Subject 3. The one-period binomial model

In finance, the binomial options model provides a generalisable numerical method for the valuation of options. The model differs from other option pricing models, in that it uses a "discrete-time" model of the varying price over time of financial instruments; the model is thus able to handle a variety of conditions for which other models cannot be applied. Essentially, option valuation here is via application of the risk neutrality assumption over the life of the option, as the price of the underlying instrument evolves.

The binomial pricing model uses a "discrete-time framework" to trace the evolution of the option's key underlying variable via a binomial lattice (tree), for a given number of time steps between valuation date and option expiration. Each node in the lattice, represents a possible price of the underlying, at a particular point in time. This price evolution forms the basis for the option valuation. The valuation process is iterative, starting at each final node, and then working backwards through the tree to the first node (valuation date), where the calculated result is the value of the option.

Option valuation using this method is, as described, a three step process:

  • price tree generation.
  • calculation of option value at each final node.
  • progressive calculation of option value at each earlier node; the value at the first node is the value of the option.

We start off by having one binomial period for a European call option.

Define:

  • r = risk-free rate
  • c+ = Max (0, S+ - X) (call price if the stock price goes up: "up state")
  • c- = Max (0, S- - X) (call price if the stock price goes down: "down-state")
  • u = (S+ / S0) ("up state" price relative)
  • d = (S- / S0) ("down-state" price relative)
  • ST = Stock price at time (T)
  • π = Greek small letter pi.

Formulas:

  • π = (1 + r - d) / (u - d) (risk-neutral "up" probability)
  • c0 = [π c+ + (1 - π) c-] / (1 + r) (the price of the call option)
  • n = (c+ - c-) / (S+ - S-) (the hedge ratio: the number of shares of stock per option to hedge).

We assume that the stock price will only take two possible values at the expiration date of the option. In our example:

  • Current stock price = S0 = $80
  • Stock price at expiration = $90 (S+) or $75 (S-)
  • Exercise price of call option = $85 (X)
  • Time to expiration = T = 1/2 year (6 months)
  • Risk-free rate of return = r = 6% (discrete and annual)

Step 1: Diagram Stock Price Dynamics and Option Values on "Trees"

Based on this information, tree diagrams for the stock value and call option payoffs (state dependent) would be drawn as follows:

Step 2: Compute Risk Neutral Probabilities of Up and Down States

μ = (90/80) = 1.125
d = (75/80) = 0.9375
π = [(1.06)0.5 - 0.9375] / (1.125 - 0.9375) = 0.4912.

Step 3: Compute Expected Value of Call Option

c0 = [0.4912 $5 + (1 - 0.4912) $0] / (1.06)0.5 = $2.385.

Therefore, today's value of the 1-period option is $2.385.

Alternatively, we can use combination of Stocks and Calls to create state-independent payoffs and then determine the no-arbitrage value of the option rights.

Step 1: Calculate the hedge ratio (shares per call)

n = ($5 - $0) / ($90 - $75) = 0.3333.

Step 2: Use the hedge ratio to construct a portfolio of stocks and calls in which terminal payoff is state-independent. Let TV denote the terminal value of the portfolio at expiration.

  • If stock price = S-, TV- = n S- - c-.
  • If stock price = S+, TV+ = n S+ - c+.
  • TV- = TV+.

Regardless of which way the underlying moves, the portfolio value should be the same (perfectly hedged).

Continuing from the previous example:

To form a perfectly hedged portfolio, an investor needs to buy 1/3 of a share of stock for each call that written (sold), or buy 1 share of stock and sell (write) 3 calls. To see that this is true, consider the position of the portfolio at the expiration of the call if the investor writes 1 call:

Note that each call is worth $5 to the buyer or holder, so the seller of the call is in a negative position.

Step 3: Value the Option

Guaranteed outcome is $25 for this portfolio, regardless of value of the stock at expiration. How much should you pay for this risk-free position?

The present value of the guaranteed $25 to be received in six months is: $25 / 1.060.5 = $24.28. To guarantee the $25 outcome, investor would have to buy 1/3 share of the stock and sell 1 call option.
$24.28 = n S0 - c0 = initial investment ⇒ c0 = $80/3 - $24.28 = $2.38.

This is the same value we ended up with using direct approach.

