Derivatives

1
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

2
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.

3
Example of Riskless HedgeStock 40, Call
Option Buys it at 35
4
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

5
Recall, Stock is now worth 40.

• So, it costs 0.75(40) 30.00 to purchase
• ¾ of a share. Then
• Price of Option Cost Stock PV Portfolio
• Price of Option 30 20.83
• 9.17

6
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.

7
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.

8
Run Simulation

• From Financial Models Using Simulation and
  Optimization
• by Wayne Winston.

9
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
Gaussian
Fat Tails
T
0
si1 si
10
How is this Modeled?

• Mertons (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/KR
  IMS2004.ppt

11
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

12
Reasons for Fraud

• Leveraging makes possible fantastic gain, but
  also horrible losses
• Gamblers Last Desperate Hope (Adverse
  Selection)
• Complexity of Derivatives make fraud harder to
  identify

13
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 ?