The Binomial Model

Question 1

What is the binomial model?

Answer 1

The binomial model is a model for the changes in the stock price over time to find the price of an option today. The binomial model assumes that, over a period of time, the price of the underlying asset can move up or down by a specified amount – that is, the asset price follows a binomial distribution. Given this assumption, it is possible to determine a no-arbitrage price for the option. Using the no-arbitrage condition, we will be using the concept of riskless hedge to derive the value of an option.

Question 2

Consider a stock priced at $20. Assume a one-year time period. You know that the price is either going to be $22 or $18. Suppose there is a European call option on the stock with an exercise price of $21. Assume that the continuously compounded risk-free rate is 3% per period. We can find the value of the option by assuming no AOs and using the concept of a riskless hedge.

Answer 2

The Stock is risky – its payoff is variable. A RF bond, on the other hand is riskless; we know its payoff at maturity. We will be combining the stock and the option in such a way that the payoff of the combination replicates the payoff of the bond. Thus, the combination also becomes a riskless one. (Photos) So you can either buy 0.25 shares and sell 1 call, or buy 1 share and sell 4 calls to perform a riskless hedge. We can then solve for C, either using discounting, or assuming no arbitrage opportunities.

u = Up, d = Down, F_u = Option Payoff State 1 ($22 - $21 = $1), F_d = Option payoff state 2 (No exercise = $0), S_u = Share price State 1 ($22), S_d = Share price State 2 ($18)

Solve for c, using either:

Discounting method: Since both portfolios have the same payoff ($18), portfolio A is riskless. This means that we can discount its payoff at the risk-free rate, which will give us the price of the portfolio today, allowing us to find c. Discounting its payoff ($18) at the RFR and solving for the call price, gives us $0.633.

No Arbitrage Method: Since both portfolios have the same payoff at maturity, their values should be the same today, so as to ensure no arbitrage. This means we can equate portfolio B with portfolio A today and solve to find c. 4c-20=17.468→c=$0.633

Question 3

What is the Riskless Payoff?

Answer 3

The riskless payoff is Delta, which is equal to: The up state value of the option - the down state value of the option (fu - fd) Divided by The up state of the share price - the down state of the share price (su - sd)

Question 4

What are the assumptions of the one-period binomial model?

Answer 4

• We assume no transaction costs.
• The continuously compounded risk-free rate per annum is r.
• There are no arbitrage opportunities.
• There is only one-period to maturity. So, T = ∆t

Question 5

What are fu and fd? And how do they change with share price changes?

Answer 5

At maturity, if the stock price increases, the value of the option is fu = max⁡(0, uS - K). (Price increased by u) If the price falls, then the value is fd = max⁡(0, dS - K). (Price decreased by d)

Question 6

What is the Current value of the option in the one-period binomial model?

Answer 6

f is the current value of the option; fu is the value in the up state and vice versa. So, we’re finding the expected value of the option, and p is the probability factor of a stock price increase. (Picture of Equation) (Picture of Tree)

Question 7

Using the binomial model, does the value of the option depend on the expected return of the stock?

Answer 7

No. There is no term in the equation for the stock price return, only for the future state of the share price.

Question 8

What is Delta (δ)?

Answer 8

Delta (δ) = The number of shares that one needs to buy for each option sold at the start of the period to create a risk-free portfolio. It’s the riskless hedge. It shows how much option price changes for a given change in the underlying instrument’s price.

Question 9

What is δ in the Black-Scholes Model?

Answer 9

In the Black-Scholes Model, δ = N(d1).

Question 10

How do you use the No Arbitrage Argument to find the option price? What are the two methods? Explain the steps.

Answer 10

Given the Option type, Stock Price, Strike price, risk-free rate, time, U and D

Method One: Riskless Hedge Equation to find δ

(1) Note the values provided.
(2) Draw the tree showing the Up and Down states of Share value and Option value.
(3) Find δ using riskless hedge Equation – the values are taken from the tree. This will give us the risk-free future payoff in (4).
(4) Substitute δ to find portfolio values at state u, which should equal state d.
(5) Find present value of portfolio value at state u (πu), state πu=πd, then use e to discount π_u back to π, the PV.
(6) Using the present value of portfolio equation, S×∆ - f=π, to find f.

