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)