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Lecture 1: Introduction to QF4102 Financial Modeling

Lecture 1: Introduction to QF4102 Financial Modeling. Dr. DAI Min matdm@nus.edu.sg , http://www.math.nus.edu.sg/~matdm/qf4102/qf4102.htm. Modern finance. Modern Portfolio Theory single-period model: H. Markowitz (1952) optimization problem

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Lecture 1: Introduction to QF4102 Financial Modeling

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  1. Lecture 1: Introduction to QF4102 Financial Modeling Dr. DAI Min matdm@nus.edu.sg, http://www.math.nus.edu.sg/~matdm/qf4102/qf4102.htm

  2. Modern finance • Modern Portfolio Theory • single-period model: H. Markowitz (1952) optimization problem • continuous-time finance: R. Merton (1969), P. Samuelson stochastic control • We take risk to beat the riskfree rate • Option Pricing Theory • continuous-time: Black-Scholes (1973), R. Merton (1973) • discrete-time: Cox-Ross-Rubinstein (1979) • We eliminate risk to find a fair price

  3. Option pricing theory • Pricing under the Black-Scholes framework • Vanilla options • Exotic options • Pricing beyond Black-Scholes • Local volatility model • Jump-diffusion model • Stochastic volatility model • Utility indifference pricing • Interest rate models

  4. Lecture outline (I) • Aims of the module • The goal is to present pricing models of derivatives and numerical methods that any quantitative financial practitioner should know • Module components • Group assignments and tutorials: (40%) • A group of 2 or 3, attending the same tutorial class. • ST01 (Thu): 18:00-19:00, LT24; (MQF and graduates) • ST02 (Wed): 17:00-18:00, S16-0304; (QF) • Final exam: (60%), held on 21 Nov (Sat)

  5. Lecture outline (II) • Required background for this module • Basic financial mathematics • options, forward, futures, no-arbitrage principle, Ito’s lemma, Black-Scholes formula, etc. • Programming • Matlab is preferred, but C language is encouraged. • For efficient programming in Matlab, use vectors and matrices • Pseudo-code: for loops, if-else statements • Course website: http://www.math.nus.edu.sg/~matdm/qf4102/qf4102.htm

  6. Numerical methods • Why we need numerical methods? • Analytical solutions are rare • Numerical methods • Monte-Carlo simulation • Lattice methods • Binomial tree method (BTM) • Modified BTM: forward shooting grid method • Finite difference • Dynamic programming • Handling early exercise

  7. Brief review: basic concepts • A derivative is a security whose value depends on the values of other more underlying variables • underlying: stocks, indices, commodities, exchange rate, interest rate • derivatives: futures, forward contracts, options, bonds, swaps, swaptions, convertible bonds

  8. Forward contracts • An agreement between two parties to buy or sell an asset (known as the underlying asset) at a future date (expiry) for a certain price (delivery price) • Contrasted to the spot contract. • Long Position / Short Position • Linear Payoff

  9. Forward contracts (continued) • At the initial time, the delivery price is chosen such that it costs nothing for both sides to take a long or short position. • A question: how to determine the delivery price?

  10. Options • A call option is a contract which gives the holder the right to buy an asset (known as the underlying asset) by a certain date (expiration date or expiry) for a predetermined price (strike price). • Put option: the right to sell the underlying • European option：exercised only on the expiration date • American option：exercised at any time before or at expiry

  11. Vanilla options • The payoff of a European (vanilla) option at expiry is ---call ---put where -- underlying asset price at expiry -- strike price • The terminal payoff of a European vanilla option only depends on the underlying price at expiry.

  12. Exotic options • Asian options： • Lookback options： • barrier options: • Multi-asset options:

  13. Option pricing problem European vanilla option: At expiry the option value is for call for put Problem:what’s the fair value of the option before expiry,

  14. No arbitrage principle • No free lunch • Assuming that short selling is allowed, we have by the no-arbitrage principle

  15. Applications of arbitrage arguments • Pricing forward (long): • Properties of option prices:

  16. Binomial tree model (BTM): CRR (1979) • Assumptions: • Model derivation • Delta-hedging • Option replication

  17. Risk neutral pricing

  18. Continuous-time model: Black-Scholes (1973) • GBM assumption

  19. Brownian motion and Ito integral

  20. Black-Scholes model (continued) • Ito lemma • Delta-hedging

  21. Black-Scholes equation • For Vanilla options • Black-Scholes formulas:

  22. Comments • In the B-S equation, S and t are independent • The B-S equation holds for any derivative whose price function can be written as V(S,t) • Hedging ratio: Delta • Risk neutral pricing and Feynman-Kac formula

  23. Another derivation: continuous-time replication

  24. Continued

  25. Module outline • Monte-Carlo simulation • Lattice methods • Multi-period BTM • Single-state BTM • Forward shooting grid method • Finite difference method • Convergence/consistency analysis • Applications of lattice methods • Lookback options • American options

  26. Module outline (continued) • Numerical methods for advanced models (beyond Black-Scholes) • Local volatility model • Jump diffusion model • Stochastic volatility model • Utility indifference (dynamic programming approach)

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