Question

Problem 3: Derivatives Valuation

A stock price is currently $36. During each three-month period for the next six months it is expected to increase by 9% or decrease by 8%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off (max[(40-S_T),0])² where St is the stock price in six months.

1. What are the payoffs at the final nodes of the tree? [1 mark]
2. Use no-arbitrage arguments (you need to show how to set up the riskless portfolios at the different nodes of the binomial tree). [2 mark]
3. Use risk-neutral valuation. [1 mark]
4. Verify whether both approaches lead to the same result. [1 mark]
5. If the derivative is of American style (S_T in the payoff function refers to the stock price when the option is exercised), should it be exercised early? [1 mark]

Answer 1

As we are instructed to answer the first question only in case of multiple questions if not specified so I am going to provide the soolution for question (c)

To value the opton using the risk-neutral approach, we need to first calculate the risk neutral probabilities by the formula

Where, "Rf" is the risk free rate

"d" is the down move factor :

"u" is the up move factor:

So we already have Rf, but for u and d we are provided in the question with the information that stock can either move up by 9% or move down 8% in each 3 month. So lets calculate u and d

For time step 1 the share can move up by 9% so you would be would be :

and now lets calculate d ,as it can move down 8% :

NOTE : I will be denoting all the risk neutral probability for up move as P1 and for down move as P2

So lets calculate P1:

With the help of P1 now we can calculate P2 as well. Hence P2:

I am making the binomial tree to help you get it more intutively and to understand the valuation better rather than just bombing the formulas.

NOTE: I will be using some notations in the Binomial tree and those are

So = Share rice today, which is $36

X = Strike price, which is 40

Po = value of the put option

Su= share price if it moves up at time step 1

Sd= share price if moves down at time step 1

Suu= Share price if it moves up at both time step 1 and 2

Sud= if share price moves up at time step 1 and the moves down at time step 2

Sdd = if share price moves down at time step 1 and 2.

Suu = 42.7716

Payoff = 0

Su=39.24

So= $36 Sud = 36.10

Po = 2.9513 = 40 - 36.10

= 3.90

Sd= 33.12

Sdd = 30.47

Payoff = 40 - 30.47

= 9.53

The formula used here is

NOTE : Puu, Pud and Pdd are same as the above notations for share price, except here they are used to denote the payoff from the put option.

So, the value of the put option is 2.9513