Binomial Trees

Read Chapters 12 and 18 in Hull

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

Simple Binomial Tree

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

Simple Binomial Tree

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

Setup a riskless portfolio

  • Portfolio value
    • Risk-free interest rate is 10%
    • The riskless portfolio is:
      • long 0.6667 shares
      • short 1 call option
    • The future value 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
  • Note: A put portfolio has the following construction
    • Short Δ shares of stock
    • Long one put option
    • Creates a delta neutral portfolio

Generalizing the Binomial Tree

  • A derivative lasts for time T and depends on a stock price, S 0
    • Percentage up, u
    • Percentage down, d

Generalize binomial tree

  • Portfolio value
    • Long Δ shares
    • Short 1 derivative

Generalize riskless portfolio

  • The portfolio is riskless when
    • S 0 u Δ – ƒ u = S 0 d Δ – ƒ d
    • Solve for Δ

Option delta

  • Future Value of the portfolio at time T
    • FV = S 0 u Δ – f u
  • Present Value of the portfolio today
    • PV = (S 0 u Δ – f u )e –rT
  • Today's portfolio value
    • PV= S 0 Δ – f
  • Hence
    • PV = FV x discount
    • S 0 Δ - ƒ = (S 0 u Δ – ƒ u )e –rT
    • ƒ = S 0 Δ – (S 0 u Δ – ƒ u )e –rT
  • Substituting for Δ to obtain
    • ƒ = [ p ƒ u + (1 – p) ƒ d ] e –rT
  • where

Option delta

  • 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 Outcome 1 + (1 – p) Outcome 2
  • The derivative's value equals its expected payoff that is discounted at the risk-free rate

Binomial distribution

Risk-Neutral Valuation

  • The probability of an up and down movements are p and 1-p
    • The expected stock price at time T is S 0e rT
    • 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

Binomial distribution

  • Calculate the parameters to the tree
    • u = 28 / 25 = 1.12
    • d = 22 / 25 = 0.88
    • p = ( e rt - d ) / ( u - d ) = ( e 0.1x0.5 - 0.88 ) / ( 1.12 - 0.88 ) = 0.7136
  • Since p is a risk-neutral probability, then
    • You can solve for p using the formula
    • 25e 0.10x0.5 = 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

Two-step binomial tree

  • 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

Two-step binomial tree

  • 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

Two-step binomial tree

  • 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

Two-step binomial tree

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

Binomial tree

  • 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
    • e rΔt = pu + (1 – p) d
      • or S 0 e rΔt = p S 0 u + (1 – p) S 0 d
    • σ 2Δt = p u 2 + (1 – p)d 2 – [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

Binomial tree

Binomial tree

  • 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, r f
        • 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
    • S 0 = 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
  • Holders can exercise American and European at Time T, maturity
    • Node G, Payoff = 10.4039
    • Node H, Payoff = 3.3189
    • Node I, Payoff = 0.0000
    • Node J, Payoff = 0.0000
  • Since a holder can exercise an American option early, we must check each node
    • Node D
      • Intrinsic value = 7.4071
      • Can holder exercise American early, payoff = 81.7847 - 75 = 6.7847
      • Thus, holder waits since payoff is higher for intrinsic value
    • Node E
      • Intrinsic value = 1.9280
      • Can holder exercise American early, payoff = 75 - 75 = 0
      • Thus, holder waits since payoff is higher for intrinsic value
    • Node F
      • Intrinsic value = 0
      • Can holder exercise American early, payoff = 0
      • Thus, holder does not exercise early
    • Node B
      • Intrinsic value = 5.0949
      • Can holder exercise American early, payoff = 78.3189 -75 = 3.3189
      • Thus, holder does not exercise early since waiting has higher payoff
    • Node C
      • Intrinsic value = 1.1200
      • Can holder exercise American early, payoff = 0
      • Thus, holder does not exercise early
    • Node A
      • Intrinsic value = 3.4198
      • Can holder exercise American early, payoff = 0
      • Thus, holder does not exercise early
  • 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, or C = c

An example of pricing an American call option using a binomial tree

  • 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