Binomial Trees
Read Chapters 12 and 18 in Hull
Outline
 A Simple Binomial Model
 Generalizing the Binomial Tree
 RiskNeutral Valuation
 TwoStep 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 6month 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
 Riskfree 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, S_{0}
 Percentage up, u
 Percentage down, d
 Portfolio value
 Long Δ shares
 Short 1 derivative
 The portfolio is riskless when
 S_{0} u Δ – ƒ_{u} = S_{0} d Δ – ƒ_{d}
 Solve for Δ
 Future Value of the portfolio at time T
 Present Value of the portfolio today
 PV = (S_{0} u Δ – f_{u} )e^{–rT}
 Today's portfolio value
 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
 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 riskfree rate
RiskNeutral Valuation
 The probability of an up and down movements are p and 1p
 The expected stock price at time T is S_{0}e^{rT}
 Thus, the stock price earns the riskfree 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 riskfree rate
 Thus, we can discount using the riskfree rate, which we call riskneutral 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 = ( e^{rt}  d ) / ( u  d ) = ( e^{0.1x0.5}  0.88 ) / ( 1.12  0.88 ) = 0.7136
 Since p is a riskneutral 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
TwoStep Binomial Tree Examples
 Refer to the tree below
 The tree represents a European call option
 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
 RiskNeutral 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 riskneutral 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^{2}Δt = 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
 Backwards Induction
 We know the value of the option at the final nodes
 We work back through the tree using riskneutral 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
 Changing the binomial tree for different cases
 Constructing the tree remains the same
 Only the probability changes
 Four cases
 For plain vanilla (icecream) options, q = 0
 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 riskfree rate, r_{f}
 For options on futures contracts q = r
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
 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 BlackScholesMerton Model
 We can derive the BlackScholesMerton 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 oneunit 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
