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-ST),0])2 where ST is the stock price in six months.

a. What are the payoffs at the final nodes of the tree? - not PV's for prices on tree

b. Use no-arbitrage arguments (you need to show how to set up the riskless portfolios at the different nodes of the binomial tree).

c. Use risk-neutral valuation. d. Verify whether both approaches lead to the same result. e. If the derivative is of American style (ST in the payoff function refers to the stock price when the option is exercised), should it be exercised early?

Answer 1

As one time step be equal to 3 months

With u=1.09 and d =0.92 the stock lattice, value of derivative at t=2  is given below

		42.7716	0
	39.24	36.1008	15.20376
36	33.12	30.4704	90.81328
t=0	t=1	t=2	Payoff

a)

Payoffs at the final nodes of the tree

When Stock price is $42.7716, payoff = max(40-42.7716,0)^2 =0

When Stock price is $36.1008, payoff = max(40-36.1008,0)^2 =$15.20376

When Stock price is $30.4704, payoff = max(40-30.4704,0)^2 =$90.81328

b)

Under No arbitrage approach,

From t=1 to t=2 when stock price is $39.24

Let X units of stock and one unit of Derivative be purchased to create the no arbitrage portfolio

So, Value of portfolio at t=2 in upside = Value of portfolio at t=2 in downside

X*42.7716 + 0 = X*36.1008 +15.20376

=> X = 2.279151

(At this node the Riskless portfolio consists of Long position in 2.279151 Stocks and Long position in 1 derivative)

So, Value of derivative (D1h) at t=1 when stock price is $39.24 is given by

2.279151*39.24 + D1h = 2.279151*42.7716/1.05^(3/12)   

D1h = 6.86722

Similarly From t=1 to t=2 when stock price is $33.12

X*36.1008+15.20376= X*30.4704 +90.81328

=> X = 13.4288

(At this node the Riskless portfolio consists of Long position in 13.4288 Stock and Long position in 1 Derivative)

So, Value of Derivative (D1L) at t=1 when stock price is $33.12 is given by

13.4288*33.12 + D1L= (13.4288*36.1008+15.20376)/1.05^(3/12)

=> D1L= 49.17067

and  From t=0 to t=1when stock price is $36

X*39.24 + 6.86722 = X*33.12 + 49.17067

=> X = 6.912329 (At this node the Riskless portfolio consists of Long position in 6.912329 Stocks and Long position in 1 Derivative)

So, Value of Derivative (D) at t=0 when stock price is $36 is given by

6.912329*36 +D= (6.912329*39.24+6.86722)/1.05^(3/12)

=> D= $25.89

So, the value of the Derivative is $25.89

c) Under risk neutral valuation ,the risk neutral probability for one period is given by

p = (1.05^(3/12)- 0.92)/(1.09-0.92) = 0.5428

So, Value of Derivative

= (p^2*value of Derivative when stock is $42.7716 + 2*p*(1-p)*value of Derivative when stock is $36.1008 + (1-p)^2*value of Derivative when stock is $30.4704) / 1.05^(6/12)

= (2*0.5428*0.4572*15.20376+0.4572^2*90.81328)/1.05^0.5

= $25.89

d) Thus, it can be seen that both the no-arbitrage approach as well as Risk neutral valuation leads to the same price of the Derivative i.e. $25.89