Transcript FE5106 Advanced Numerical Methods

Lecture 1: Introduction to QF4102
Financial Modeling
Dr. DAI Min
[email protected],
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
(ST  K )  max(ST  K ,0) ---call
---put
( K  ST )
where ST -- underlying asset price at expiry
K -- strike price
• The terminal payoff of a European vanilla option only
depends on the underlying price at expiry.
Exotic options
1
• Asian options： ( AT  K ) , where AT 
T


T
0
St dt

St
• Lookback options： ( M T  K ) , where M T  max
0t T
• barrier options:
(ST  K ) I{St H , t[0,T ]}
• Multi-asset options: (S1T  S2T ) ,
max(S1T , S2T )  K 
Option pricing problem
European vanilla option:
At expiry the option value is


( ST  K )
VT  


(
K

S
)
T

for call
for put
Problem: what’s the fair value of the option before expiry,
Vt  ? for 0  t  T
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 BlackScholes)
– Local volatility model
– Jump diffusion model
– Stochastic volatility model
– Utility indifference (dynamic programming approach)