You are pricing a derivative using the risk-neutral approach on a Binomial tree.

The length of each branch in the tree (delta t) is 9 months.

The riskfree rate of interest is 5% per annum.

The proportional up movement in stock price is u = 1.1.

Calculate the risk-neutral probability (p*) that share price will move up.

Note: if the probability is (say) 51.23%, enter 0.5123

The possible movement in share price over the next 12 months is depicted in the 4-step Binomial tree shown below.

Key details are:

• the proportional up and down movements on each branch of the tree are u = 1.2214 and d = 0.8187 respectively

• the riskfree interest rate is 6% pa continuously compounded

Let S_T denote the share value one year from now (far right side of the Binomial tree).

A strange derivative security is based on this share. It has a payoff = sqrt(S_T).

That is, if you buy this derivative today, one year from now you receive a payoff equal to the square root of whatever the share price is.

Required:

Using the 4-step Binomial tree, calculate the fair value of this derivative today.

Give your answer to at least 2 decimal places.

A single branch on a Binomial tree is represented by the following diagram. The riskfree rate is 4% and the branch length (delta t) = 6 months

Calculate how much money is required to invest in the bank ("B") to replicate the derivative.

Enter your answer to two decimal places.

Do not enter the dollar sign "$".

A positive (negative) number means invest (borrow).

What is the upper bound of a European call option with 10 months to expiry and a strike price of $ 9.3. The underlying stock is currently trading at $ 13.3, and the risk-free rate is 9.6%.

What is the upper bound of an American put option with 1 months to expiry and a strike price of $ 6. The underlying stock is currently trading at $ 4.6, and the risk-free rate is 6.3%.

Do not enter the dollar sign "$".

The volatility (sigma) of a stock is 70% per annum.

We will model stock-price movements using a Binomial tree, where each branch of the tree has a step length (delta t) of 1 months.

Calculate the proportion up movement (u) that applies to each branch of the tree.

Your answer should have at least two decimal places.

A single branch on a Binomial tree is represented by the following diagram.

Calculate how many shares of the underlying asset (delta) are required to delta hedge.

Enter your answer to two decimal places.  If your answer is negative, then enter the negative sign.

What is the lower bound of a European call option with 5 months to expiry and a strike price of $1.7. The underlying stock is currently trading at $3.3, and the risk-free rate is 7.8%.

Do not enter the dollar sign "$".

What is the lower bound of an American put option with 5 months to expiry and a strike price of 15.8. The underlying stock is currently trading at 12.2, and the risk-free rate is 4.9%.

We model the movement of stock prices using a Binomial tree.

Each branch of the tree spans a time horizon of 4 months.

The proportional up movement on the tree (u) is 1.72.

What is the volatility of the stock implied by u? 

If your answer is (say) 40%, enter 0.40.