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# Outline

• A Simple Binomial Model
• Generalizing the Binomial Tree
• Risk-Neutral Valuation
• Two-Step Binomial Tree Examples
• Binomial Trees in Practice
• An American Call Example
• Calculating the Greeks

# A Simple Binomial Model

• A stock price is currently \$25
• In three months, it will be either \$28 or \$22

• A 6-month call option on the stock has a strike price of 24.

• Set up a riskless portfolio
• We create the portfolio
• Long Δ shares
• Short 1 call option
• Refer to binomial tree below
• Portfolio is riskless when
• 28 Δ – 4 = 22 Δ – 0
• Δ = 0.6667
• Both scenarios yield the same portfolio value

• Portfolio value
• Risk-free interest rate is 10%
• The riskless portfolio is:
• long 0.6667 shares
• short 1 call option
• The futurevalue of the portfolio in 6 months is
• FV = 28 Δ – 4 = 28x0.6667 – 4 = \$14.6676
• The value of the discount
• discount = e–0.10x0.5 = 0.9512
• The value of the portfolio today
• PV= \$25 Δ – f = \$25x0.6667 – f = \$16.6675 – f
• The option's value equals
• PV = FV x discount
• \$16.6675 – f = \$14.6676 x 0.9512
• f = 2.7157

# Generalizing the Binomial Tree

• A derivative lasts for time T and depends on a stock price, S0
• Percentage up, u
• Percentage down, d

• Portfolio value
• Long Δ shares
• Short 1 derivative

• The portfolio is riskless when
• S0 u Δ – ƒu = S0 d Δ – ƒd
• Solve for Δ

• Future Value of the portfolio at time T
• FV = S0 u Δ – fu
• Present Value of the portfolio today
• PV = (S0 u Δ – fu )e–rT
• Today's portfolio value
• PV= S0 Δ – f
• Hence
• PV = FV x discount
• S0 Δ - ƒ = (S0 u Δ – ƒu )e–rT
• ƒ = S0 Δ – (S0 u Δ – ƒu )e–rT
• Substituting for Δ to obtain
• ƒ = [ p ƒu + (1 – p) ƒd ] e–rT
• where

• Refer to handout for derivation
• Define probability, p, as going up while 1 − p is probability of moving down
• Binomial distribution – two outcomes: Up or down
• Average = p Outcome1 + (1 – p) Outcome2
• The derivative's value equals its expected payoff that is discounted at the risk-free rate

## Risk-Neutral Valuation

• The probability of an up and down movements are p and 1-p
• The expected stock price at time T is S0erT
• Thus, the stock price earns the risk-free rate
• We calculate the average of the outcomes
• Binomial trees show how to value a derivative as the expected return of the underlying asset is assumed to earn the risk-free rate
• Thus, we can discount using the risk-free rate, which we call risk-neutral valuation
• Irrelevance of Stock's Expected Return
• When we value an option in terms of the underlying stock, the expected return on the stock is irrelevant
• Returning to the original example

• Calculate the parameters to the tree
• u = 28 / 25 = 1.12
• d = 22 / 25 = 0.88
• p = ( ert - d ) / ( u - d ) = ( e0.1x0.5 - 0.88 ) / ( 1.12 - 0.88 ) = 0.7136
• Since p is a risk-neutral probability, then
• You can solve for p using the formula
• 25e0.10x0.5 = 28 p + 22 ( 1 – p )
• p = 0.7136
• The value of the option is
• f = [ 0.7136x4 + 0.2864x0 ] e–0.10x0.5
• f = 0.2.7152

# Two-Step Binomial Tree Examples

• Refer to the tree below
• The tree represents a European call option
• K=26
• Each time step is 3 months
• Discount rate, r =10%
• Probability, p = 0.7091
• u = 1.0583
• d = 0.9449

• Calculate the payoff for each node
• Determine whether you can exercise or not
• Node D:
• payoff = max ( 28 - 26, 0 ) = 2.0000
• Node E:
• payoff = max ( 25 - 26, 0 ) = 0.0000
• Node F:
• payoff = max ( 22.3214 - 26, 0 ) = 0.0000
• The other nodes, we use the discounted binomial average
• You cannot exercise an European option at these nodes
• You are calculating the intrinsic value
• Node B
• f = ( 0.7091 x 2.0000 + 0.2909 x 0.00) e–0.10x0.25 = 1.3832
• Node C
• f = ( 0.7091 x 0.00 + 0.2909 x 0.00) e–0.10x0.25 = 0.00
• Node A
• f = ( 0.7091 x 1.3832 + 0.2909 x 0.00) e–0.10x0.25 = 0.9566

• Refer to the tree below
• The tree represents a European put option
• K = 160
• Each time step is 3 months
• Discount rate, r =10%
• Probability, p = 0.7270
• u = 1.0541
• d = 0.9487
• Calculate the payoff for each node
• Determine whether you can exercise or not
• Node D:
• payoff = max ( 160 - 166.6667, 0 ) = 0.0000
• Node E:
• payoff = max ( 160 - 150, 0 ) = 10.0000
• Node F:
• payoff = max ( 160 - 135, 0 ) = 25.0000
• The other nodes, we use the discounted binomial average
• You cannot exercise an European option at these nodes
• You are calculating the intrinsic value
• Node B
• f = ( 0.7270 x 0.00 + 0.2730 x 10.00) e–0.10x0.25 = 2.6627
• Node C
• f = ( 0.7270 x 10.00 + 0.2730 x 25.00) e–0.10x0.25 = 13.7471
• Node A
• f = ( 0.7270 x 2.6627 + 0.2730 x 13.7471) e–0.10x0.25 = 5.5483

