© 2007 the mathworks, inc. ® ® pricing derivatives securities using matlab mayeda reyes-kattar...
TRANSCRIPT
© 2
007
The
Mat
hWor
ks, I
nc.
® ®
Pricing Derivatives Securities using MATLAB
Mayeda Reyes-Kattar
March 2007
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
® ®
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
® ®
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
® ®
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
® ®
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
® ®
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
® ®
Pricing Derivatives Securities using MATLAB
Closed form formula : Black-ScholesPricing 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
® ®
Pricing Derivatives Securities using MATLAB
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
11
® ®
Pricing Derivatives Securities using MATLAB
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
13
® ®
Pricing Derivatives Securities using MATLAB
Binomial and Black-Scholes ConvergencePricing Example: Vanilla Option
14
® ®
Pricing Derivatives Securities using MATLAB
Monte Carlo SimulationPricing Example: Vanilla Option
Price $50
Strike $60
Rate 4%
Time (Months) 24
Volatility 30%
Dividend Yield 0%
# of simulations 15,000
500,000
# of steps 50
60
15
® ®
Pricing Derivatives Securities using MATLAB
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
16
® ®
Pricing Derivatives Securities using MATLAB
Type of Equity Tree Models
CRR: Cox-Ross-Rubinstein
EQP: Equal Probability
ITT: Implied Trinomial Tree
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
® ®
Pricing Derivatives Securities using MATLAB
Creating an ITT
ITTTree = itttree (StockSpec, RateSpec, TimeSpec, StockOptSpec)
StockSpec Stock’s original price, its volatility, and its dividend information
RateSpec Interest rate environment
TimeSpec Tree time layout specification
StockOptSpec
Parameters of European stock options (eg Strike, Maturity)
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
® ®
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
® ®
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
® ®
Pricing Derivatives Securities using MATLAB
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%
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
® ®
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
® ®
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
© 2
007
The
Mat
hWor
ks, I
nc.
® ®
Questions?