Derivatives & Risk Management

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 toderive 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 Empirical pdf St Jump S0 Fat Tails Gaussian 0 T si–1– si 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 www.na.cse.nagoya-u.ac.jp/~yamamoto/work/KRIMS2004.ppt 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-Datingof 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 Concernthan 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 ? Stodder: Derivatives

Derivatives & Risk Management