1. © 2007 The MathWorks, Inc. ® ® Pricing Derivatives Securities using MATLAB Mayeda Reyes-Kattar March 2007

2. 2 ® ® Pricing Derivatives Securities using MATLAB Outline  What is a Derivative Instrument?  Type of Derivatives  Why use Derivatives securities? How are they used?  How to price Derivatives  Type of Equity Tree models  Implied Trinomial Tree  What is hedging?  Examples of hedging using Equity Derivatives  Interest Rate Derivatives  What are customers doing? Why are they doing it?  Why are our tools a good fit?

3. 3 ® ® Pricing Derivatives Securities using MATLAB What is a Derivative Instrument?  A security which derives its value from the value of an underlying asset.  Common underlying assets: - stocks - bonds - currencies - interest rates  Example: An European put (derivative) on a given stock (underlying) is described in terms of its Strike and its Maturity. Purchasing the put gives you the (non-binding) right to sell the stock only at the Maturity date, at a price equal to the Strike price.

4. 4 ® ® Pricing Derivatives Securities using MATLAB Types of Derivatives  Interest Rate Derivatives  Options: calls/put  Caps / Floors  Swaps  Futures / Forwards  Equity Derivatives  Vanilla options: calls/puts  Exotic options:  Asian  Barrier  Compound  Lookback

5. 5 ® ® Pricing Derivatives Securities using MATLAB Why use Derivative Securities?  Manage and hedge risk :  interest rate risk  price risk  currency risk  How are Derivative Securities used?  Expose you to more or less risk  Generally used as a risk management tool:  hedge risk  But can also be used for speculative purposes

6. 6 ® ® Pricing Derivatives Securities using MATLAB Main Methods of Pricing Derivatives  Closed form formula (not available for all securities)  Trees (binomial and trinomial)  Monte Carlo simulation

7. 7 ® ® Pricing Derivatives Securities using MATLAB Pricing Example: Vanilla Option  Call or Put Option: Right to buy or sell an underlying at a specified price (strike).  Types: American, European and Bermuda

8. 8 ® ® Pricing Derivatives Securities using MATLAB Closed form formula : Black-Scholes Pricing Example: Vanilla Option Price Current price of the underlying asset $50 Strike Strike (i.e., exercise) price of the option. $60 Rate Annualized continuously compounded risk-free rate of return over the life of the option 4% Time Time to expiration of the option, expressed in years. 24 Months Volatility Annualized asset price volatility 30% [Call, Put] = blsprice(50, 60, 0.04, 24/12, 0.30) Call = 6.4109 Put = 11.7979

9. 9 ® ® Pricing Derivatives Securities using MATLAB Binomial Tree : Cox-Ross-Rubinstein Model Pricing Example: Vanilla Option Valuation Date1/1/2006 End Date1/1/2008 Risk free rate (annual)4.00% The underlying’s price$50 The underlying’s volatility (sigma) 30% Number of time steps4 Setting up the Stock Tree

10. 10 ® ® Pricing Derivatives Securities using MATLAB Binomial Tree : Cox-Ross-Rubinstein Model Pricing Example: Vanilla Option Setting up the stock tree 1. Define the Interest Rate Term Structure Rate = 0.04; ValuationDate = '01-01-2006'; EndDate = '01-01-2008'; RateSpec = intenvset('StartDates', ValuationDate, 'EndDates', EndDate,... 'ValuationDate', ValuationDate, 'Rates', Rate, 'Compounding', -1) 2. Define the stock’s characteristics Sigma = 0.30; AssetPrice = 50; StockSpec = stockspec(Sigma, AssetPrice) 3. Define the tree structure ValuationDate = '01-01-2006'; EndDate = '01-01-2008'; NumPeriods = 4; TimeSpec = crrtimespec(ValuationDate, EndDate, NumPeriods) 4. Create and view the CRR Tree CRRTree = crrtree(StockSpec, RateSpec, TimeSpec); treeviewer(CRRTree);

11. 11 ® ® Pricing Derivatives Securities using MATLAB Binomial Tree : Cox-Ross-Rubinstein Model Pricing Example: Vanilla Option Pricing Options on the Tree Valuation Date1/1/2006 End Date1/1/2008 InstrumentsEuropean Call European Put Strike$60

