1. IS-1 Financial Primer Stochastic Modeling Symposium By Thomas S.Y. Ho PhD Thomas Ho Company, Ltd Tom.ho@thomasho.com April 3, 2006

2. 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. 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. 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. 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. 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. 7 One-Period Binomial Model  S u /S > exp(rT)> S d /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.   =(C u – C d )/(S u -S d )  Π u =(S- exp(-rT) S d )/(S u - S d )  C = π u C u + π d C d  S= π u S u + π d S d  1 = π u exp(rT)+ π d exp(rT)

8. 8 Numerical Example: Call Option Pricing Stock Price($)S100 Strike Price ($)X100 Stock VolatilityσSσS 0.2 Time to expiration (year)T1 Risk-free rater0.05 dividend yieldsdN/A the number of periodsn6dt = T/n upward movementu1.0851= exp(σ√dt) downward movementd0.9216= 1/u risk-neutral probability of up0.5308= (exp(rdt)-d)/(u-d)

9. 9 Stock lattice 163.21 4965 150.41 8059 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 stock lattice100 92.159 4775 84.933 693 78.274 4477 72.137 3221 66.481 3791 61.268 8917 time0123456

10. 10 Call Option Lattice 63.214965 51.24793038.624497 40.27735228.58548317.738905 30.22462119.3917599.3374300.000000 21.72363412.4945334.9150500.000000 15.0554607.7807622.5871910.000000 10.1255734.7293441.3618490.000000

11. 11 Martingale Processes, p and q measures  C/R = p u C u /R u + p d C d /R d  S/R = p u S u /R u + p d S d /R d  1 = p u + p d  C/S = q u C u /S u + q d C d /S d  R/S = q u R u /S u + q d R d /S d  1 = q u + q d  Probability measure: assigning prob  Denominator: numeraire  Martingale: “expected” value= current value

12. 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 = g t dt + g X dX + 1/2 g xx (dX) 2 Geometric Brownian motion  dS/S =µdt + σdB(t)  S(t) = S(0)exp (µt - σ 2 t/2 + σ B(t))

13. 13 Numeraires and Probabilities  dS/S = µ s dt + σ s dB s (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 + σ s dB(t)  V as numeraire Z(t) = R(t)/V(t) dS/S = (r – q + σ s 2 )dt + σ s dB’

14. 14 Numeraire General Case  Y as numeraire Z(t) = V(t)/Y(t) dS/S = (r – q + ρσ s σ y )dt + σ s dB’’  Volatility invariant

15. 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. 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. 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. 18 Numerical Example of the Greeks CallPut Price9.925056.00400 Δ(Delta)0.61791-0.38209 Γ(Gamma)0.01907 v (Vega)38.13878 Θ(Theta)-5.88852-2.04536 ρ (Rho)51.86609-44.21286

19. 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. 20 Interest Rate Model: Setting Up year012345 initial yield curve0.060 0.0650.0700.0750.080 initial discount function p(n) 1.00000 0 0.94176 5 0.87809 5 0.81058 4 0.74081 8 0.67032 0 one period forward curve0.060 0.0700.0800.0900.100 lognormal spot volatility (σ S )00.0775 lognormal forward volatility (σ f )00.0775

21. 21 Ho –Lee (basic) Model 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 lattice 0.94176 5 0.93672 9 0.93313 6 0.93094 6 0.93012 6 0.93064 2 year 012345

22. 22 Ho-Lee One Period Rates 0.14938 06 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 year012345

23. 23 0.86673 1 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 0.926800 0.9 23 04 4 0.91976 5 0.91654 9 0.91299 4 Discount function lattice 0.94176 5 0.937988 0.9 34 84 1 0.93189 3 0.92872 7 0.92494 0 year012345

24. 24 Ho-Lee model rates with term structure of volatilities 0.1430264 0.12674020.1300264 0.10983680.11354020.1170264 0.09277840.09673680.10034020.1040264 0.0760180.08007840.08363680.08714020.0910264 0.060.0640180.06737840.07053680.07394020.0780264 012345 lognormal spot volatility (σ S )00.10.0950.090.0850.08 lognormal forward volatility (σ f )00.10.09071430.0818750.07333330.065 1

25. 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. 26 Alternative Valuation Algorithms  Discounting along the spot curve  Backward substitution  Pathwise valuation monte-carlo Antithetic, control variate Structured sampling  Finite difference methods

27. 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. 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. 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. 30 References  Ho and Lee (2005) The Oxford Guide to Financial Modeling Oxford University Press  Excel models (185 models) www.thomasho.com www.thomasho.com  Email: tom.ho@thomasho.comtom.ho@thomasho.com