Method Two: Equate the States

(1) Note the values provided
(2) Choose to go long on Shares, short the option such that π=δS-f; note this equation down (it’s the portfolio’s PV).
(3) Find portfolio in u and d states; e.g. π_u=δS_u-f_u
(4) Create a risk-free future payoff by equating the future payoffs such that πu=πd, then solving for δ (negative when put). δ is negative for put options, positive for call options, so with shorting put options, we short the stock too.
(5) Substitute δ into πu or πd to find the future value of π.
(6) Discount π_u back using e^(-rt∆) to discount (negative to discount, positive to project) to find π.
(7) Sub π, δ and S into π=δS - f to find f.

Question 11

Interpret the valuation formula of the binomial model.

Answer 11

The formula calculates the current value of the option based on the present value of its expected future payoff in a risk- neutral world using the risk-free rate as the discount rate.

Question 12

Explain Risk-Neutral Valuation

Answer 12

Assuming the world is risk-neutral, the expected return is the risk-free rate, simplifying calculations and allowing us to find the correct theoretical price of an option.

Question 13

Explain the steps of the Expected payoff = Risk Free ROR form of the RNV Method.

Answer 13

Method One: Expected payoff = Risk-free ROR

(1) Assume that the expected return of the underlying asset is the risk-free rate, making the future value pSu + Sd (1 - p) equal to the current value (S) compounded at the risk-free interest rate. Solve for p, the probability of a positive payoff – this is the risk-neutral probability.
(2) Let f = option value. Compute the expected payoff of the option: p(fu)+(1 - p)(fd) = fT
(3) Discount the expected payoff at the risk-free rate: fT e^r∆t = f

Example: Expected payoff = Risk-free ROR

(1) The expected return is the risk-free rate (12%). Thus, p must satisfy the following equation. We can solve for p to find the risk-neutral probability: 22p+18 (1 - p ) = 20e^(0.12)(3/12) 4p = 20e^(0.12)(3/12) -18 p = 0.6523
(2) The expected value of the call at the end of three months is (0.6523)(1) + (1 - 0.6523)(0) = $0.6523
(3) The discount rate in a risk-neutral world is the risk-free rate. The price of the call is then: Call = 0.6523e^((-0.12)(3/12))

Question 14

Explain the steps of the equation form of the RNV Method.

Answer 14

Method Two: Equation Method

(1) State the values, finding u and d.
(2) Substitute them into the p equation to find the risk-neutral probabilities
(3) Substitute p and other relevant values into the equation for f, solving for f.

Example:

(1) u = 1.1, d = 0.9, Su = 20*1.1 = 22, Sd = 20*0.9 = 18, fu = 22 - 21= 1, fd = 18 - 21 = 0, r = 0.12, t = -0.25
(2) p = (e^rΔt - d) / (u - d) = (e^(0.12)(0.25)-0.9) / (1.1- 0.9) = 0.6523
(3) f = e^(-rΔt) (pfu+(1-p) fd )=e^(-0.12)(0.25) ((0.6523)(1)-(1-0.6523)(0))=$0.633

Question 15

Show that the Risk Neutral Valuation Model is a simplified version of the No Arbitrage Argument.

Answer 15

Question 16

Write out the equation for the current value of the option using a two-period binomial model. Then, find the option’s value given: European Call Option Strike Price $21 t = 6 months = 0.5, 0.25 per period. r = 12% Two time periods Stock price moves up or down 10% S(time 0) = $20

Answer 16

The value of the option when there are two nodes is given by:

f = e^(-2rΔt) (p^2 fuu+2p(1 - p) fud + (1 - p)^2 fdd)

fuu = 20(1.1^2) - 21 = 24.2 - 21 = 3.20

fdu = fud = max⁡(0, 19.8 - 21) = 0

fdd = 0 p = (e^((0.12)(0.25))-0.9) / (1.1-0.9)=0.6523

f = e^(-2(0.12)(0.25)) ((0.6523)^2(3.2) + 2(0.6523)(1 - 0.6523)(0)+(1 - 0.6523)^2(0)) = $1.282

Question 17

Explain how to value American Options using the two-period binomial model.

Answer 17

There is no analytical formula for American options like the one for European options, but we still can use the binomial model to find the price of an American option. To do that, we start at maturity and work backwards.

Note: Node value is the higher of the exercise and the unexercised value

(1) Calculate the value of the option at each node.
(2) Find the value of the option if it were exercised immediately. This means (for calls) find (S – K).
(3) Take the higher of the two as the option value at that node.
(4) Work our way backwards as before until we reach the initial node (t = 0). Do this by calculating p, and using the one period binomial model equation at each node.