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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

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 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
option pricing theory
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
lecture outline i
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)
lecture outline ii
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
numerical methods
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
brief review basic concepts
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
forward contracts
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
forward contracts continued
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?
options
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
vanilla options
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.
exotic options
Exotic options
  • Asian options:
  • Lookback options:
  • barrier options:
  • Multi-asset options:
option pricing problem
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,

no arbitrage principle
No arbitrage principle
  • No free lunch
  • Assuming that short selling is allowed, we have by the no-arbitrage principle
applications of arbitrage arguments
Applications of arbitrage arguments
  • Pricing forward (long):
  • Properties of option prices:
binomial tree model btm crr 1979
Binomial tree model (BTM): CRR (1979)
  • Assumptions:
  • Model derivation
    • Delta-hedging
    • Option replication
black scholes model continued
Black-Scholes model (continued)
  • Ito lemma
  • Delta-hedging
black scholes equation
Black-Scholes equation
  • For Vanilla options
  • Black-Scholes formulas:
comments
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
module outline
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
module outline continued
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)