Stodder: Derivatives Derivatives & Risk Management Derivatives are mostly used to ‘hedge’ (limit) risk But like most financial instruments, they can also be used for ‘speculation’ – taking on added risk in the expectation of gain

Stodder: Derivatives Basics of Option Pricing Basic to Option Pricing is the idea of a ‘Riskless Hedge’ A Riskless Hedge would be a situation in which you can buy some form of insurance that guarantees you the same money -- whether the market goes up or down.

Stodder: Derivatives Example of Riskless Hedge: Stock = $40, ‘Call’ Option Buys it at $35

Stodder: Derivatives What is this Call Option Worth? Since this hedge is riskless, it should be evaluated at the risk-free rate. Say “risk-free rate” (on US Bonds) is 8%. In one year, Portfolio of $22.50 has Present Value of PV = $22.50/1.08 = $20.83

Stodder: Derivatives Recall, Stock is now worth $40. So, it costs 0.75($40) = $30.00 to purchase ¾ of a share. Then PV Portfolio = Cost Stock – Value of Option $20.83 = $30 – Value of Option => V.o.O. = $9.17, what you sell it for

Stodder: Derivatives We have just derived the price We take as ‘known’ the present and future prices of the underlying asset. We ‘know’ the probabilities of these future prices. From this knowledge of future prices and probabilities, we ‘derive’ the price of the derivative.

Stodder: Derivatives In the simulation to follow, we will ‘Go in Both Directions’ We will use knowledge of future prices and volatility on underlying asset to derive the current price of the option. We can also use knowledge of the current price to derive future prices and volatility.

Stodder: Derivatives Run Simulation From Financial Models Using Simulation and Optimization by Wayne Winston.

Stodder: Derivatives Limitations of Log-Normal Assumption Log-Normality fails to reproduce some of the important features of empirical asset price dynamics such as Jumps in the asset price “Fat Tails” of the Probability Distribution Function StSt S0S0 T 0 Jump Gaussian Empirical pdf s i–1 – s i Fat Tails

Stodder: Derivatives How is this Modeled? Merton’s (1976) “Jump Diffusion” Process –Size of Jumps is itself Log-Normally Distributed and added to the model. –Timing of Jumps is Poisson Distributed. - Yusaku Yamamoto: Application of the Fast Gauss Transform to Option Pricing Application of the Fast Gauss Transform to Option Pricing

Stodder: Derivatives Derivatives get a ‘Bad Name’ Most Financial Scandals of the last decade in the US and UK were linked to derivatives, some combination of excessive speculation and fraud: Barrings Bank Enron World-Com Back-Dating of Options CDOs on Sub-Prime Mortgages

Stodder: Derivatives Reasons for Fraud Leveraging makes possible fantastic gain, but also horrible losses Gambler’s ‘Last Desperate Hope’ (Adverse Selection) Complexity of Derivatives make fraud harder to identify

Stodder: Derivatives Greater Long-Term Concern than Fraud: Systemic Risk The Moral Hazard of Insurance If you had a car that is less damaged by any given car crash – would that make you drive faster? If you (and everybody else) drove faster, could this actually wind up making you less safe ?