Download IS-1 Financial Primer Stochastic Modeling

IS-1 Financial Primer
Stochastic Modeling Symposium
By
Thomas S.Y. Ho PhD
Thomas Ho Company, Ltd
[email protected]
April 3, 2006
Purpose
Overview of the basic principles in the
relative valuation models
 Overview of the basic terminologies



Equity derivatives
Fixed income securities
Practical implementation of the models
 Examples of applications

2
“Traditional Valuation”
Net present value
 Expected cashflows
 Cost of capital as opposed to cost of
funding
 Capital asset pricing model
 Cost of capital of a firm as opposed to cost
of capital of a project (or security)

3
Relative Valuation
Law of one price: extending to nontradable financial instruments
 Applicability to insurance products and
annuities (loans and GICs)
 Arbitrage process and relative pricing

4
Stock Option Model
Modeling approach: specifying the
assumptions, types of assumptions
 Description of an option
 Economic assumptions:





Constant risk free rate
Constant volatility
Stock return distribution
Efficient capital markets
5
Binomial Lattice Model
Generality of the model in describing the
equity return distribution
 Market lattice and risk neutral lattice
 Dynamic hedging and valuation
 Intuitive explanation of the model results
 Comparing the relative valuation approach
and the traditional approach – the case of
a long dated equity put option

6
One-Period Binomial Model







Su/S > exp(rT)> Sd/S
In the absence of arbitrage opportunities, there
exist positive state prices such that the price of
any security is the sum across the states of the
world of its payoff multiplied by the state price.
=(Cu – Cd)/(Su -Sd )
Πu =(S- exp(-rT) Sd )/(Su - Sd )
C = πuCu + πdCd
S= πuSu + πdSd
1 = πuexp(rT)+ πdexp(rT)
7
Numerical Example: Call Option Pricing
Stock Price($)
S
100
Strike Price ($)
X
100
Stock Volatility
σS
0.2
Time to expiration (year)
T
1
Risk-free rate
r
0.05
dividend yields
d
N/A
the number of periods
n
6
dt = T/n
upward movement
u
1.0851
= exp(σ√dt)
downward movement
d
0.9216
= 1/u
risk-neutral probability of u
p
0.5308
= (exp(rdt)-d)/(u-d)
8
Stock lattice
163.21
4965
stock lattice
time
150.41
8059
138.62
4497
138.62
4497
127.75
5612
117.738
905
127.75
5612
117.738
905
108.50
756
100
117.738
905
108.50
756
100
92.159
4775
84.933
693
108.50
756
100
92.159
4775
84.933
693
78.274
4477
72.137
3221
100
92.159
4775
84.933
693
78.274
4477
72.137
3221
66.481
3791
61.268
8917
0
1
2
3
4
5
6
9
Call Option Lattice
63.214965
10.125573
51.247930
38.624497
40.277352
28.585483
17.738905
30.224621
19.391759
9.337430
0.000000
21.723634
12.494533
4.915050
0.000000
0.000000
15.055460
7.780762
2.587191
0.000000
0.000000
0.000000
4.729344
1.361849
0.000000
0.000000
0.000000
0.000000
10
Martingale Processes, p and q measures









C/R = puCu/Ru + pdCd/Rd
S/R = puSu/Ru + pdSd/Rd
1 = pu + pd
C/S = quCu/Su + qdCd/Sd
R/S = quRu/Su + qdRd/Sd
1 = qu + qd
Probability measure: assigning prob
Denominator: numeraire
Martingale: “expected” value= current value
11
Continuous Time Modeling

Ito process

dX(t) = µ(t)dt + σ(t)dB(t)




Z = g( t, X)


(dt)2 =0
(dt)(dB)=0
(dB)2 =dt
dZ = gt dt + gXdX + 1/2 gxx (dX)2
Geometric Brownian motion


dS/S =µdt + σdB(t)
S(t) = S(0)exp (µt - σ2t/2 + σ B(t))
12
Numeraires and Probabilities
dS/S = µs dt + σsdBs(t) dividend paying
 dV/V = qdt + dS/S dividend re-invested
 dY/Y = µ* dt + σ*dB*(t) any asset
 R(t) = integral of r(s) stochastic rates
 Risk neutral measure




Z(t) = V(t)/R(t)
dS/S = (r- q) dt + σsdB(t)
V as numeraire


Z(t) = R(t)/V(t)
dS/S = (r – q + σs2)dt + σs dB’
13
Numeraire General Case

Y as numeraire



Z(t) = V(t)/Y(t)
dS/S = (r – q + ρσs σy)dt + σs dB’’
Volatility invariant
14
Risk Neutral Measure
Martingale process
 Examples of measures


p measure, forward measure, market measure
Generalization of the Black-Scholes Model
 Applications in the capital markets
 Applications to the insurance products




Life products
Fixed annuities
Variable annuities
15
Sensitivity Measures







Delta , S
Gamma Г, 
Theta θ (time decay) t
Vega v measure σ
Rho  , r
Relationships of the sensitivity measures
Intuitive explanation of the greeks


