Derivative Financial Products
Donald C. WilliamsDoctoral Candidate
Department of Computational and Applied Mathematics, Rice University
Thesis AdvisorsDr. Richard A. Tapia, Department of Computational and Applied Mathematics
Dr. Jeff Fleming, Jesse H. Jones Graduate School of Management
10 September 2003
Computational Finance Seminar
Outline
• Motivation– Derivative Securities Markets
– Nature of Derivative Financial Products
• Modeling– Relevant Parameters
– Option Valuation Problem
• Mathematical Machinery• Concluding Remarks
Derivative Securities Markets
The valuation of various derivative securities is an area of extreme importance in modern finance theory and practice:
• Market Growth. Gross market value of outstanding over-the-counter derivative contracts stood in excess of $US 128 trillion (BIS, 2002)
• Product Innovations. New derivative products are becoming more complex to fit desired exposure of clients.
• Quantitative Evolution. Valuation and hedging techniques must evolve to effectively manage financial risk.
• Decision Science. Corporate decision strategy.
Nature of Derivatives
A derivative (or derivative security) is a financial instrument whose value depends on the value of other, more basic underlying assets.
What is a derivative?
Nature of Derivatives
• Equity (e.g., common stock)
• Agricultural (e.g., corn, soybeans)
• Energy (e.g., oil, gas, electricity)
• Bandwidth (e.g., communication)
100 sharesIBM
class ACommon
Basic Underlying Assets
General Derivative Contracts
• Forward Contracts• Futures Contracts• Swaps• Options
An option is a particular type of derivative security that gives the owner the right (without the obligation) to trade the underlying asset for a specific price (the strike or exercise price) at some future date.
Market Structure
Financial Institution
A
Exchange
General Financial Market
Over-The-Counter (OTC)
Financial Institution
B
Financial Institution
C
Individual A
Individual B
Individual B
Option Contract Specification
An American-style option is a financial contract that provides the holder with the right, without obligation, to buy or sell an underlying asset, S, for a strike price K, at any exercise time where T denotes the contract maturity date.
T,0
Basic Financial Contracts:
An European-style option is similarly defined with exercise restricted to the maturity date, T .
Option Types
A call option gives the holder the right to buy the underlying asset.
A put option gives the holder the right to sell the underlying asset.
Two Basic Option Types:
Payoff: Fundamental Constructs
Payoff Functions
ST STKK
ST STKK
Call Option)0,max(),( KStS
)0,max(),( SKtS Put Option
Long position Short position
Ex. : Put Option - Hedge
Example. ABC is an oil company that will produce a 1,000 barrels of oil this
year and sell them in December. The expected selling price is $20/bbl. Assume
ABC can buy a put option contract (hedge) on a thousand barrels of oil for
$500, with strike, X=$20/bbl and December expiration.
Unhedged-2,500-5001,5003,5005,500
Hedged1,0001,0001,0003,0005,000
Cash flow variations with the price of oil:
Oil Price16.0018.0020.0022.0024.00
Revenue16,00018,00020,00022,00024,000
Fixed Cost18,50018,50018,50018,50018,500
Put Payoff3,5001,500-500-500-500
Profits
Ex.: Call Option - Speculation
Profit from buying an IBM European call option: option price = $5, strike price = $100, option life = 2 months
30
20
10
0-5
70 80 90 100
110 120 130
Profit ($)
Long Call on IBM:
)0,max(),( KSTS TT
Maturity @
PriceAsset
TS
Modeling
How do we mathematically model the value of option contracts?
Basic Question:
Financial specificationsand intuition
Highly sophisticatedquantitative models
Modeling
Option valuation models establish a functional relationship between the traded option contract, the underlying asset, and various market variables (e.g., asset price volatility).
Modeling
Idea: Express the value of the option as a function of the underlying asset price and various market parameters, e.g.,
drtSTKSftSU ,,,,,,),(
• S and t are asset price and time
• volatility of underlying asset price
• K and T are contract specific parameters
• r is the interest rate associated with underlying currency
• d is the expected dividend during the life of the option
Modern Option Valuation
• Uses continuous-time methodologies– (1900) Louis Bachelier (one of the 1st analytical treatments)– (1973) Fisher Black and Myron Scholes– (1973) Robert Merton
>>>> Black-Scholes-Merton PDE <<<< >>> 1998 Nobel Prize in Economic <<<• Employs mathematical machinery that derives from
– Stochastic calculus– Probability– Differential equations, and– Other related areas.
Modeling Building Framework
• Define State Variables. Specify a set of state variables (e.g., asset price, volatility) that are assumed to effect the value of the option contract.
• Define Underlying Asset Price Process. Make assumptions regarding the evolution of the state variables.
• Enforce No-Arbitrage. Mathematically, the economic argument of no-arbitrage leads to a deterministic partial differential equation (PDE) that can be solved to determine the value of the option.
Asset Price Evolution
At the heart of any option pricing model is the assumed mathematical representation for underlying asset price evolution.
By postulating a plausible stochastic differential equation (SDE) for the underlyinging price process, a suitable mathematical model for asset price evolution can be established.
Asset Price: Single Factor Model
This fundamental model decomposes asset price returns into two components, deterministic and stochastic, written in terms of the following SDE
This geometric Brownian motion is the reference model from which the Black-Scholes-Merton approach is based.
tt
t dWdtS
dS
Asset Price Trajectory
Assume: SDE models asset price dynamics
tttt dWSdtSdS
x
t
t
x
t
t
xt dWSdxSSS 00
0
00with SSt
European Option Valuation Problem
Black-Scholes-Merton Equation
0)2
( 22
rUrSUUSU SSSt
where appropriate initial and boundary conditions are specified.
Concluding Remarks
• Black-Scholes Assumption– The market is frictionless
– There are no arbitrage opportunities
– Asset price follows a geometric Brownian motion
– Interest rate and volatility are constant
– The option is European
• Circumventing the limitations inherent in the aforementioned assumption is a large part of option pricing theory.
The value, U(S,t), of an American option must satisfy the followingpartial differential complementarity problem (PDCP):
rateinterest freerisk denotes
payoffoption specifies ),(
)2
()],(L[ 22
r
tS
rUrSUUSUtSU SSSt
0)]),())(L[,(),((
0),(),((
0)],(L[
tSUtStSV
tStSV
tSU
where,
American Option Valuation Problem