FEC Financial Engineering Club

1. FECFinancial Engineering Club

2. Pricing European OPtions

3. Agenda • Stochastic Processes • Stochastic Calculus • Black-Scholes Equation

4. Stochastic Processes

5. 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

6. A Simple Process • Note that ’s are iid with and • Then and

7. 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 incrementsif, for the increments are independent

8. 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

9. 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

10. Brownian Motion • By the central limit theorem as , where is a Brownian motion Properties of B.M. • has independent increments • for (we have been using B.M. with = 1)

11. Brownian Motion Ex) What is assuming (suppose W has parameter ) Ex) What is ? , independent is a martingale

12. 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

13. Brownian Motion and Quadratic Variation • Note if has a continuous derivative, = (by MVT) Then = =

14. Brownian Motion and Quadratic Variation • For a B.M. , consider the random variable + + =

15. Brownian Motion and Quadratic Variation • Let . Choose large so that . Then and thus • Then since by LLN. • Conclusion • Similarly, and

16. Stochastic Calculus

17. Ito Integral Let and note that Thus

18. Ito Integral Quadratic Variation

19. Ito’s Lemma • We seek an approximation By Taylor’s formula we have higher higher-order terms Note that Then, using the expansion above:

20. Ito’s Lemma • Now taking limits, since and • In differential form, Ito’s formula is with the last two terms cancelling out to zero

21. Ito’s Lemma • Ex) Suppose . What is ? Then

22. Ito’s Lemma • Ex) Suppose . What is ? Then

23. 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 ?

24. Ito’s Lemma • Here, • Note that this is actually just a function of a single variable x Then

25. 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.

26. Black-Scholes Equation

27. 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

28. Black Scholes • We need to take the present value of this so we consider the function: Again, by Ito’s formula

29. 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

30. 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 :

31. 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 :

32. 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

33. 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

34. 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,

35. 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.

36. Black-Scholes • Solving the Black-Scholes-Merton PDE gives us the familiar results: is the standard-normal CDF of x

37. 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

38. Thank you! • Facebook: http://www.facebook.com/UIUCFEC • LinkedIn: http://www.linkedin.com/financialengineeringclub • Email: uiuc.fec@gmail.com President Greg Pastorek gfpastorek@gmail.com • Internal Vice President Matthew Reardon mreardon5@gmail.com