IS-1 Financial Primer Stochastic Modeling Symposium

IS-1 Financial PrimerStochastic Modeling

SymposiumBy

Thomas S.Y. Ho PhDThomas Ho Company, [email protected]

April 3, 2006

2

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

3

“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)

4

Relative Valuation Law of one price: extending to non-

tradable financial instruments Applicability to insurance products and

annuities (loans and GICs) Arbitrage process and relative pricing

5

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

6

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

7

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)

8

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)

9

Stock lattice 163.214965

150.418059

138.624497

138.624497

127.755612

117.738905

127.755612

117.738905

108.50756

100

117.738905

108.50756

10092.1594775

84.933693

108.50756

10092.1594775

84.933693

78.2744477

72.1373221

stock lattice 10092.1594775

84.933693

78.2744477

72.1373221

66.4813791

61.2688917

time 0 1 2 3 4 5 6

10

Call Option Lattice            63.214965

          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

10.125573 4.729344 1.361849 0.000000 0.000000 0.000000 0.000000

11

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

12

Continuous Time Modeling Ito process

dX(t) = µ(t)dt + σ(t)dB(t) (dt)2 =0 (dt)(dB)=0 (dB)2 =dt

Z = g( t, X) 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))

13

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 + σs

2)dt + σs dB’

14

Numeraire General Case Y as numeraire

Z(t) = V(t)/Y(t) dS/S = (r – q + ρσs σy)dt + σs dB’’

Volatility invariant

15

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

16

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

17

??

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

18

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

19

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

20

Interest Rate Model: Setting Up

year 0 1 2 3 4 5

initial yield curve 0.060 0.060 0.065 0.070 0.075 0.080

initial discount function p(n)1.00000

00.94176

50.87809

50.81058

40.74081

80.67032

0

one period forward curve 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

21

Ho –Lee (basic) Model

( 1)(1) 2

( ) (1 )

ini n

P nP

P n

         0.86124

1

       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 lattice0.94176

5 0.93672

9 0.93313

6 0.93094

6 0.93012

6 0.93064

2

year 0 1 2 3 4 5

22

Ho-Lee One Period Rates

ln ( )( )

nn ii

P Tr T

T

         0.14938

06

       0.12823

510.13388

06

     0.10875

380.11428

510.11838

06

   0.09090

470.09635

380.10033

510.10288

06

 0.07466

080.08005

470.08395

380.08638

510.08738

06

Interest rate lattice 0.060.06536

080.06920

470.07155

380.07243

510.07188

06

year 0 1 2 3 4 5

23

1 2 1 1

1 2

1 1 2( 1)(1)

( ) 1 1 1n n nn i

i nn n n

P nP

P n

         0.86673

1

       0.88096

3 0.87807

2

     0.89598

0 0.89266

8 0.88956

2

0.911395

0.907795

0.904530

0.901201

  0.926800

0.923044

0.919765

0.916549

0.912994

Discount function lattice0.94176

5 0.937988

0.934841

0.931893

0.928727

0.924940

year 0 1 2 3 4 5

24

1 2 1 1

1 2

1 1 2( 1)(1)

( ) 1 1 1n n nn i

i nn n n

P nP

P n

 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

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

25

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

26

Alternative Valuation Algorithms Discounting along the spot curve Backward substitution Pathwise valuation

monte-carlo Antithetic, control variate Structured sampling

Finite difference methods

27

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

28

Examples of Applications Corporate bonds (liquidity and credit risks)

Option adjusted spreads Mortgage-backed securities

Prepayment models CMOs

Capital structure arbitrage valuation Insurance products

29

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

30

References Ho and Lee (2005) The Oxford Guide to

Financial Modeling Oxford University Press Excel models (185 models)

www.thomasho.com Email: [email protected]