pricing derivatives securities using matlab

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Pricing Derivatives Securities using MATLAB

Mayeda Reyes-KattarMarch 2007

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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?

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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.

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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

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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

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Main Methods of Pricing Derivatives

Closed form formula (not available for all securities)

Trees (binomial and trinomial)

Monte Carlo simulation

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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

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Closed form formula : Black-ScholesPricing Example: Vanilla Option

Price Current price of the underlying asset $50Strike Strike (i.e., exercise) price of the option. $60Rate Annualized continuously compounded risk-free rate

of return over the life of the option4%

Time Time to expiration of the option, expressed in years. 24 MonthsVolatility

Annualized asset price volatility 30%

[Call, Put] = blsprice(50, 60, 0.04, 24/12, 0.30)Call = 6.4109Put = 11.7979

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Binomial Tree : Cox-Ross-Rubinstein ModelPricing Example: Vanilla Option

Valuation Date 1/1/2006

End Date 1/1/2008

Risk free rate (annual) 4.00%

The underlying’s price $50

The underlying’s volatility (sigma)

30%

Number of time steps 4

Setting up the Stock Tree

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Binomial Tree : Cox-Ross-Rubinstein ModelPricing Example: Vanilla Option

Pricing Options on the Tree

Valuation Date 1/1/2006

End Date 1/1/2008

Instruments European Call European Put

Strike $60

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Binomial and Black-Scholes ConvergencePricing Example: Vanilla Option

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Monte Carlo SimulationPricing Example: Vanilla Option

Price $50Strike $60Rate 4%Time (Months) 24Volatility 30%Dividend Yield 0%# of simulations 15,000

500,000# of steps 50

60

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Monte Carlo SimulationPricing Example: Vanilla Option

100000 1000000

6.35

6.4

6.45

6.5

6.55

6.6

6.65

Simulations

Comparison of European Call PricingMonte Carlo Method with Black Scholes Formula

6.41076.4006

6.4109

60Steps50 Steps

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Type of Equity Tree Models

CRR: Cox-Ross-Rubinstein

EQP: Equal Probability

ITT: Implied Trinomial Tree

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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.

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Creating an ITT

ITTTree = itttree (StockSpec, RateSpec, TimeSpec, StockOptSpec)

StockSpec Stock’s original price, its volatility, and its dividend information

RateSpec Interest rate environmentTimeSpec Tree time layout specificationStockOptSpec

Parameters of European stock options (eg Strike, Maturity)

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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.

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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.

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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.

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Hedging using BarriersExample: 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%

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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

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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

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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

pricing derivatives securities using matlab

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