Ch 12 - Introduction to Binomial Trees

Question 1

Why do we use the simple binomial model?

Answer 1

􀁺 Model can be extended to cover all possible future prices
􀁺 Binomial model allows us to price American options
􀁺 Simple model that requires minimum of mathematics – used often by market practitioners

Question 2

What are the two ways of valuing options using binomial trees?

Answer 2

No arbitrage approach

Risk-neutral valuation

Question 3

What is the no-arbitrage approach of valuing options using binomial trees?

Answer 3

􀁺 Set up a risk-less portfolio consisting of a position in the option and a position in the stock (Delta stock)
􀁺 By setting the return on the portfolio equal to the risk-free interest rate, we are able to value the option (risk-neutral probabilities up & down)

Question 4

What is the risk-neutral approach of valuing options using binomial trees?

Answer 4

􀁺 Choose probabilities for the branches of the tree so that the EXPECTED RETURN on the stock equals the RFR
􀁺 Value the option by calculating its EXPECTED PAYOFF and DISCOUNTING this expected payoff at the RFR

Question 5

What is the only assumption of the no arbitrage approach?

Answer 5

Only assumption: no arbitrage opportunities exist

Question 6

How do we set up our portfolio in the no arbitrage approach?

Answer 6

􀁺 We set up the portfolio of the stock and the option in such a way that there is NO UNCERTAINTY about the VALUE of the portfolio AT the END of the period
􀁺 Because the portfolio has NO RISK then the RETURN it earns MUST = RFR

Question 7

How can we calculate Delta Δ?

Answer 7

Δ = option’s delta (change option price/change stock price)

Question 8

When is a portfolio riskless (mathematically - no arbitration)

Answer 8

S0uΔ – ƒu = S0dΔ – ƒd

Question 9

What is the secondary way of calculating Delta Δ?

Answer 9

Δ = (ƒu - ƒd) / (S0u - S0d)

Question 10

What is the value of the portfolio at time T? (mathematically)

Answer 10

Su Δ – ƒu

Question 11

Value of the portfolio today? (mathematically)

Answer 11

(Su Δ – ƒu )e^(–rT)
or
S Δ – ƒ so S Δ – ƒ = (Su Δ – ƒu )e^(–rT)
Hence: ƒ = S Δ – (Su Δ – ƒu )e^(–rT)

Question 12

How do we calculate the risk-neutral probability up?

Answer 12

p = [e^(rT) - d] / [u - d]

Question 13

How do we calculate the risk-neutral probability down?

Answer 13

1-p

Question 14

What is the value of an option using the risk-neutral approach? (mathematically)

Answer 14

ƒ = [ p ƒu + (1 – p )ƒd ]e^(–rT)

Question 15

How is d calculated?

Answer 15

d = Sd/S

Question 16

How is u calculated?

Answer 16

u = Su/S

Question 17

Are p and (1-p) are probabilities of the up and down movement?

Answer 17

p and (1-p) are the probabilities that would exist if ALL investors were RISK NEUTRAL

Question 18

Explain the no-arbitrage approach to valuing a European option using the one-step binomial tree.

Answer 18

We set up a riskless portfolio consisting of a position in the option and a position in the stock. By setting the return on the portfolio equal to the RFR, we are able to value the option.

Question 19

Explain the risk-neutral valuation approach to valuing a European option using the one-step binomial tree

Answer 19

We first choose probabilities for the branches of the tree so that the expected return on the stock equals the RFR. We then value the option by calculating its expected payoff and discounting this expected payoff at the RFR.