1. Lecture 1: Introduction to QF4102 Financial Modeling
Dr. DAI Min

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, S ; (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:

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)