Question

Based on a spot price of $46 and strike price of $48 as well as the fact that the risk-free interest rate is 6% per annum with continuous compounding, please undertake option valuations and answer related questions according to following instructions:

risk-free interest rate is 6% per annum with continuous compounding

Spot price: 46

Strike Price: 48

Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach

Additionally, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%.

1. Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach.
2. Use a two-step binomial tree to calculate the value of an eight-month European put option using the no-arbitrage approach.
3. Show whether the put-call-parity holds for the European call and the European put prices you calculated in a. and b.
4. Use a two-step binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation.
5. Use a two-step binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation.
6. Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results.
7. Use a two-step binomial tree to calculate the value of an eight-month American put option.
8. Calculate the deltas of the European put and the European call at the different nodes of the binomial three.

9. Note: When you use no-arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the different nodes of the binomial tree.

Answer 1

Let one time step be equal to 4 months

With u=1.11 and d =0.9 the stock lattice, value of call option at t=2  is given below

		56.68	8.68	0.00
	51.06	45.95	0.00	2.05
46.00	41.40	37.26	0.00	10.74
t=0	t=1	t=2	Value of Call option at t=2	Value of Put option at t=2

For European Call option

Under No arbitrage approach,

From t=1 to t=2 when stock price is $51.06

Let X shares be purchased and one call option be shorted to create the no arbitrage portfolio

So, X*56.68- 8.68 = X*45.95 -0

=> X = 0.8092

So, Value of option(C1h) at t=1 when stock price is $51.06 is given by

0.8092*51.06 - C1h = 0.8092*45.95/exp(0.06*4/12)

=> C1h= $4.868

Similarly From t=1 to t=2 when stock price is $41.4

X*45.95- 0 = X*37.26 -0

=> X = 0

So, Value of option(C1L) at t=1 when stock price is $41.4 is given by

0*41.4 - C1L= 0*37.26/exp(0.06*4/12)

=> C1L= 0

and  From t=0 to t=1when stock price is $46

X*51.06- 4.868 = X*41.4 -0

=> X = 0.5039

So, Value of option(C) at t=0 when stock price is $46 is given by

0.5039*46 - C= 0.5039*41.4/exp(0.06*4/12)

=> C= $2.73

For European Put option

Under No arbitrage approach,

From t=1 to t=2 when stock price is $51.06

Let X shares be purchased and one put option be purchased to create the no arbitrage portfolio

So, X*56.68 + 0 = X*45.95 +2.05

=> X = 0.1908

So, Value of option(P1h) at t=1 when stock price is $51.06 is given by

0.1908*51.06 + P1h = 0.1908*56.68/exp(0.06*4/12)

P1h = 0.8575

Similarly From t=1 to t=2 when stock price is $41.4

X*455.95+2.05= X*37.26 +10.74

=> X = 1

So, Value of option(P1L) at t=1 when stock price is $41.4 is given by

1*41.4 + P1L= (1*45.95+2.05)/exp(0.06*4/12)

=> P1L= 5.6495

and  From t=0 to t=1when stock price is $46

X*51.06 + 0.8575 = X*41.4 + 5.6495

=> X = 0.4961

So, Value of option(P) at t=0 when stock price is $46 is given by

0.4961*46 +P= (0.4961*51.06+0.8575)/exp(0.06*4/12)

=> P= $2.85

From put call parity

C+K*exp(-rt) = P +S

LHS = 2.73+ 48*exp(-0.06*8/12) = $48.85

RHS = 2.85+46 = $48.85

As LHS = RHS , the Put call parity holds

d) Under risk neutral valuation ,the risk neutral probability is given by

p = (exp(0.06*4/12)- 0.9)/(1.11-0.9) = 0.5724

So, Value of European call option

= (p^2*value of option when stock is $56.68 + 2*p*(1-p)*value of option when stock is $45.95 + (1-p)^2*value of option when stock is $37.26) / exp(0.06*8/12)

= (0.5724^2*8.68)/exp(0.06*8/12) = $2.73

e) Under risk neutral valuation

Value of European put option

= (p^2*value of option when stock is $56.68 + 2*p*(1-p)*value of option when stock is $45.95 + (1-p)^2*value of option when stock is $37.26) / exp(0.06*8/12)

= (2*0.5724*0.4276*2.05+0.4276^2*10.74)/exp(0.06*8/12)

=$2.85

Thus, we get the same value of the Call and the Put option from both no-arbitrage approach and the risk-neutral valuation.