IS-1 Financial Primer Stochastic Modeling Symposium

1. IS-1 Financial PrimerStochastic Modeling Symposium By Thomas S.Y. Ho PhD Thomas Ho Company, Ltd Tom.ho@thomasho.com 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

9. Stock lattice

10. Call Option Lattice

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 + σs2)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

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18. Numerical Example of the Greeks

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

21. Ho –Lee (basic) Model

22. Ho-Lee One Period Rates

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: tom.ho@thomasho.com