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Pricing Derivatives Securities using MATLAB

Pricing Derivatives Securities using MATLAB . Mayeda Reyes-Kattar March 2007. 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?

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Pricing Derivatives Securities using MATLAB

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  1. Pricing Derivatives Securities using MATLAB Mayeda Reyes-Kattar March 2007

  2. 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. 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. 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. 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. Main Methods of Pricing Derivatives • Closed form formula (not available for all securities) • Trees (binomial and trinomial) • Monte Carlo simulation

  7. 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. Closed form formula : Black-ScholesPricing Example: Vanilla Option [Call, Put] = blsprice(50, 60, 0.04, 24/12, 0.30) Call = 6.4109 Put = 11.7979

  9. Binomial Tree : Cox-Ross-Rubinstein ModelPricing Example: Vanilla Option Setting up the Stock Tree

  10. Binomial Tree : Cox-Ross-Rubinstein ModelPricing Example: Vanilla Option Pricing Options on the Tree

  11. Binomial and Black-Scholes ConvergencePricing Example: Vanilla Option

  12. Monte Carlo SimulationPricing Example: Vanilla Option

  13. Monte Carlo SimulationPricing Example: Vanilla Option

  14. Type of Equity Tree Models • CRR: Cox-Ross-Rubinstein • EQP: Equal Probability • ITT: Implied Trinomial Tree

  15. 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.

  16. Creating an ITT ITTTree = itttree (StockSpec, RateSpec, TimeSpec, StockOptSpec)

  17. 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.

  18. 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.

  19. 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.

  20. 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% Hedging using BarriersExample: Portfolio Insurance

  21. 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

  22. 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

  23. 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

  24. Questions?

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