied
A SIMPLE PROCESS
Note that ’s are iid with and
Then and
A SIMPLE PROCESS Generally, we care about the increments of a process:
So that , and
The symmetric random walk is defined to have independent increments A process X is said to have independent increments if, for the increments are
independent
QUADRATIC VARIATION Define the quadratic variation of a sequence up to time as
This is a path-dependent measure of variation (thus it is random) For some unique processes, it may not be random
For our symmetric random walk, note that a one step increment, , is either or . Thus
SCALED SYMMETRIC RANDOM WALK
Let be a scaled symmetric random walk If is not an integer, is interpolated between the two neighboring
integers of
Like a the symmetric r.w., the scaled symmetric r.w. has independent increments
BROWNIAN MOTION By the central limit theorem as , where is a Brownian motion
Properties of B.M.
1) has independent increments2) for (we have been using B.M. with = 1)
BROWNIAN MOTION
Ex) What is assuming (suppose W has parameter )
Ex) What is ?
, independent
is a martingale
BROWNIAN MOTION
Note that B.M. is a function and not a sequence of random variables and so our definition of quadratic variation must be altered:
Let be a partition of the interval : with Let . For a function , the quadratic variation of up to time T is
BROWNIAN MOTION AND QUADRATIC VARIATION Note if has a continuous derivative, = (by MVT)
Then = =
BROWNIAN MOTION AND QUADRATIC VARIATION
For a B.M. , consider the random variable
+ + =
BROWNIAN MOTION AND QUADRATIC VARIATION Let . Choose large so that . Then and thus
Then since by LLN.
Conclusion
Similarly, and
STOCHASTIC CALCULUS
ITO INTEGRAL
Let and note that
Thus
ITO INTEGRAL
Quadratic Variation
ITO’S LEMMA We seek an approximation
By Taylor’s formula we have higher higher-order terms
Note that
Then, using the expansion above:
ITO’S LEMMA
Now taking limits, since and
In differential form, Ito’s formula is with the last two terms cancelling out to zero
ITO’S LEMMA Ex) Suppose . What is ?
Then
ITO’S LEMMA Ex) Suppose . What is ?
Then
ITO’S LEMMA More generally, if is a stochastic process
We have been using Ito’s formula to construct stochastic differential equations (SDE’s)—that is, differential equations with a random term.
Consider the SDE: If , what is ?
ITO’S LEMMA Here,
Note that this is actually just a function of a single variable x
Then
ITO’S LEMMA Note that =
Then
This is a model for an asset that has return and volatility and whose randomness is driven by a single risk factor(Brownian motion)—it can be applied to roughly any asset.
BLACK-SCHOLES EQUATION
BLACK-SCHOLES Let the underlying follow this SDE with constant rate and volatility: The only variable inputs to an options price are the time until maturity and the
price of the stock, so we start by considering the function
Ito’s formula tells us
BLACK SCHOLES We need to take the present value of this so we consider the function:
Again, by Ito’s formula
BLACK SCHOLES Meanwhile, we try to replicate the option contract as we did in the binomial option
pricing model. That is, by investing some money in a stock position and some in some money market account (a bond):
Let be the value of our portfolio at time At time we invest a necessary amount into the stock and the remainder, , into the
money market instrument. Then we gain from our investment in the stock And from our investment in the money market instrument Thus
By Ito’s lemma, the differential of the PV(stock) is Likewise, the differential of our discounted portfolio is
BLACK SCHOLES At each time , we want the replicating portfolio to match the value of the option We do this by ensuring that for all and that :
BLACK SCHOLES At each time , we want the replicating portfolio to match the value of the option We do this by ensuring that for all and that :
BLACK-SCHOLES At each time , we want the replicating portfolio to match the value of the option We do this by ensuring that for all and that :
Need
BLACK-SCHOLES At each time , we want the replicating portfolio to match the value of the option We do this by ensuring that for all and that :
Need Need
BLACK-SCHOLES At each time , we want the replicating portfolio to match the value of the option We do this by ensuring that for all and that :
Need Need
Simplifying this we need,
BLACK-SCHOLES
With
Is the Black-Scholes-Merton partial differential equation. Its is a backward parabolic equation, which are known to have solutions. Using the fact that, we solve this ODE: . This gives us our first boundary condition at :
Additionally,
That is, the fact that as the underlying approaches , the call option begins to look like the underlying minus the discounted strike. This serves as the second boundary condition.
BLACK-SCHOLES Solving the Black-Scholes-Merton PDE gives us the familiar results:
is the standard-normal CDF of x
BLACK-SCHOLES Why doesn’t this method work for American options?
Early exercise is not modeled!
Pros Gives an analytical (no algorithms necessary!) solution to the value of a
European option This is simple enough to be extended The resulting PDE’s can be solved numerically
Cons Some unrealistic assumptions about rates and volatilities does not
match data Normal distribution has thin tales under-approximates large returns in
stocks
THANK YOU!
Facebook: http://www.facebook.com/UIUCFEC LinkedIn: http://www.linkedin.com/financialengineeringclub Email: [email protected]
Internal Vice PresidentMatthew [email protected]
PresidentGreg Pastorekgfpastorek@gmail.
com
