Transcript Presentation on 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, July
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, July
Example of Riskless Hedge:
Stock = $40, ‘Call’ Option Buys it at $35
Ending Stock Price
$30
$50
Difference:
$20
minus Strike Price
- $35
- $35
$0
= Option Value
= $0
= $15
$15
Ending Stock Value
$30 x 0.75 = $22.50
$50 x 0.75 = $37.50
Difference:
$15
minus Strike Price
- $35
- $35
$0
= Option Value
$0
$15
$15
Ending Stock Value
$30 x 0.75 = $22.50
$50 x 0.75 = $37.50
(No one will buy)
(No one will buy)
minus Option Value = Value of Porftolio
$0
$22.50
$15
$22.50
Stodder: Derivatives, July
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, July
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
Stodder: Derivatives, July
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, July
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, July
Run Simulation
• From Financial Models Using
Simulation and Optimization
by Wayne Winston.
Stodder: Derivatives, July
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
S0
Jump
Fat
Tails
0
Gaussian
T
si–1– si
Stodder: Derivatives, July
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, July
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
Stodder: Derivatives, July
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, July
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 ?
Stodder: Derivatives, July