12. 12 ® ® Pricing Derivatives Securities using MATLAB Binomial Tree : Cox-Ross-Rubinstein Model Pricing Example: Vanilla Option Pricing options on the tree 1. Define the instrument portfolio (things to price). Add a call and put with Strike 60, Expry 1/1/08. InstSet = instadd('OptStock',{'call';'put'},60,'1/1/06', '1/1/08'); 2. Price the options and view tree [Price, PTree] = crrprice(CRRTree,InstSet) treeviewer(PTree)

13. 13 ® ® Pricing Derivatives Securities using MATLAB Binomial and Black-Scholes Convergence Pricing Example: Vanilla Option

14. 14 ® ® Pricing Derivatives Securities using MATLAB Monte Carlo Simulation Pricing Example: Vanilla Option Price$50 Strike$60 Rate4% Time (Months)24 Volatility30% Dividend Yield0% # of simulations15,000 500,000 # of steps50 60

15. 15 ® ® Pricing Derivatives Securities using MATLAB Monte Carlo Simulation Pricing Example: Vanilla Option

16. 16 ® ® Pricing Derivatives Securities using MATLAB Type of Equity Tree Models  CRR: Cox-Ross-Rubinstein  EQP: Equal Probability  ITT: Implied Trinomial Tree

17. 17 ® ® Pricing Derivatives Securities using MATLAB Idea behind the ITT model  Recognize market price of vanilla options play a key role in market expectations.  Build a tree consistent with the market prices of the vanilla European options and therefore consistent with the implied volatility smile.

18. 18 ® ® Pricing Derivatives Securities using MATLAB Creating an ITT ITTTree = itttree (StockSpec, RateSpec, TimeSpec, StockOptSpec) StockSpecStock’s original price, its volatility, and its dividend information RateSpecInterest rate environment TimeSpecTree time layout specification StockOptSpecParameters of European stock options (eg Strike, Maturity)

19. 19 ® ® Pricing Derivatives Securities using MATLAB Example  Assume that the interest rate is fixed at 4% annually between the valuation date of the tree until its maturity.  Build an implied trinomial tree.  Price a portfolio of equity derivatives using the ITT model.

20. 20 ® ® Pricing Derivatives Securities using MATLAB What is Hedging?  The idea behind hedging is to minimize exposure to market movements. As the underlying changes, the proportions of the instruments forming the portfolio may need to be adjusted to keep the sensitivities within the desired range.  Traders and portfolio managers must evaluate the cost of achieving their target sensitivities, which involves a tradeoff between the portfolio insurance and the cost of insurance coverage.

21. 21 ® ® Pricing Derivatives Securities using MATLAB Examples of hedging analysis  Asset allocation: use futures to re-allocate portfolio.  Portfolio insurance: use put options or up-and-out put options to generate minimum amount of cash in the future.  Debt obligation: Use interest rate swaps to convert a variable rate obligation to a fixed rate obligation.

22. 22 ® ® Pricing Derivatives Securities using MATLAB Hedging using Barriers Example: Portfolio Insurance Scenario #1: Long asset Premium vanilla put = $0.53 Premium knock-out put barrier = $0.26 Barrier reduces the cost of the hedge by 50% Scenario #2: Short asset Premium vanilla call = $17.88 Premium knock-In call barrier (110) = $16.74  6% Premium Knock-Out call barrier (120) = $6.62  62%

23. 23 ® ® Pricing Derivatives Securities using MATLAB Interest Rate Derivatives  Create a portfolio of instruments  Price the portfolio using a Zero Curve  Price the portfolio using Trees  Show some hedging strategies to minimize exposure to market movements

24. 24 ® ® Pricing Derivatives Securities using MATLAB Customers are using our financial platform for …  Modeling the underlying assets  Computing ‘fair’ price and Greeks (sensitivities) of derivatives  Understanding how sensitive a portfolio is to changes in the underlying assets  Performing sensitivity analyses to manage risk

25. 25 ® ® Pricing Derivatives Securities using MATLAB Why are our tools a good fit?  Powerful math and graphics engine  Pre-built financial functionality for Fixed-Income and Derivatives  Flexible and inexpensive deployment options

26. © 2007 The MathWorks, Inc. ® ® Questions?