FINANCIAL INSTRUMENT MODELING

IT FOR FINANCIAL SERVICES (IS356)

The content of these slides is heavily based on a Coursera course taught by Profs. Haugh and Iyengar from the Center for Financial Engineering at the Columbia Business School, NYC. I attended the course in Spring 2013 and again in Fall 2013 and Spring 2014 when the course was offered in 2 parts.

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Options… The Basics

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Payoff and Intrinsic Value of a Call

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Payoff and Intrinsic Value of a Put

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Put-Call Parity

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European Options(Using Simple Binomial Modeling)

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Profit Timing and Determination

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Stock Price Dynamics – binomial lattice

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Stock price goes up/down by the same amount each time period. In this example: 1.07 and 1/1.07

Options Pricing – call option formula

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The value of the option at expiration is Max(ST - K,0). You will only exercise a European option if it is in-the-money at expiration, in which case the price of the stock (ST) at expiration is greater than the strike price K. We will move backwards in the lattice to compute the value of the option at time 0.

European Call Option Pricing Example

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15.48 = 1/R( 22.5q + 7(1-q))R=1.01Q=(R-d)/(u-d)d=1/1.07u=1.07

A European put option uses the same formula. The only difference is in the last column: max(0, K-ST). You only exercise a put option if the strike price > current price. You can buy shares at the current price and sell them at the higher strike K.

European Options: Excel Modeling

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Does Put Call Parity Hold?

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American Options(Using Simple Binomial Modeling)

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Pricing American Options

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Reverse through the Lattice

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American Put vs. Call – early or not?

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Black-Scholes Model

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Geometric Brownian MotionModels random fluctuations in stock prices

Black-Scholes Model… continued

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Black-Scholes Model in Excel

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Implied Volatility

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Futures and Forwards

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Forwards Contracts

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Futures and Forwards…

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Problems with Forwards

Futures Contracts

Mechanics of a Futures Contract

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Excel Example with Daily Settlement

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Hedging using Futures

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A Perfect Hedge Isn’t Always Possible…

Term Structure of Interest Rates

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Yield Curves (US Treasuries)

29Source: http://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/TextView.aspx?data=yieldYear&year=2013

Rates are climbing – highest in Dec 2013

Sample Short Rate Lattice

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9.375% = 7.5% x 1.25

Pricing a Zero-coupon Bond (ZCB)

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9.375% comes from the last slide

Assumes a 50:50 chance of rates increasing/decreasing

Excel Modeling

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Again, we work backwards through the lattice.

89.51 = 1/1.1172 * ( 100 x 0.5 + 100 x 0.5)

Pricing European Call Option on ZCB

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Max(0, 83.08-84)Max(0, 87.35-84)

Max(0, 90.64-84)

Pricing American Put Option on ZCB

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Pricing Forwards on Bonds

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Pricing Forwards on Bonds - excel

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Start at the end and work back to t=4

Then work from t=4 backwards

Mortgage Backed Securities (MBS)Collateralized Debt Obligations

(CDO)

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Mortgage Backed Securities Markets

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The Logic of Tranches (MBS)

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The Logic of Tranches (CDO)

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A Simple Example: A 1-period CDO

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Excel model of CDO

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Credit # Default Prob1 0.22 0.23 0.064 0.35 0.46 0.657 0.38 0.239 0.0210 0.1211 0.13412 0.2113 0.0814 0.115 0.116 0.0217 0.318 0.01519 0.220 0.03

1-probability of default = probability of survival

Expected number of losses in the CDO = sum of all probabilities of individual defaults 3.668

Probability of losses P(0) 0.010989 0 0.000P(1) 0.064562 1 0.065P(2-20) 0.924448 2 1.849

Tranche (0-2) 1.913

calculations are not shown for Tranche (2-4) 1.283these other tranches in this file Tranche (4-20) 0.472

3.668

CDON

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Portfolio Optimization

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Return on Assets and Portfolios

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Two-asset Example

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Optimization Example (solver)

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Mean returnREITs US Large Growth US Small Growth2.40 4.10 5.20

Covariance matrixREITs US Large Growth US Small Growth

REITs 0.0010 -0.0006 0.0001US Large Growth -0.0006 0.0599 0.0635US Small Growth 0.0001 0.0635 0.1025

REITs US Large Growth US Small GrowthVolatility 3.17 24.46 32.01

Porfolio x1 x2 x3 x00.05 0.00 0.00 0.95 1.00 = 1.00

Interest rate (%) 1.5

risk aversion (tau) 1

Net rate of return (%) 1.55

Volatility (%) 0.16

Risk-adjusted return 1.52

= maximum risk adjusted return, no shorts permitted, x0 permitted= maximum risk adjusted return, no shorts permitted, x0 prohibited= maximum risk adjusted return, no shorts permitted, x0 permitted, no more than 50% of portfolio in any one bucket

Optimization with trading costs

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Trading cost parametersalpha 1 0.0035 Average trade volume / Total daily volume (proportion of daily volume in each trade)alpha 2 0.3 volatility termalpha 3 0.0015 basic commission estimate - constantbeta 0.65 power to which alpha 1 is raised: higher power means a disproportionate impact of a single tradeeta 0.1 random error term

Initial position 10 10 10 10 10 10 10 10 10 10 100Final position 11.213 0.000 14.364 28.714 6.465 17.509 18.647 0.000 3.088 0.000 100.000Trading cost 0.0421 14.1066 15.9215 39.4942 0.4136 1.0781 3.9883 0.8577 0.2038 2.1567

Mean return 649.5468Variance 3.8588Total trading cost 78.2626

Objective 603.1325