European, American, Bermudian, Asian put/call options
Comparing with the equilibrium models

Continual adjustment of the implied volatility
16
Stock Price (S)
100
Strike Price (K)
100
?
Time to expiration (T)
1
Stock volitility (σ)
0.2
Risk-free rate (r)
0.04
Dividend yields (δ)
0
17
Numerical Example of the Greeks
Call
Put
Price
9.92505
6.00400
Δ(Delta)
0.61791
-0.38209
Γ(Gamma)
0.01907
0.01907
v (Vega)
38.13878
38.13878
Θ(Theta)
-5.88852
-2.04536
ρ (Rho)
51.86609
-44.21286
18
Interest Rate Modeling
Lattice models
 Yield curve estimation
 Yield curve movements
 Dynamic hedging of bonds
 Term structure of volatilities
 Sensitivity measures


Duration, key rate duration, convexity
19
Interest Rate Model: Setting Up
year
initial yield curve
initial discount function p(n)
one period forward curve
0
1
2
3
4
5
0.060
0.060
0.065
0.070
0.075
0.080
1.00000
0
0.94176
5
0.87809
5
0.81058
4
0.74081
8
0.67032
0
0.060
0.060
0.070
0.080
0.090
0.100
lognormal spot volatility (σS)
0
0.0775
0.0775
0.0775
0.0775
0.0775
lognormal forward volatility (σf)
0
0.0775
0.0775
0.0775
0.0775
0.0775
20
Ho –Lee (basic) Model
0.86124
1
 P ( n  1)   i
Pi (1)  2 

n
 P ( n)  (1   )
n
0.87964
7
0.87469
5
0.89695
1
0.89200
4
0.88835
8
0.91310
5
0.90814
3
0.90453
4
0.90223
5
0.92805
8
0.92306
6
0.91947
4
0.91724
1
0.91632
8
Discount function lattice
0.94176
5
0.93672
9
0.93313
6
0.93094
6
0.93012
6
0.93064
2
year
0
1
2
3
4
5
21
Ho-Lee One Period Rates
0.14938
06
 ln Pi n (T )
ri (T ) 
T
n
0.12823
51
0.13388
06
0.10875
38
0.11428
51
0.11838
06
0.09090
47
0.09635
38
0.10033
51
0.10288
06
0.07466
08
0.08005
47
0.08395
38
0.08638
51
0.08738
06
Interest rate lattice
0.06
0.06536
08
0.06920
47
0.07155
38
0.07243
51
0.07188
06
year
0
1
2
3
4
5
22
 P (n  1)   1   n 1 n  2 1  1   n 1  2  i
Pi n (1)  
 n

 P (n)   1   n 1 1   n  2  1   n  
0.86673
1
0.926800
0.94176
0.88096
3
0.87807
2
0.89598
0
0.89266
8
0.88956
2
0.9
11
39
5
0.90779
5
0.90453
0
0.90120
1
0.9
23
04
4
0.91976
5
0.91654
9
0.91299
4
0.9
34
23
0.93189
0.92872
0.92494
1  1   n 1  2  i
 P ( n  1)   1   n 1 n  2
Pi n (1)  


 n

P
(
n
)
1



1



1









 
n
1
n
2
n

lognormal spot volatility (σS)
0
0.1
0.095
0.09
0.085
0.08
lognormal forward volatility (σf)
0
0.1
0.0907143
0.081875
0.0733333
0.065
1
Ho-Lee
model rates
with term
structure of
volatilities
0.1430264
0.1267402
0.1300264
0.1098368
0.1135402
0.1170264
0.0927784
0.0967368
0.1003402
0.1040264
0.076018
0.0800784
0.0836368
0.0871402
0.0910264
0.06
0.064018
0.0673784
0.0705368
0.0739402
0.0780264
0
1
2
3
4
5
24
Alternative Arbitrage-free Interest Rate
Modeling Techniques
These are not economic models but
techniques
 Spot rate model
 N-factor model
 Lattice model
 Continuous time model
 Calibrations

25
Alternative Valuation Algorithms
Discounting along the spot curve
 Backward substitution
 Pathwise valuation





monte-carlo
Antithetic, control variate
Structured sampling
Finite difference methods
26
Example of Interest Rate Models
Ho-Lee, Black-Derman-Toy, Hull-White
 Heath-Jarrow-Morton model
 Brace-Gatarek-Musiela/Jamshidian model
(Market Model)
 String model
 Affine model

27
Examples of Applications

Corporate bonds (liquidity and credit risks)


Option adjusted spreads
Mortgage-backed securities


Prepayment models
CMOs
Capital structure arbitrage valuation
 Insurance products

28
Conclusions
Comparing relative valuation and the NPV
model
 Imagine the world without relative
valuation
 Beyond the Primer:



Importance of financial engineering
Identifying the economics of the models
29
References
Ho and Lee (2005) The Oxford Guide to
Financial Modeling Oxford University Press
 Excel models (185 models)
www.thomasho.com
 Email: [email protected]

30