• How to use binomial tree to calculate the price of an American option
• Using the put example from the last example
• A holder can exercise the American put at maturity just like the European
• So Nodes D, E, and F stay the same
• A holder can exercise an American option anywhere on the tree
• Node B
• Intrinsic value: f = ( 0.7270 x 0.00 + 0.2730 x 10.00) e–0.10x0.25 = 2.6627
• Option value if holder does not exercise
• Exercise early: payoff = max ( 160 - 158.1139, 0 ) = 1.8861
• Holder goes for the higher value, which means he does not exercise
• Node C
• Intrinsic value: f = ( 0.7270 x 10.00 + 0.2730 x 25.00) e–0.10x0.25 = 13.7471
• Exercise early: payoff = max ( 160 - 142.3025, 0 ) = 17.6975
• Holder earns greater value by exercising early
• Node A
• Intrinsic value: f = ( 0.7270 x 2.6627 + 0.2730 x 17.6975) e–0.10x0.25 = 5.5483
• Note, Node C caused a number change in the payoff
• Exercise early: payoff = max ( 160 - 150.0000, 0 ) = 10.0000
• Holder exercises immediately earns a payout of 10
• If he or she waits, he or she only earns 5.5483

# Binomial Trees in Practice

• Analysts frequently use binomial trees to approximate the price movements of a stock or other asset
• In each small interval of time, the stock price is assumed to
• move up by a proportional amount u
• or to move down by a proportional amount d
• Movements in Time Δt

• Risk-Neutral Valuation
• We choose the tree parameters p, u, and d so the tree gives correct values for the mean and standard deviation of the stock price changes in a risk-neutral world
• Two conditions are
• erΔt = pu + (1 – p) d
• or S0 erΔt = p S0 u + (1 – p) S0 d
• σ2Δt = p u2 + (1 – p)d2 – [p u + (1 – p) d ]2
• where σ is the volatility
• The following condition is often imposed
• u = 1/ d
• Cox, Ross, and Rubinstein (1979) used this approach
• Be careful on exams whether this condition holds
• For a small time step, Δt, a solution to the equations is

• Backwards Induction
• We know the value of the option at the final nodes
• We work back through the tree using risk-neutral valuation to calculate the value of the option at each node
• For European options, keep working backwards
• For American options, you must check whether you can exercise at each node
• Choose greater payoff
• Changing the binomial tree for different cases
• Constructing the tree remains the same
• Only the probability changes
• a = e(r - q)Δt
• Four cases
• For plain vanilla (ice-cream) options, q = 0
• a = e(r - 0)Δt
• If a stock price pays continuous dividends at rate q
• For options on stock indices, q equals the dividend yield on the index
• a = e(r - q)Δt
• For options on a foreign currency, q equals the foreign risk-free rate, rf
• a = e(r - rf)Δt
• For options on futures contracts q = r
• a = e(q - q)Δt = 1

# An American Call Example

• Parameters for example
• S0 = 75
• K = 75
• r =10%
• σ = 15%
• T = 3 months or 0.25 year
• Time step: Δt = 1 month or 1 /12 = 0.0833 year
• Calculate the parameters
• u = 1.0443
• d = 0.9576
• p = 0.5858
• 1 – p = 0.4142
• discount = e–rt = 0.9917
• The top number at each node is the stock price
• The value of the option is below the stock price
• The holder does not exercise the American option early at any nodes
• So, in this case, the American call option equals the European call option

• Increasing the Time Steps
• A tree should have 30 time steps or more to give good option values
• DerivaGem calculates up to 500 time steps
• The Black-Scholes-Merton Model
• We can derive the Black-Scholes-Merton model by looking at what happens to the price of a European call option as the time step tends to zero
• Means the number of branches increases to infinity
• The binomial tree for European options converges to Black Scholes

# Calculating the Greeks

• Calculating delta
• Delta means how much the value of the option changes if the spot price increases by \$1
• Delta (Δ) is the ratio of the change in the price of a stock option to a one-unit change in the price of the underlying stock
• Once can calculate delta at any adjacent nodes
• The value of Δ varies from node to node
• At node B, stock price = 78.3189 and call price = 5.0949
• At node C, stock price = 71.8217 and call price = 1.1200
• delta = ( 5.0949 – 1.1200 ) / ( 78.3189 – 71.8217 ) = 0.6118
• Delta hedging – used delta to make our portfolio riskless
• Calculating Gamma
• Gamma – how delta changes to a one \$1 increase in the spot price
• Gamma requires two deltas from the same time point
• Calculate the following from Nodes D, E, and F
• delta 1 = 0.8076
• delta 2 = 0.3099
• denominator = average ( 81.7847 75.0000, 75.0000 68.7781 ) = 6.5033
• gamma = ( 0.8076 0.3099 ) / 6.5033 = 0.0765
• If the stock prices rises by \$1, then delta increases by 0.0765
• Calculating Theta
• Theta is how the option's value changes if you increase the time by one unit
• Theta is calculated from the central nodes at 0 months and 2 months
• That way, the stock price remains constant
• Only the time and option prices differ
• Denominator starts at 0 and ends at 2 months (or 0.1667 year)
• theta = ( 1.9280 3.4198 ) / ( 0.1667 0 ) = –8.9506
• If time increases by one year, then the option value decreases by 8.9506
• Calculating Vega
• Vega is is how the option's value change if volatility increases by one unit
• We use the following procedure
• From the original tree, volatility equals 15% and the option price equals 3.4198
• Construct a new tree with a volatility of σ = 16%
• Option price equals 3.5744
• vega = ( 3.5744 3.4198 ) / 1 = 0.1546
• A 1% increase in volatility raises the option's value by approximately 0.1516.
• More volatility means the holder will more likely exercise the option
• Note: Delta, Gamma, Theta, and Vega apply to Black Scholes as well.
• Same ideas but calculated differently

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