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.
2
Options… The Basics
3
Payoff and Intrinsic Value of a Call
4
Payoff and Intrinsic Value of a Put
5
Put-Call Parity
6
European Options
(Using Simple Binomial Modeling)
7
Profit Timing and Determination
8
Stock Price Dynamics – binomial lattice
Stock price goes up/down by the
same amount each time period.
In this example: 1.07 and 1/1.07
9
Options Pricing – call option formula
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.
10
European Call Option Pricing Example
15.48 = 1/R( 22.5q + 7(1-q))
R=1.01
Q=(R-d)/(u-d)
d=1/1.07
u=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.
11
European Options: Excel Modeling
12
Does Put Call Parity Hold?
13
American Options
(Using Simple Binomial Modeling)
14
Pricing American Options
15
Reverse through the Lattice
16
American Put vs. Call – early or not?
17
Black-Scholes Model
Geometric Brownian Motion
Models random fluctuations in stock prices
18
Black-Scholes Model… continued
19
Black-Scholes Model in Excel
20
Implied Volatility
21
Futures and Forwards
22
Forwards Contracts
23
Futures and Forwards…
Problems with Forwards
Futures Contracts
24
Mechanics of a Futures Contract
25
Excel Example with Daily Settlement
26
Hedging using Futures
A Perfect Hedge Isn’t Always Possible…
27
Term Structure of Interest Rates
28
Yield Curves (US Treasuries)
Rates are climbing – highest in Dec 2013
Source: http://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/TextView.aspx?data=yieldYear&year=2013
29
Sample Short Rate Lattice
9.375% = 7.5% x 1.25
30
Pricing a Zero-coupon Bond (ZCB)
9.375% comes from the last slide
Assumes a 50:50 chance of
rates increasing/decreasing
31
Excel Modeling
Again, we work backwards through the lattice.
89.51 = 1/1.1172 * ( 100 x 0.5
+ 100 x 0.5)
32
Pricing European Call Option on ZCB
Max(0, 83.08-84)
Max(0, 87.35-84)
Max(0, 90.64-84)
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Pricing American Put Option on ZCB
34
Pricing Forwards on Bonds
35
Pricing Forwards on Bonds - excel
Start at the end and work back to t=4
Then work from
t=4 backwards
36
Mortgage Backed Securities (MBS)
Collateralized Debt Obligations (CDO)
37
Mortgage Backed Securities Markets
38
The Logic of Tranches (MBS)
39
The Logic of Tranches (CDO)
40
A Simple Example: A 1-period CDO
41
Excel model of CDO
Credit # Default Prob
1
0.2
2
0.2
3
0.06
4
0.3
5
0.4
6
0.65
7
0.3
8
0.23
9
0.02
10
0.12
11
0.134
12
0.21
13
0.08
14
0.1
15
0.1
16
0.02
17
0.3
18
0.015
19
0.2
20
0.03
Expected number of losses in the CDO = sum of all probabilities of individual defaults
Probability of losses P(0)
P(1)
P(2-20)
calculations are not shown for
these other tranches in this file
0.010989
0
0.064562
1
0.924448
2
Tranche (0-2)
0.000
0.065
1.849
1.913
Tranche (2-4)
Tranche (4-20)
1.283
0.472
3.668
3.668
1-probability of default = probability of survival
42
CDON
43
Portfolio Optimization
44
Return on Assets and Portfolios
45
Two-asset Example
46
Optimization Example (solver)
Mean return
REITs
2.40
US Large Growth US Small Growth
4.10
5.20
Covariance matrix
REITs
US Large Growth
US Small Growth
Volatility
Porfolio
Interest rate (%)
risk aversion (tau)
REITs US Large Growth US Small Growth
0.0010
-0.0006
0.0001
-0.0006
0.0599
0.0635
0.0001
0.0635
0.1025
REITs
3.17
US Large Growth US Small Growth
24.46
32.01
x1
0.05
x2
0.00
x3
0.00
x0
0.95
1.00
=
1.00
1.5
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
47
Optimization with trading costs
Trading cost parameters
alpha 1
alpha 2
alpha 3
beta
eta
0.0035
0.3
0.0015
0.65
0.1
Initial position
Final position
Trading cost
10
11.213
0.0421
Mean return
Variance
Total trading cost
649.5468
3.8588
78.2626
Objective
603.1325
Average trade volume / Total daily volume (proportion of daily volume in each trade)
volatility term
basic commission estimate - constant
power to which alpha 1 is raised: higher power means a disproportionate impact of a single trade
random error term
10
0.000
14.1066
10
14.364
15.9215
10
28.714
39.4942
10
6.465
0.4136
10
17.509
1.0781
10
18.647
3.9883
10
0.000
0.8577
10
3.088
0.2038
10
0.000
2.1567
100
100.000
48