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Derivative Financial ProductsDonald C WilliamsDoctoral CandidateDepartment of Computational and Applied Mathematics Rice UniversityThesis Advisors.

Dr Richard A Tapia Department of Computational and Applied MathematicsDr Jeff Fleming Jesse H Jones Graduate School of Management10 September 2003Computational Finance Seminar Motivation.

Derivative Securities Markets Nature of Derivative Financial Products Modeling Relevant Parameters Option Valuation Problem.

Mathematical Machinery Concluding Remarks Derivative Securities MarketsThe valuation of various derivative securities is an area of extremeimportance in modern finance theory and practice .

Market Growth Gross market value of outstanding over the counterderivative contracts stood in excess of US 128 trillion BIS 2002 Product Innovations New derivative products are becoming morecomplex 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 DerivativesWhat is a derivative A derivative or derivative security is a financial instrument.

whose value depends on the value of other more basicunderlying assets Nature of DerivativesBasic Underlying Assets100 shares.

Equity e g common stock class A Agricultural e g corn soybeans Energy e g oil gas electricity Bandwidth e g communication General Derivative Contracts.

Forward Contracts Futures Contracts OptionsAn option is a particular type of derivative security thatgives the owner the right without the obligation to trade.

the underlying asset for a specific price the strike orexercise price at some future date Market StructureGeneral Financial MarketIndividual A.

Over The Counter OTC Individual BIndividual BFinancial Financial FinancialInstitution Institution Institution.

Option Contract SpecificationBasic Financial Contracts An American style option is a financial contract thatprovides the holder with the right without obligation to buy or sell an underlying asset S for a strike price.

K at any exercise time 0 T where T denotes thecontract maturity date An European style option is similarly defined withexercise restricted to the maturity date T Option Types.

Two Basic Option Types A call option gives the holder the right to buy theunderlying asset A put option gives the holder the right to sell theunderlying asset .

Payoff Fundamental ConstructsPayoff FunctionsLong position Short positionCall Option S t max S K 0 .

Put Option K S t max K S 0 Ex Put Option HedgeExample ABC is an oil company that will produce a 1 000 barrels of oil thisyear 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 Cash flow variations with the price of oil Oil Price Revenue Fixed Cost Put Payoff Unhedged Hedged16 00 16 000 18 500 3 500 2 500 1 000.

18 00 18 000 18 500 1 500 500 1 00020 00 20 000 18 500 500 1 500 1 00022 00 22 000 18 500 500 3 500 3 00024 00 24 000 18 500 500 5 500 5 000 Ex Call Option Speculation.

Long Call on IBM Profit from buying an IBM European call option option price 5 strike price 100 option life 2 months30 Profit 70 80 90 100 ST.

5 110 120 130 Asset Price Maturity ST T max ST K 0 ModelingBasic Question .

How do we mathematically model the value of option contracts Highly sophisticatedspecificationsquantitative modelsand intuition.

ModelingOption valuation models establish a functional relationshipbetween the traded option contract the underlying asset andvarious market variables e g asset price volatility Modeling.

Express the value of the option as a function of theunderlying asset price and various market parameters e g U S t f S K T S t r d 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 thevalue of the option contract Define Underlying Asset Price Process Make assumptions.

regarding the evolution of the state variables Enforce No Arbitrage Mathematically the economicargument of no arbitrage leads to a deterministic partialdifferential equation PDE that can be solved to determinethe value of the option .

Asset Price EvolutionAt the heart of any option pricing model is the assumedmathematical representation for underlying asset priceevolution By postulating a plausible stochastic differential equation.

SDE for the underlyinging price process a suitablemathematical model for asset price evolution can beestablished Asset Price Single Factor ModelThis fundamental model decomposes asset price returns into.

two components deterministic and stochastic written interms of the following SDE dt dWtThis geometric Brownian motion is the reference modelfrom which the Black Scholes Merton approach is based .

Asset Price TrajectoryAssume SDE models asset price dynamics with S t0 S 0dSt St dt St dWtSt S 0 S x dx S x dWx European Option Valuation Problem.

Black Scholes Merton EquationU t S U SS rSU S rU 0where appropriate initial and boundary conditions arespecified 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 theaforementioned assumption is a large part ofoption pricing theory American Option Valuation Problem.

The value U S t of an American option must satisfy the followingpartial differential complementarity problem PDCP L U S t 0 V S t S t 0 V S t S t L U S t 0.

L U S t U t S U SS rSU S rU S t specifies option payoffr denotes risk free interest rate Results from Stochastic Volatility Model.

Idea: Express the value of the option as a function of the underlying asset price and various market parameters, e.g., 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 ...

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