Suppose the option of the previous example is selling for $3 - a clear case of price not equaling value. Investors would exploit this opportunity by selling the option and buying the underlying. The number of units of the underlying purchased for each option sold would be the hedge ratio: n = (c+ - c-) / (S+ - S-) = 0.3333. Suppose we sell 300 calls and buy 100 shares. The initial outlay would be 100 x $80 - 300 x $3 = $7100. Six months later, the portfolio value will be:

  • 100 x $75 - 0 = $7500, if stock price = $75.
  • 100 x $90 - 300 x $5 = $7500, if stock price = $90. Note the 300 here is the number of options bought, not the hedge ratio.

Our six-month return is 7500/7100 - 1 = 5.63%, and the annualized return is (1.0563)2 - 1 = 11.58%. This risk-free return is much higher than the actual risk-free return of 6%.

If the option sells for less than $2.38, an investor would buy the option and sell short the underlying, which would generate cash upfront. At expiration, the investor would have to pay back an amount less than 7%. All investors would perform this transaction, generating a demand for the option that would push its price back to $2.38.

Therefore, when the option is trading at the price given by the model, a hedge portfolio would earn the risk-free rate.

If the option is a put, please note the following differences:

  • Hedge ratio n = (p- - p+) / (S+ - S-).
  • A risk-free hedge has the same positions in the two instruments (underlying and the put).

Practice Question 1

Assume a stock price is $55 and in the next year it will either rise by 20% or fall by 16%. The risk-free interest rate is 5%. A call option on this stock has an exercise price of $60. What is the price of a call option that expires in one year?
Correct Answer: $3.33

We have μ = 1.2 and d = 0.84.
π = (1.05 - 0.84) / (1.2 - 0.84) = 0.5833.
S+ = 55 x 1.2 = $66.
S- = 55 x 0.84 = $46.2
c+ = Max (0, 66 - 60) = $6.
c- = Max (0, 46.2 - 60) = $0.
c = (0.5833 x $6 + 0.4167 x $0) / 1.05 = $3.33.

Practice Question 2

Continue with question 1. Assume a stock price is $55 and in the next year it will either rise by 20% or fall by 16%. The risk-free interest rate is 5%. A call option on this stock has an exercise price of $60. What would you hold to form a risk-free portfolio if the call option is selling for $3.33.

I. Sell 1000 shares of this stock and buy 303 call options.
II. Buy 1000 shares of this stock and sell 303 call options.
III. Sell 1000 call options and buy 303 shares.
IV. Buy 1000 call options and sell 303 shares.
Correct Answer: III and IV

n = (c+ - c-) / (S+ - S-) = (6 - 0) / (66 - 46.2) = 0.3030.

For every option sold (bought) we should purchase (sell) 0.303 shares of stock. The hedge portfolio will yield a risk-free rate of return.

Practice Question 3

Continue with question 1. Assume a stock price is $55 and in the next year it will either rise by 20% or fall by 16%. The risk-free interest rate is 5%. A call option on this stock has an exercise price of $60. Suppose the call option is selling for 4. Show how to execute an arbitrage transaction that will earn more than the risk-free rate. Use 1000 call options.
Correct Answer: As the current price is higher than 3.33, it is overpriced. We should sell the call and buy the underlying stock. As n is 0.303 (from question 2), for every option sold we should purchase 0.303 shares of stock.
  • Sell 1000 calls at 4: 4,000.
  • Buy 303 shares at 55: -16,665.
  • Net cash flow: -12,665.
At expiration the value of this combination will be
  • 303 x 66 - 6 x 1000 = 13,998 if ST= 66.
  • 303 x 46.2 - 0 x 1000 = 13,998 if ST= 46.2
The rate of return is (13,998 - 12,665)/ 12,665 = 10.53%, which is higher than risk-free rate of 5%.

Practice Question 4

Continue with question 1. Assume a stock price is $55 and in the next year it will either rise by 20% or fall by 16%. The risk-free interest rate is 5%. A put option on this stock has an exercise price of $60. Determine its price.
Correct Answer: $5.48

μ = 1.2 and d = 0.84.
π = (1.05 - 0.84) / (1.2 - 0.84) = 0.5833.
S+ = 55 x 1.2 = $66.
S- = 55 x 0.84 = 46.2.
p+ = Max (0, $60 - $66) = $0.
p- = Max (0, $60 - $46.2) = $13.8.
p = (0.5833 x 0 + 0.4167 x 13.8) / 1.05 = $5.48.

