lecture 1: introduction to qf4102 financial modeling dr. dai min matdm@nus.edu.sgmatdm@nus.edu.sg,...
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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– 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
• 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)
• 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)
• 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
• 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
• 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
• 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)
• 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
• 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
• 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.
TS
)0,max()( KSKS TT
)( TSK
K
Exotic options
• Asian options:
• Lookback options:
• barrier options:
• Multi-asset options:
T
tTT dtST
AKA0
1 where,)(
tTt
TT SMKM
0max where,)(
]},0[ ,{)( TtHST tIKS
KSSSS TTTT ),max( ,)( 2121
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,
)(
)(
T
TT
SK
KSV
TtVt 0for ?
No arbitrage principle
• No free lunch• Assuming that short selling is allowed, we have by the
no-arbitrage principle
Applications of arbitrage arguments
• Pricing forward (long):
• Properties of option prices:
Binomial tree model (BTM): CRR (1979)
• Assumptions:
• Model derivation– Delta-hedging
– Option replication
Risk neutral pricing
Continuous-time model: Black-Scholes (1973)
• GBM assumption
Brownian motion and Ito integral
Black-Scholes model (continued)
• Ito lemma
• Delta-hedging
Black-Scholes equation
• For Vanilla options
• Black-Scholes formulas:
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
Another derivation: continuous-time replication
Continued
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)
• 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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