Derivative Financial Products Donald C. Williams Doctoral Candidate Department of Computational and Applied Mathematics, Rice University Thesis Advisors Dr. 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 shares IBM class A Common 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

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. 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 STST STST K K STST STST K K Call Option 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 3,500 5,500 Hedged 1,000 3,000 5,000 Cash flow variations with the price of oil: Oil Price Revenue 16,000 18,000 20,000 22,000 24,000 Fixed Cost 18,500 Put Payoff 3,500 1, Profits

Ex.: Call Option - Speculation Profit from buying an IBM European call option: option price = $5, strike price = $100, option life = 2 months Profit ($) Long Call on IBM:

Modeling How do we mathematically model the value of option contracts? Basic Question: Financial specifications and intuition Highly sophisticated quantitative 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., 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 1 st 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.

Asset Price Trajectory Assume: SDE models asset price dynamics

European Option Valuation Problem Black-Scholes-Merton Equation 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 following partial differential complementarity problem (PDCP): where, American Option Valuation Problem

Results from Stochastic Volatility Model