Hedging

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Hedging
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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Hedging
Hedging is about the reduction of risk.
We will consider dynamic hedging in which a
portfolio is dynamically traded in order to
reduce risk.
Simply put, a portfolio is hedged against a
certain risk if the portfolio value is not
sensitive to that risk.
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The Basic Idea
Choose the amounts of the other assets, a1...a2,
in order to eliminate the risk in the
portfolio.
Itos lemma will tell us how much risk the
portfolio has over the next instantaneous dt.
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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Example Hedging a call option with the
underlying stock
We are long the option and would like to hedge
our risk with the stock.
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Example Hedging a call option with the
underlying stock
Underlying stock
Option
We are long the option and would like to hedge
our risk with the stock.
Portfolio
Portfolio change
The portfolio is hedged over the next
instantaneous dt.
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Black-Scholes
Provided we can trade continuously, we have
formed a riskless portfolio
Since this is riskless, it must earn the risk
free rate
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Example Hedging an interest rate derivative
We are long B1 and would like to hedge with B2.
Portfolio change
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Example Hedging an interest rate derivative
Short rate
Asset 1
Asset 2
We are long B1 and would like to hedge with B2.
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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Hedging assets with Poisson jumps
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Hedging assets with Poisson jumps
Portfolio change
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Hedging assets with Poisson jumps
This hedge doesnt really work so well. With big
jumps in assets values, we cant exactly say that
we have eliminated risk.
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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Complete versus Incomplete markets
Broadly speaking, a complete market is one in
which you can replicate any desired payoff by
trading assets in the market.
A market is incomplete when this is not possible.
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Incompleteness generally comes from two main
sources

• There are not enough assets in the market to
  span the uncertainty. (An example would be
  standard stochastic volatility.)
• Trading strategies are limited or not ideal
• discrete trading
• transaction costs, short selling constraints, etc.

When we cannot replicate a payoff perfectly, we
cannot argue for a unique price determined by a
replicating portfolio.
Any price will implicitly depend on risk
preferences. This is why we saw the market price
of risk emerging in certain problems. The market
was incomplete.
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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Example Hedging a call option with the
underlying stock
So lets set the partial derivative of the
portfolio with respect to S equal to zero.
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Example Hedging an interest rate derivative
Set the partial derivative of the portfolio with
respect to r equal to zero
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More generally, we can look at the sensitivity of
a portfolio over time Dt through a Taylor
expansion
Approach eliminate as many random terms as
possible
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Delta hedged call option
Still left with higher order risk...
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Delta hedge in pictures
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A delta-gamma hedge
A Taylor Expansion
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A delta-gamma hedge
We have a derivative which is a function of a
factor
We will hedge the stock and another derivative on
it
and
Eliminate DS and (DS)2 terms
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Delta-Gamma hedge in pictures
Current Price S 10, Risk Free Rate r 0.05
Hedged Call Option Parameters (K10, T0.2,
s0.3)
2nd Call Option Parameters (K8, T0.4, s0.25)
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Delta-Gamma hedge in pictures
Current Price S 10, Risk Free Rate r 0.05
Hedged Call Option Parameters (K10, T0.2,
s0.3)
2nd Call Option Parameters (K8, T0.4, s0.25)
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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The Greeks
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European Call Option Price
(S10, K10, T0.2, r0.05, s0.2)
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European Call Option Delta
(S10, K10, T0.2, r0.05, s0.2)
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European Call Option Gamma
(S10, K10, T0.2, r0.05, s0.2)
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European Call Option Theta
(S10, K10, T0.2, r0.05, s0.2)
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European Call Option Rho
(S10, K10, T0.2, r0.05, s0.2)
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European Call Option Vega
(S10, K10, T0.2, r0.05, s0.2)
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Just as we set up Delta and Delta-Gamma hedges,
we can hedge against changes in other parameters.
However, normally it is difficult and expensive
to try to hedge all these parameters. Instead,
you might delta hedge and monitor the other
Greeks, only taking action when needed.
In general, the Greeks provide an important
summary of the sensitivity of a portfolio to
underlying uncertainties.
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Basic Idea Hedging under Ito Processes Hedging
Poisson Jumps Complete vs. Incomplete
markets Delta and Delta-Gamma hedges Greeks and
Taylor expansions Complications with Hedging
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Options with discontinuous payoffs tend to be
very difficult to hedge, especially close to the
discontinuities.
The problem is that the holdings in the hedged
portfolio are extremely sensitive to small
changes in the underlying asset.
A digital option provides a good example of
this...
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Digital Option Price
T0.01
T0.05
(S10, K10, r0.05, s0.25)
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Digital Option Delta
T0.01
T0.05
(S10, K10, r0.05, s0.25)
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Digital Option Gamma
T0.01
T0.05
(S10, K10, r0.05, s0.25)
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The End!