Question

Spot Price: $66 Strike price $68 RFR 6% Binomial trees: 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%

a. Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach. [2.5 marks]

b. Use a two-step binomial tree to calculate the value of an eight-month European put option using the no-arbitrage approach. [2.5 marks]

c. Show whether the put-call-parity holds for the European call and the European put prices you calculated in a. and b. [1 mark]

d. Use a two-step binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation. [1 mark]

e. Use a two-step binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation. [1 mark]

f. Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results. [1 mark]

g. Use a two-step binomial tree to calculate the value of an eight-month American put option. [1 mark]

h. Calculate the deltas of the European put and the European call at the different nodes of the binomial three. [1 mark]

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

As 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

		81.3186	13.3186	0.0000
	73.26	65.9340	0.0000	2.0660
66.00	59.40	53.4600	0.0000	14.5400
t=0	t=1	t=2	Value of Call option at t=2	Value of Put option at t=2

a) For European Call option

Under No arbitrage approach,

From t=1 to t=2 when stock price is $73.26

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

So, X*81.3186- 13.3186 = X*65.934 -0

=> X = 0.8657

(At this node the Riskless portfolio consists of Long position in 0.8657 Stocks and Short position in 1 Call option)

So, Value of option(C1h) at t=1 when stock price is $73.26 is given by

0.8657*73.26 - C1h = 0.8657*65.934/exp(0.06*4/12)

=> C1h= $7.4724

Similarly From t=1 to t=2 when stock price is $59.40

X*65.934- 0 = X*53.46 -0

=> X = 0

(At this node the Riskless portfolio consists of Long position in 0 Stocks and Short position in 1 Call option)

So, Value of option(C1L) at t=1 when stock price is $59.40 is given by

0*59.4 - C1L= 0*53.46/exp(0.06*4/12)

=> C1L= 0

and  From t=0 to t=1when stock price is $66

X*73.26- 7.4724 = X*59.4 -0

=> X = 0.5391

(At this node the Riskless portfolio consists of Long position in 0.5391 Stocks and Short position in 1 Call option)

So, Value of option(C) at t=0 when stock price is $66 is given by

0.5391*66 - C= 0.5391*59.4/exp(0.06*4/12)

=> C= $4.19 (Value of 8 month European Call option)

b) For European Put option

Under No arbitrage approach,

From t=1 to t=2 when stock price is $73.26

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

So, X*81.3186 + 0 = X*65.934 +2.066

=> X = 0.1343

(At this node the Riskless portfolio consists of Long position in 0.1343 Stocks and Long position in 1 Put option)

So, Value of option(P1h) at t=1 when stock price is $73.26 is given by

0.1343*73.26 + P1h = 0.1343*81.3186/exp(0.06*4/12)

P1h = 0.8660

Similarly From t=1 to t=2 when stock price is $59.40

X*65.934+2.066= X*53.46 +14.54

=> X = 1

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

So, Value of option(P1L) at t=1 when stock price is $59.4 is given by

1*59.40 + P1L= (1*65.934+2.066)/exp(0.06*4/12)

=> P1L= 7.2535

and  From t=0 to t=1when stock price is $66

X*73.26 + 0.8660 = X*59.40 + 7.2535

=> X = 0.4609

(At this node the Riskless portfolio consists of Long position in 0.4609 Stocks and Long position in 1 Put option)

So, Value of option(P) at t=0 when stock price is $66 is given by

0.4609*66 +P= (0.4609*73.26+0.8660)/exp(0.06*4/12)

=> P= $3.53 (Value of 8 month European Put option)

c) From put call parity

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

LHS = 4.19+ 68*exp(-0.06*8/12) = $69.53

RHS = 3.53+66 = $69.53

As LHS = RHS , the Put call parity holds

d) Under risk neutral valuation ,the risk neutral probability for one period 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 $81.3186 + 2*p*(1-p)*value of option when stock is $65.934 + (1-p)^2*value of option when stock is $53.46) / exp(0.06*8/12)

= (0.5724^2*13.3186)/exp(0.06*8/12) = $4.19

e) Under risk neutral valuation

Value of European put option

= (p^2*value of option when stock is $81.3186 + 2*p*(1-p)*value of option when stock is $65.934 + (1-p)^2*value of option when stock is $53.46) / exp(0.06*8/12)

= (2*0.5724*0.4276*2.066+0.4276^2*14.54)/exp(0.06*8/12)

=$3.53

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