Lecture 3

Question 1

What is the binomial tree model in option pricing?

Answer 1

A discrete-time model where asset prices can move up or down, used to estimate the value of options.

Question 2

How does the binomial tree model approximate option prices?

Answer 2

By choosing time intervals and simulating price moves, with smaller intervals improving accuracy.

Question 3

Define “up” or “down” factors in the binomial model.

Answer 3

They represent the potential multipliers for asset prices moving up or down in a time step.

Question 4

What is a risk-neutral probability?

Answer 4

The probability q used in binomial models, reflecting pricing as if all investors were indifferent to risk.

Question 5

How is the risk-neutral probability calculated?

Answer 5

q = (e^rT - d) / (u - d), where u and d are up and down factors, and r is the risk-free rate.

Question 6

Why is the real-world probability irrelevant in option pricing?

Answer 6

Because option prices are based on the asset’s current price, incorporating real-world probabilities.

Question 7

What is the purpose of constructing a replicating portfolio in the binomial model?

Answer 7

To match option payoffs by holding a mix of assets and options, ensuring no-arbitrage pricing.

Question 8

What are the basic components needed for binomial tree option pricing?

Answer 8

The asset price, up/down factors, risk-free rate and the strike price.

Question 9

Describe a single-step binomial model for a call option.

Answer 9

Determine up/down asset prices, calculate option payoffs, then discount using risk-neutral probabilities.

Question 10

How do you calculate a call option’s value in a single-step binomial model?

Answer 10

By finding the expected payoff, weighted by risk-neutral probabilities, and discounting at the risk-free rate.

Question 11

What is the “no-arbitrage” condition in the binomial model?

Answer 11

It requires that the cost of a replicating portfolio equals the option price, ensuring no riskless profit.

Question 12

Define a “complete market”.

Answer 12

A market where the payoffs of all assets can be replicated by a combination of basic securities.

Question 13

Why is market completeness important in binomial pricing?

Answer 13

It allows for accurate option pricing by replicating payoffs with known securities.

Question 14

What is the difference between risk-neutral and real-world measures?

Answer 14

Risk-neutral assumes investors are indifferent to risk, while real-world includes risk aversion and premiums.

Question 15

How do you price a derivative under the risk-neutral measure?

Answer 15

Calculate expected payoffs using risk-neutral probabilities and discount at the risk-free rate.

Question 16

What does it mean if an asset price follows a “random walk”?

Answer 16

Prices evolve based on probabilities of moving up or down in each time period.

Question 17

How does a multi-period binomial tree differ from a single-step model?

Answer 17

It uses multiple time steps, creating a lattice of possible price paths for better approximation.

Question 18

Describe a recombining tree in binomial models.

Answer 18

A structure where paths converge, meaning the same ending price can be reached through different paths.

Question 19

How does increasing the number of steps in a binomial tree affect accuracy?

Answer 19

More steps lead to a more accurate estimation of the option price.

Question 20

What is the role of volatility in the binomial model?

Answer 20

Volatility determines the magnitude of the up and down moves, affecting price range and option value.

Question 21

Define a “binary option”.

Answer 21

An option with a fixed payoff if the asset price meets a certain condition, like exceeding a strike price.

Question 22

How does the binomial model handle American options?

Answer 22

Through backward induction, comparing exercise and continuation values at each node.

Question 23

Why is early exercise typically suboptimal for American call options?

Answer 23

Exercising forfeits time value unless dividends are involved.

Question 24

What is backward induction in binomial option pricing?

Answer 24

Starting from the final payoff nodes and working back to calculate the option’s present value.

Question 25

When would early exercise of an American put option be optimal?

Answer 25

When the option’s intrinsic value exceeds the continuation value, especially deep-in-the-money.

Question 26

Explain calibration in the binomial model.

Answer 26

Adjusting parameters like u and d to align with observed asset volatility.

Question 27

How does the Cox-Ross -Rubinstein model define up and down factors?

Answer 27

u = e^volatility*SQRT(time step)
d = 1 / u

Question 28

What is a delta in the context of the binomial model?

Answer 28

The number of shares needed to hedge one option, ensuring riskless payoff.

Question 29

How does the replicating portfolio work in a two-period binomial model?

Answer 29

Calculate delta and bond holdings at each node to match option payoffs without risk.