Real Estate Finance Australia

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Real Estate FinanceAustralia

• Investment Management Finance 5B
• Tutorial One Presentation August 2007
• Jasmine Lee
• Course Leader
• Brian Ang
• The Royal Bank of Scotland

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Agenda

• Forwards and Futures
• Binomial Model
• Black Scholes Model
• Monte Carlo Simulation
• Parameter Estimation
• Constant Proportion Portfolio Insurance Technique
• Delta Hedging
• Study Tips

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Forwards and Futures
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Forward Contracts
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Futures Contracts
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Binomial Model
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Binomial Model

• Share Price Model
• Assumes stock price follows the binomial
  distribution, set out as binomial branch model
  where the stock price increases or decreases over
  any period.
• Risk Neutral Pricing
• Option price based on arbitrage free condition
• The above portfolio is a risk less portfolio
  (arbitrage free).

Time 0 Set up portfolio comprising units of
share and bonds Time 1 Share price
increases or decreases and therefore and

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Binomial Model (Cont.)

• Risk Neutral Pricing
• Hence it must earn the risk free rate of return,
  ie
• The measure p is the risk neutral probability. It
  is not a real probability.
• Risk neutral probabilities exist only if there
  are no risk-less arbitrage opportunities.

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Black Scholes Model
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Black-Scholes Model Assumptions

• Share Price Model
• Share price follows the Geometric Brownian
  motion
• Share price has a log-normal distribution
• Share price return has a normal distribution
• Risk Neutral Pricing
• When pricing options, we assume a risk-neutral
  world or no-arbitrage condition similar to the
  binomial model.
• No-arbitrage condition has the impact of changing
  the growth rate or the drift term of the process
  followed by the share price as follows
  The term U is the expected
  return in the risk neutral world, ie r the risk
  free rate.
• This is analogous to the binomial model. In the
  binomial model, a change in the probability
  measure (to a risk neutral) results in adjustment
  to the level of increase/decrease in the expected
  stock price.

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Black-Scholes Model (Cont.)

• Black-Scholes Model
• Option value calculated by integrating the
  pay-off within the relevant limits. For call
  option
• f(S) is the probability density function of the
  log-normal distribution with risk neutral
  parameters
• Note this is not the expected value of the
  option, but is the risk neutral value, ie the
  arbitrage free price

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Monte Carlo Simulation
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Approach

• Monte Carlo Simulation
• Procedure involves simulating random asset paths
  up to expiry
• At expiry compute option payoff for each path
• Option value is the discounted average of these
  pay-offs

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Steps
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Option Pricing Methods

• Monte Carlo Simulation
• Problems with Monte Carlo Simulation
• Slow compared to numerical analytical methods
• American style derivatives cannot be easily
  handled
• Advantages of Monte Carlo Simulation
• Can handle complex path-dependency
• Can handle large number of underlying assets
• e.g. basket options

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Parameter Estimation
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Parameter Estimation
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Parameter Estimation
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Parameter Estimation

• VOLATILITY
• Estimate volatility going forward
• Look at
• Historical volatility
• Implied volatility
• Volatility implied by the options market
• Judgement call

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Parameter Estimation

• Dividend
• Estimate dividends going forward
• Discrete dividends (dollar dividends)
• Dividend yield (percentage of future spot price)
• Source of information
• Historical dividends
• Company reports
• Company guidance
• Analyst reports
• Judgement call

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Parameter Estimation

• YIELD CURVE
• Estimate spot yield curve
• Look at pages on Reuters
• Overnight rate
• BBSW
• Bank bill futures
• Swap spreads
• Bond futures

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Constant Proportion Portfolio Insurance
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CPPI

• Model
• CPPI is an asset allocation technique
• CPPI is used when the Risky Asset is illiquid or
  can not be shorted.
• CPPI model takes the following general form
• Bond Floor (t) PV of Protected Amount
• Distance (t) NAV(t) Bond Floor
  (t)/NAV(t)
• NAV (t) Risky Asset Risk Free Asset
• Multiplier M
• Risky percentage MDistance (t)
• Risk free percentage 100 - MDistance (t)

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Delta Hedging
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Delta Hedging

• APPROACH
• Option payoff replication
• Portfolio consisting of cash and D underlying
  asset
• No arbitrage principle leads to Fair Value
• Can use replicating portfolio to hedge asset
  exposure
• Cost of hedging is the fair value of the option

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Delta Hedging

• IMPLICATIONS
• For a delta hedged long option position we profit
  on both rises and falls in asset price.
• Why? We can answer that question by looking at
  two typical hedging trades.
• One important consideration in our analysis will
  be that, as we are long a European call we are
  also long gamma.

Premium
9.57
4.57
-0.43
80.7
101.4
122.1
-5.43
Asset price ()
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Delta Hedging
GREEKS We are long delta. Therefore we will
hedge by selling the underlying asset in order to
obtain an overall zero-delta position.
1. Asset price has increased. As were long
gamma - delta increases.
Asset price
Time
3. Asset price has decreased. As were long
gamma - delta decreases.
Delta
2. We are short the asset. As delta has
increased we need to be more short. We sell the
asset.
4. We are short the asset. As delta has
decreased we need to be less short. We buy the
asset.
Time
Notice that we sell on rises and buy on falls.
So on both asset price rises and falls we make a
gapping profit. Have we discovered a
money-making machine?
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Delta Hedging
GREEKS Sadly - no. Most of what we gain on
gamma gapping we must expect to lose on theta
decay of our long option position. In other
words Greeks are interdependent. A balance
equation describes how the Greeks depend on each
other.
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Delta Hedging
THE BALANCE EQUATION
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Delta Hedging
THE BALANCE EQUATION
Note that the above applies only to
non-path-dependent options. So the equation
doesnt apply to options like American options
and average rate options.
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Delta Hedging
THE BALANCE EQUATION
Another way of looking at the balance equation is
to look at the non-Greek terms.
These terms can be thought of as being
normalising factors which scale Greeks from
their various unique dimensions to a common /day
scale.
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Delta Hedging
THE BALANCE EQUATION
The balance equation is true for a portfolio of
options and assets In the case where we
continuously delta hedge, and the portfolio delta
stays zero we can simplify
So there is a direct relationship between making
money on theta and losing money on gamma (other
things being constant).
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Delta Hedging
THE BALANCE EQUATION
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Study Tips
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Study Tips

• Study timetable for the course
• Discussion Forum
• Study groups
• Attempt questions in the prescribed texts
• Review difficult material
• Attempt past exam papers

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Questions