Practice Question 5

If the market price of a European put option is lower than the price suggested by the one-period binomial model, what is the appropriate arbitrage strategy?

A. Sell the put option and short the underlying.
B. Buy the put option and short the underlying.
C. Buy the put option and long the underlying.
Correct Answer: C

As the put option is under-priced, we should buy it. To create a hedge portfolio we should also buy the underlying as well. The portfolio will give us a risk-free rate of return higher than the risk-free rate.

Note that the arbitrage strategy is to long or short positions in BOTH instruments.

Practice Question 6

According to the binomial model, the value of a call option is NOT determined by:

A. The probabilities of the up and down moves.
B. The volatility of the underlying.
C. The risk-free rate.
Correct Answer: A

The actual probabilities of up and down moves do not matter. The binomial model specifies two possible prices of the underlying asset one period later, and enables the construction of a risk-free hedge consisting of the option and the underlying.

Practice Question 7

Assume a stock price is $55 and in the next year it will either rise by 30% or fall by 20%. The risk-free interest rate is 5%. A call option on this stock has an exercise price of $60. It is selling for $5 in the market. How would you execute an arbitrage transaction to take advantage of this situation?

A. Sell the option and buy the underlying.
B. Buy the option and sell the underlying.
C. There is no arbitrage opportunity in this case.
Correct Answer: B

We need to calculate the price of the call option.

μ = 1.3 and d = 0.8.
π = (1.05 - 0.8) / (1.3 - 0.8) = 0.5.
S+ = 55 x 1.3 = 71.5.
S- = 55 x 0.8 = 44.
c+ = Max (0, 71.5 - 60) = 11.5.
c- = Max (0, 44 - 60) = 0.
c = (0.5 x 11.5 + 0.5 x 0) / 1.05 = 5.48.

As the option is under-priced, we should buy the option and sell the underlying to replicate a loan that would charge us less than the risk-free rate.

Practice Question 8

Assume a stock price is $50 and in the next year it will either rise by 20% or fall by 10%. The risk-free interest rate is 6%. A put option on this stock has an exercise price of $50. If we use a one-period binomial model, what is the price of this put option?

A. 4.4
B. 2.52
C. 2.2
Correct Answer: C

μ = 1.2 and d = 0.9.
π = (1.06 - 0.9) / (1.2 - 0.9) = 0.5333.
S+ = 50 x 1.2 = 60.
S- = 50 x 0.9 = 45.
p+ = Max (0, 50 - 60) = 0.
p- = Max (0, 50 - 45) = 5.
p = (0.5333 x 0 + 0.4667 x 5) / 1.06 = 2.2.

Practice Question 9

Assume a stock price is $50 and in the next year it will either rise by 20% or fall by 10%. The risk-free interest rate is 6%. A put option on this stock has an exercise price of $50. If we use a one-period binomial model, what should be the hedge ratio?

A. -0.33
B. 0.5333
C. 1.89
Correct Answer: A

μ = 1.2 and d = 0.9.
S+ = 50 x 1.2 = 60.
S- = 50 x 0.9 = 45.
p+ = Max (0, 50 - 60) = 0.
p- = Max (0, 50 - 45) = 5.
n = (p+ - p-) /(S+ - S-) = -0.333.

Practice Question 10

If the market price of a European put option is higher than the price suggested by the one-period binomial model, what is the appropriate arbitrage strategy?

A. Sell the put option and short the underlying.
B. Sell the put option and long the underlying.
C. Buy the put option and long the underlying.
Correct Answer: A

This strategy would replicate a loan that would charge us less than the risk-free rate.

Practice Question 11

According to the binomial model, the value of a put option is NOT determined by:

A. The volatility of the underlying.
B. The probabilities of the up and down moves.
C. The risk-free rate.
Correct Answer: B

The actual probabilities of the up and down moves are irrelevant to pricing options. The risk-free rate and underlying volatility are highly relevant.

Practice Question 12

The binomial model values options using the:

A. risk-free rate.
B. required yield of the underlying.
C. risk-free rate + risk premium.
Correct Answer: A

The option is valued as though investors are risk neutral.