# Option Pricing Using Binomial Trees

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• 1.

### Using the Black Scholes formula, calculate the price of a 4-month European call option on the British pound. You are given the following details: The current exchange rate is 1.3, the exercise price is 1.3. The risk free interest rate in the United States is 3% per annum whereas the risk free rate in Britain is 4% per annum. The annualized implied volatility in the exchange rate is 20%. The European call option price is:

• A.

6.1 cents

• B.

5.7 cents

• C.

5.3 cents

• D.

4.9 cents

B. 5.7 cents
Explanation
The Black Scholes formula is used to calculate the price of options. In this case, we are calculating the price of a European call option on the British pound. The formula takes into account the current exchange rate, exercise price, risk-free interest rates in both countries, and the implied volatility in the exchange rate. By plugging in these values, we can calculate the price of the option. In this case, the calculated price is 5.7 cents.

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• 2.

### What are the values of u, d and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is 1 month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.

• A.

1.035, 0.966, 0.455

• B.

1.035, 0.966, 0.527

• C.

1.039, 0.963, 0.451

• D.

1.039, 0.963, 0.530

A. 1.035, 0.966, 0.455
• 3.

### Use the conventional binomial tree method with n=3 steps to calculate the price of a 4-month American put option on the British pound. You are given the following details: The current exchange rate is 1.3, the exercise price is 1.3. The risk free interest rate in the United States is 3% per annum whereas the risk free rate 4% per annum. The annualized implied volatility in the exchange rate is 10%. The price of the American put option is:

• A.

2.7 cents

• B.

2.9 cents

• C.

3.2 cents

• D.

3.4 cents

D. 3.4 cents
Explanation
The price of the American put option can be calculated using the binomial tree method. With 3 steps, the time interval between each step is 4/3 months. The up factor is calculated as e^(0.1 * sqrt(4/3)) = 1.099. The down factor is calculated as 1/up factor = 0.909. The risk-neutral probability of an up movement is calculated as (e^(0.03 * 4/3) - down factor) / (up factor - down factor) = 0.515. The risk-neutral probability of a down movement is 1 - risk-neutral probability of an up movement = 0.485. Using these probabilities, the option price at each node of the tree can be calculated. The final option price is 3.4 cents.

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• 4.

• A.

3.1 cents

• B.

3.2 cents

• C.

3.3 cents

• D.

3.4 cents

B. 3.2 cents
• 5.

### The difference between the value of an European Call option calculated by the Black-Scholes formula and that calculated using Binomial trees arises because the Binomial Tree method uses a discrete number of times steps where Black Scholes assumes continuous hedging and rebalancing of the portfolio.

• A.

True

• B.

False

A. True
Explanation
The explanation for the given correct answer is that the Black-Scholes formula assumes continuous hedging and rebalancing of the portfolio, while the Binomial Tree method uses a discrete number of time steps. This difference in methodology leads to a discrepancy in the calculated value of an European Call option between the two methods. Therefore, the statement is true.

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• 6.

### For an American option the value of the option at a node other than the terminal node is:

• A.

The expected present value of option at successor nodes

• B.

The payoff from early exercise

• C.

The minimum of the expected present value of option values at successor nodes or the payoff from early exercise

• D.

The maximum of the expected present value of option values at successor nodes or the payoff from early exercise

• E.

None of the above

D. The maximum of the expected present value of option values at successor nodes or the payoff from early exercise
Explanation
For an American option, the value of the option at a node other than the terminal node is determined by comparing the expected present value of option values at successor nodes with the payoff from early exercise. The option holder will choose the option value that provides the highest value, whether it is the expected present value or the immediate payoff. Therefore, the correct answer is "The maximum of the expected present value of option values at successor nodes or the payoff from early exercise."

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• 7.

### For a European up-and-out  call option the option will only be exercisable and exercised at expiry if:

• A.

The barrier is reached during the tenor of the option and the spot price on maturity is less than the strike price

• B.

The barrier is reached during the tenor of the option and the spot price on maturity exceeds the strike price

• C.

The barrier is not reached during the tenor of the option and the spot price on maturity exceeds the strike price

• D.

The barrier is not reached during the tenor of the option and the spot price on maturity is less than the strike price

• E.

None of the above

C. The barrier is not reached during the tenor of the option and the spot price on maturity exceeds the strike price
Explanation
For a European up-and-out call option, the option will only be exercisable and exercised at expiry if the barrier is not reached during the tenor of the option and the spot price on maturity exceeds the strike price. This means that if the spot price reaches or goes above the barrier at any point during the option's lifespan, the option becomes null and void. Therefore, the correct answer is that the barrier is not reached during the tenor of the option and the spot price on maturity exceeds the strike price.

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• 8.

### The accuracy of calculating the value of a American call option using the binomial tree method can always be improved by increasing the number of time steps used in the calculation.

• A.

True

• B.

False

A. True
Explanation
Increasing the number of time steps used in the calculation of the binomial tree method allows for a more precise approximation of the option's value. This is because a higher number of time steps allows for a finer division of time intervals, leading to a more accurate representation of the underlying asset's price movement. As a result, the calculated value of the American call option becomes more accurate as well. Therefore, increasing the number of time steps improves the accuracy of the calculation.

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• 9.

### The accuracy of calculating the value of a down and out call option using the binomial tree method can always be improved by increasing the number of time steps used in the calculation.

• A.

True

• B.

False

B. False
Explanation
Increasing the number of time steps used in the calculation does not necessarily improve the accuracy of calculating the value of a down and out call option using the binomial tree method. While increasing the number of time steps can lead to a more precise approximation, it also increases the computational complexity and time required for the calculation. Therefore, it is not always true that increasing the number of time steps will improve accuracy.

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• 10.

### The sensitivity of an option’s price to volatility of the underlying’s price is measured by the following Greek:

• A.

Delta

• B.

Gamma

• C.

Rho

• D.

Theta

• E.

Vega

E. Vega
Explanation
Vega is the Greek that measures the sensitivity of an option's price to changes in volatility of the underlying asset. It indicates how much the option's price will change for a one-point increase in implied volatility. A higher Vega value suggests that the option's price is more sensitive to changes in volatility, while a lower Vega value suggests less sensitivity.

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• Current Version
• Mar 19, 2023
Quiz Edited by
ProProfs Editorial Team
• Sep 22, 2011
Quiz Created by
Uzmas

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