portfolio insurance

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portfolio insurance

• another use of options eliminate extreme
  downside risk (at the cost of eliminating some of
  the potential for gain)
• 2 ways to do portfolio insurance
• buy a put and hold it to maturity
• trade the portfolio so as to duplicate the effect
  of buying a put and hedging it

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Buying a put
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• Now lets look at the current value of the same
  portfolio, i.e., its value as a function of the
  current stock price.
• Uninsured portfolio looks the same.
• However, the current value of the put is a smooth
  function of stock price, so the current value of
  the insured portfolio is smooth as well.

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Add these together

• You get the value of the insured portfolio as a
  function of current stock price

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who sells the puts?

• Anyone who want to investment banks
• They need to hedge their risk too
• How do they do this?
• Selling the stock short

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what happens in practice?

• If stock price drops, the hedge ratio rises
• insurer must sell short more shares to stay
  hedged.

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The second way to do portfolio insurance

• combine the functions of the investor and insurer
• synthetic puts -- trade the position so as to
  duplicate the effect of a put, together with its
  hedge
• portfolio insurance is equivalent to gradually
  selling out a position as the price drops.
• usually index futures are used for this.

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Effect of portfolio insurance on market stability

• at first glance, this depends on whether the
  insured investor insures via a real put or a
  synthetic put
• if a real put, the insured investor doesnt have
  to do anything -- the put value rises as the
  market drops.
• If a synthetic put, the insured investor sells as
  the market drops destabilizing.

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However

• with a real put, the portfolio insurer needs to
  rebalance his hedge.
• As the price drops, the hedge ratio rises the
  insurer must take a bigger short position in
  stock to hedge the puts he has written.
• So hes selling as the market drops
  destabilizing.

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• In fact, when you aggregate the behavior of the
  insured and the insurer, synthetic portfolio
  insurance is equivalent to real portfolio
  insurance both involve selling as the market
  drops, buying as market rises
• Thererfore both are destabilizing!
• Portfolio insurance may have been a major factor
  in the 1987 and 1997 selloffs.

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A misnomer?

• It looks like portfolio insurance is not really
  insurance
• One party is just shifting the risk onto someone
  else
• Someone has to be long the stock thus the risk
  of a drop must be borne by someone.

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Black-Scholes Model

• example p. 664.
• S 100
• X 95
• r .1
• T .25
• sigma .5
• Black-Scholes value of call 13.70

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Binomial model

• stock price can take on only 2 values in the
  future (i.e., at the exercise date of the
  option).
• Then its easy to find a portfolio of stock and
  the risk-free security that will duplicate the
  payoff on a call.
• The current value of this portfolio must equal
  the value of the call -- otherwise theres
  arbitrage.

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binomial model
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• How do you choose the parameters of the binomial
  model?
• you need to make assumptions about the mean
  growth rate of the stock and its standard
  deviation.
• In the example, these parameters are already
  given

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• choose the mean growth rate to equal the interest
  rate.
• This is .1 per year, or .025 per quarter (simple
  interest).
• Standard deviation is .5 per year, so variance is
  .25 per year.

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• variance per quarter is .25/4 .0625.
• standard deviation per quarter is the square root
  of variance, or 0.25.
• so under the binomial model, the stock price in
  the next quarter is 100 plus 2.5 plus or minus 25
• 127.5 if up, or 77.5 if down.

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binomial model -- stock
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solve for duplicating portfolio

• 32.5 127.5 s 1.025 b
• 0 77.5 s 1.025 b
• s .65 (hedge ratio)
• b -49.1

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• cost of hedge .65 100 - 49.1 15.9
• If you believe the model, this has to be the
  price of the call.
• otherwise you can earn an arbitrage profit.
• compare Black-Scholes price of 13.7

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This is very crude

• but you can make it more sophisticated.
• Suppose that there are 2 jumps instead of one.
  Then the stock can take on 3 values rather than
  2.
• You can still price the call by going backwards.
• Suppose 4 instead of 2.
• In the limit, you get the Black-Scholes model

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