1. Primbs, MS&E 345 1 Applications of the Linear Functional Form: Pricing Exotics

2. Primbs, MS&E 345 2 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

3. Primbs, MS&E 345 3 The Black-Scholes formula: where under Q Basically, we just have to calculate the expectation by brute force! This time we use risk neutral pricing for a European call option:

4. Primbs, MS&E 345 4 The Black-Scholes formula: under Q So: But so by symmetry where where N( ) is the distribution function for a standard Gaussian. Now, note that under Q we have:

5. Primbs, MS&E 345 5 The Black-Scholes formula: Let’s also rewrite this in terms of z T. We actually just did that calculation: where corresponds to Brownian motion. This looks like a Gaussian with mean  T.

6. Primbs, MS&E 345 6 The Black-Scholes formula: This looks like a Gaussian with mean  T. where z T is Gaussian with mean  T and variance T. is standard Gaussian where by symmetry

7. Primbs, MS&E 345 7 The Black-Scholes formula: where: distribution function for a standard Normal (i.e. N(0,1))

8. Primbs, MS&E 345 8 under Qor Question:Where does the Black-Scholes partial differential equation come from in this framework???? But we know that where under Q So by Ito’s lemma: Hence: The Black-Scholes pde! We know that all portfolios must earn the risk free rate under the risk neutral probabilities Q.

9. Primbs, MS&E 345 9 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

10. Primbs, MS&E 345 10 Continuous dividends If the stock pays a continuous dividend at rate q then: So That is, the price of the stock has mean return (r-q). So, under Q, the stock price follows: If we purchase 1 share at time 0. Price = S 0 We have e qT shares at time T: Payoff = e qT S T Under reinvestment in the stock

11. Primbs, MS&E 345 11 Continuous dividends Hence, we could compute the price of a European call option as: where under Q where: distribution function for a standard Normal (i.e. N(0,1))

12. Primbs, MS&E 345 12 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

13. Primbs, MS&E 345 13 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 What do we do with early cash flows: Reinvest in the bond! We can use our pricing formula at time T: at time t at time T Reinvest in the bond becomes We just discount from time t!

14. Primbs, MS&E 345 14 Early cash flows are just discounted from the time they occur. This makes sense, because there was nothing particularly special about the time we called T. Furthermore, if we hold an asset, we could decide to sell it at any time of our choosing, and not wait until a specified expiration or maturity date. Our pricing formula must hold no matter what time we decide to sell, otherwise we could arbitrage by selling at smart times...

15. Primbs, MS&E 345 15 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

16. Primbs, MS&E 345 16 European Digital (Binary) Option: where under Q K Payoff payoff = STST Risk Neutral Pricing: But: where (We did this calculation for the Black-Scholes Formula!) So: Cash or nothing call

17. Primbs, MS&E 345 17 European Digital (Binary) Option: where under Q K payoff = STST Risk Neutral Pricing: So: Asset or nothing call K But: where (We did this calculation for the Black-Scholes Formula!)

18. Primbs, MS&E 345 18 It should be clear that a standard European call option is a portfolio of 1 asset-or-nothing call and –K cash-or-nothing calls. One difficulty with digital options is that they have discontinuous payoffs. Theoretically, this is not a problem. However, in practice it can make hedging these options very difficult and risky. Notes:

19. Primbs, MS&E 345 19 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

20. Primbs, MS&E 345 20 Asian Options: Note that under Q, S follows: or hence whereis Gaussian Recall that Asian options involve averages. Normally, we do not have closed form solutions for Asian options. The exception is when the option is on a geometric average: so I T is actually log-normally distributed! Geometric Average:

21. Primbs, MS&E 345 21 If we can calculate the mean and variance of then we know everything about I T. Mean:

22. Primbs, MS&E 345 22 If we can calculate the mean and variance of then we know everything about I T. Variance : T Tt s t > s by symmetry

23. Primbs, MS&E 345 23 If we can calculate the mean and variance of then we know everything about I T. Variance : T Tt s t > s by symmetry 0 by independent increment.

24. Primbs, MS&E 345 24 If we can calculate the mean and variance of then we know everything about I T. So, the geometric average I T is where Therefore, to price an average price call whose payoff is: we just need to compute: where You may check that this is the same as if I t follows the geometric Brownian Motion: in a risk neutral world. Which just looks like Black-Scholes on an asset paying a continuous dividend of and with volatility of

25. Primbs, MS&E 345 25 Arithmetic Asian Options: The majority of Asian options involve arithmetic means: In this case, I T, is not log-normally distributed, and hence we cannot fit it into the Black-Scholes formula framework. However, it is common to compute the first two moments of I T and assume that its distribution is log-normal with the same first two moments. In this case, the Black-Scholes formula provides a quick and closed form approximation to the true price. This is sometimes referred to as the method of moments in pricing.

26. Primbs, MS&E 345 26 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

27. Primbs, MS&E 345 27 Barrier Options x Knock-outs: The option is worthless if it hits the barrier. Knock-ins: The option is worthless until it hits the barrier. Barrier options are like normal European options, except that they are either activated, or become worthless when the underlying asset hits a pre-specified barrier. The basic types are: Also, the barrier can be hit on the way down (down-and-out, down-and-in) or it can be hit on the way up (up-and-out, up-and-in).

28. Primbs, MS&E 345 28 Lookback Options max time 0 time T Lookback options depend on the maximum or minimum price achieved during the life of the option. A European lookback call option pays off: A European lookback put option pays off: To price these, we need to be able to compute the statistics of the maximum and minimum...

29. Primbs, MS&E 345 29 American Digitals K time 0 time T American digital options payoff $1 the moment the strike price is hit. To evaluate this, we need to know the statistics of the hitting time... $1 where  represent the first hitting time of the strike price K.

30. Primbs, MS&E 345 30 The statistics that I need to know for Barriers, Lookbacks, and American digitals are all related: (Densities can then be obtained by differentiation) The: max = Probability that the max is greater than B first hitting time . = Probability that the time to hit B is less that T B time 0 time T Probability that a barrier B is hit before time T

31. Primbs, MS&E 345 31 The key trick here is the “reflection principle”. For every path that crosses the barrier but ends up below the barrier, there is a reflected path that ends up above the barrier. x C The Reflection Principle B time 0 time T Basic Idea: The only twist that we have to deal with is that the probability of the original path and reflected path may be different. However, if we start with Brownian motion, we don’t have this problem. 2B-C

32. Primbs, MS&E 345 32 The reflection principle for Brownian motion: B x C time 0 time T B time 0 time T Assume that is a Brownian motion  Denote Then for C < B, let’s calculate: 2B-C 2B-dC dC x

33. Primbs, MS&E 345 33 The reflection principle for Brownian motion with drift: Assume that Denote Then for C < B, let’s calculate: Use Girsanov to change the mean of z  to zero B x C time 0 time T 2B-C

34. Primbs, MS&E 345 34 In a similar manner: Assume: (i.e. S follows a geometric Brownian motion) etc... where Then you can calculate: These provide the building blocks for closed form solutions...

35. Primbs, MS&E 345 35 The calculations become quite messy, so I won’t go through them The main points are: There are closed form solutions for many barriers and lookbacks. American digitals almost have a closed form solution (you need to evaluate an integral by quadrature). These are a consequence of risk neutral pricing combined with the reflection principle. For some options, risk neutral pricing provides the most direct route to closed form solutions. Consult a good text (such as Hull) for the messy formulas...