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Stodder: DerivativesStodder: Derivatives

Derivatives & Risk ManagementDerivatives & 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


Basics of Option PricingBasics 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.


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

Ending Stock Price minus Strike Price = Option Value$30 - $35 = $0 (No one will buy)$50 - $35 = $15

Difference: $20 $0 $15

Ending Stock Value minus Strike Price = Option Value$30 x 0.75 = $22.50 - $35 $0 (No one will buy)$50 x 0.75 = $37.50 - $35 $15

Difference: $15 $0 $15

Ending Stock Value minus Option Value = Value of Porftolio$30 x 0.75 = $22.50 $0 $22.50$50 x 0.75 = $37.50 $15 $22.50


What is this Call Option Worth?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


RecallRecall, Stock is now worth , Stock is now worth $40$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


We have just We have just derivedderived the price 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.


In the simulation to follow, we will In the simulation to follow, we will ‘Go in Both Directions’‘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.


Run SimulationRun Simulation

• From Financial Models Using Simulation and Optimization

by Wayne Winston.


LimitationsLimitations of Log-Normal Assumption 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

St

S0

T0

Jump

Gaussian

Empirical pdf

si–1– si

FatTails


How is this How is this Modeled?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


Derivatives get a Derivatives get a ‘Bad Name’‘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


Reasons for FraudReasons 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


Greater Long-Term ConcernGreater Long-Term Concernthan Fraud: than Fraud: Systemic RiskSystemic 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 ?

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