FEC Financial Engineering Club

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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

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 ?

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

ITO’S LEMMA Note that =

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 :

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

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: uiuc.fec@gmail.com

Internal Vice PresidentMatthew Reardonmreardon5@gmail.com

PresidentGreg Pastorekgfpastorek@gmail.

FEC Financial Engineering Club

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