Basic Numerical Procedure

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Basic Numerical Procedure
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Content

• 1 Binomial Trees
• 2 Using the binomial tree for options on
  indices,
• currencies, and futures contracts
• 3 Binomial model for a dividend-paying stock
• 4 Alternative procedures for constructing trees
• 5 Time-dependent parameters
• 6 Monte Carlo simulation
• 7 Variance reduction procedures
• 8 Finite difference methods

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Binomial Trees

• In each small interval of time (?t)the stock
  price is assumed to move up by a proportional
  amount u or to move down by a proportional amount
  d

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Risk-Neutral Valuation

• 1. Assume that the expected return from all
  traded assets is the risk-free interest rate.
• 2. Value payoffs from the derivative by
  calculating their expected values and discounting
  at the risk-free interest rate.

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Determination of p, u, and d

• Mean e(r-q)Dt pu (1 p )d
• Variance s2Dt pu2 (1 p )d 2 e2(r-q)Dt
• A third condition often imposed is u 1/ d

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• A solution to the equations, when terms of higher
  order than Dt are ignored, is

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Tree of Asset Prices

• At time i?t

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Working Backward through the Tree

• Example American put option
• S0 50 K 50 r 10 s 40
• T 5 months 0.4167
• Dt 1 month 0.0833
• The parameters imply
• u 1.1224 d 0.8909
• a 1.0084 p 0.5073

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Example (continued)
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Example (continued)

• In practice, a smaller value of ?t, and many more
  nodes, would be used. DerivaGem shows

steps 5 30 50 100 500
f0 4.49 4.263 4.272 4.278 4.283
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Expressing the Approach Algebraically
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Estimating Delta and Other Greek Letters

• delta(?)at time ?t

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• gamma(G) at time 2?t

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• theta(T)

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• Vega(?)
• Rho(?)

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Example
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Using the binomial tree for options on
indices, currencies, and futures contracts

• As with Black-Scholes
• For options on stock indices, q equals the
  dividend yield on the index
• For options on a foreign currency, q equals the
  foreign risk-free rate
• For options on futures contracts q r

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Example
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Example
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Binomial model for a dividend-paying
stock

• Known Dividend Yield
• before
• after
• Several known dividend yields

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Known Dollar Dividend

• i?k
• ik1
• ik2

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Simplify the problem

• The stock price has two componentsa part that is
  uncertain and a part that is the present value of
  all future dividends during the life of the
  option.
• Step 1A tree can be structured in the usual way
  to model .
• Step 2By adding to the stock price at each
  nodes, the present value of future dividends, the
  tree can be converted into model S.

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Example
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Control Variate Technique

• 1. Using the same tree to calculate both the
  value of the American option( )and the value of
  the European option( ).
• 2. Calculating the Black-Scholes price of the
  European option( ).
• 3. This gives the estimate of the value of the
  American option as

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Example

• B-S model
• ?

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Alternative procedures for constructing
trees

• Instead of setting u 1/d we can set each of the
  2 probabilities to 0.5 and

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Example
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Trinomial Trees
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Adaptive mesh model(Figlewski and Gao,1999)
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Time-dependent parameters
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Monte Carlo simulation

• When used to value an option, Monte Carlo
  simulation uses the risk-neutral valuation
  result. It involves the following steps
• 1. Simulate a random path for S in a risk neutral
  world.
• 2. Calculate the payoff from the derivative.
• 3. Repeat steps 1 and 2 to get many sample values
  of the payoff from the derivative in a risk
  neutral world.
• 4. Calculate the mean of the sample payoffs to
  get an estimate of the expected payoff.
• 5. Discount this expected payoff at risk-free
  rate to get an estimate of the value of the
  derivative.

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Monte Carlo simulation (continued)

• In a risk neutral world the process for a stock
  price is
• We can simulate a path by choosing time steps of
  length ?t and using the discrete version of this
• where e is a random sample from f (0,1)

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Monte Carlo simulation (continued)
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Derivatives Dependent on More than One Market
Variable

• When a derivative depends on several underlying
  variables we can simulate paths for each of them
  in a risk-neutral world to calculate the values
  for the derivative

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Generating the Random Samples from Normal
Distributions

• How to get two correlated samples e1 and e2 from
  univariate standard normal distributions x1 and
  x2?

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Cholesky decomposition
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Number of Trials

• Denote the mean by µ and the standard deviation
  by ?.
• The standard error of the estimate is
• where M is
  the number of trials.
• A 95 confidence interval for the price f of the
  derivative is
• To double the accuracy of a simulation, we must
  quadruple the number of trials.

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Applications

• Advantage
• 1. It tends to be numerically more efficient
  (increases linearly)than other procedures(
  increases exponentially)when there are more
  stochastic variables.
• 2. It can provide a standard error for the
  estimates.
• 3. It is an approach that can accommodate
  complex payoffs and complex stocastic processes.

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Applications (continued)

• An estimate for the hedge parameter is
• Sampling through a Tree

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Variance reduction procedures

• Antithetic Variable Techniques
• standard error of the estimate is
• Control Variate Technique

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Variance reduction procedures
(continued)

• Importance Sampling
• Stratified Sampling
• Moment Matching
• Using Quasi-Random Sequences

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Finite difference methods

• Define ƒi,j as the value of ƒ at time iDt when
  the stock price is jDS
• ?TT/N ?SSmax /M

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Implicit Finite Difference Method

• Forward difference approximation
• backward difference approximation

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Implicit Finite Difference Methods(continued)
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Implicit Finite Difference Methods(continued)
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Implicit Finite Difference Methods(continued)
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Explicit Finite Difference Methods
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Explicit Finite Difference Methods(continued)
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Explicit Finite Difference Methods(continued)
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Difference between implicit and explicit finite
difference methods
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Change of Variable
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Change of Variable (continued)
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Relation to Trinomial Tree Approaches
The three probabilities sum to unity.
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Relation to Trinomial Tree Approaches (continued)
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Other Finite Difference Methods

• Hopscotch method
• Crank-Nicolson scheme
• Quadratic approximation

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Summary

• We have three different numerical procedures for
  valuing derivatives when no analytic solution
  trees, Monte Carlo simulation, and finite
  difference methods.
• Trees derivative price are calculated by
  starting at the end of the tree and working
  backwards.
• Monte Carlo simulation works forward from the
  beginning, and becomes relatively more efficient
  as the number of underlying variables increases.
• Finite difference method similar to tree
  approaches. The implicit finite difference method
  is more complicated but has the advantage that
  does not have to take any special precautions to
  ensure convergence.