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
( S
T
( K

K )


S
T
)

 max( S
T

K , 0 ) ---call
---put where S
T
-- 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
• Asian options ：
( A
T

K )

, where A
T

1
T
 T
0
S t dt
• Lookback options ： ( M
T

K )

, where M
T
 max
0
 t

T
S t
• barrier options: ( S
T

K )

I
{ S t

H , t

[ 0 , T ]}
• Multi-asset options:
( S
1 T

S
2 T
)

,
 max( S
1 T
, S
2 T
)

K
 

Option pricing problem
European vanilla option:
At expiry the option value is
V
T

( S
T
 ( K

K

S
T
)

)
 for call for put
Problem: what’s the fair value of the option before expiry,
V t

?
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

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

Brownian motion and Ito integral

• 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)