Question

In this question, you need to price options with various approaches. You will consider puts and calls on a share. Please read following instructions carefully:

• The spot price of the share will be 56.
• the strike price of the options will be 58.

Based on this spot price and this strike price 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:

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 risk-neutral valuation. [1 mark]
2. Use a two-step binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation. [1 mark]
3. Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results. [1 mark]
4. Use a two-step binomial tree to calculate the value of an eight-month American put option. [1 mark]
5. 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

4 month is equivalent to one period . The stock price and the options value at maturity is shown below

		68.9976	10.9976	0.0000
	62.16	55.9440	0.0000	2.0560
56.00	50.40	45.3600	0.0000	12.6400
t=0	t=1	t=2	Value of Call option at t=2	Value of Put option at t=2

a)

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 $68.9976 + 2*p*(1-p)*value of option when stock is $55.944 + (1-p)^2*value of option when stock is $45.36) / exp(0.06*8/12)

= (0.5724^2*10.9976)/ exp(0.06*8/12)

= $3.46

b)

Under risk neutral valuation

Value of European put option

= (p^2*value of option when stock is $68.9976 + 2*p*(1-p)*value of option when stock is $55.944 + (1-p)^2*value of option when stock is $45.36) / exp(0.06*8/12)

= (2*0.5724*0.4276*2.056+0.4276^2*12.64)/exp(0.06*8/12)

=$3.1876

c)

Under No arbitrage approach,

From t=1 to t=2 when stock price is $62.16

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

So, X*68.9976- 10.9976 = X*55.944-0

=> X = 0.8425

So, Value of option(C1h) at t=1 when stock price is $62.16 is given by

0.8425*62.16 - C1h = 0.8425*55.944/exp(0.06*4/12)

=> C1h= $6.1703

Similarly From t=1 to t=2 when stock price is $50.4

X*55.944- 0 = X*45.36 -0

=> X = 0

So, Value of option(C1L) at t=1 when stock price is $50.4 is given by

0*50.4 - C1L= 0*45.36/exp(0.06*4/12)

=> C1L= 0

and  From t=0 to t=1when stock price is $56

X*62.16- 6.1703 = X*50.4 -0

=> X = 0.5247

So, Value of option(C) at t=0 when stock price is $56 is given by

0.5247*56 - C= 0.5247*50.4/exp(0.06*4/12)

=> C= $3.46

Under No arbitrage approach,

From t=1 to t=2 when stock price is $62.16

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

So, X*68.9976 + 0 = X*55.944 +2.056

=> X = 0.1575

So, Value of option(P1h) at t=1 when stock price is $62.16 is given by

0.1575*62.16 + P1h = 0.1575*68.9976/exp(0.06*4/12)

P1h = 0.8617

Similarly From t=1 to t=2 when stock price is $50.4

X*55.944+2.056= X*45.36 +12.64

=> X = 1

So, Value of option(P1L) at t=1 when stock price is $50.4 is given by

1*50.4 + P1L= (1*45.36+12.64)/exp(0.06*4/12)

=> P1L= 6.4515

and  From t=0 to t=1when stock price is $56

X*62.16 + 0.8617 = X*50.4 + 6.4515

=> X = 0.4753

So, Value of option(P) at t=0 when stock price is $56 is given by

0.4753*56 +P= (0.4753*50.4+6.4515)/exp(0.06*4/12)

P = $3.1876

THUS ,iT CAN BE SEEN THAT VALUE OF CALL AND PUT OPTIONS AS DERIVED FROM RISK NEUTRAL VALUATION IS THE SAME AS THAT DERIVED FROM NO-ARBITRAGE APPROACH

d) For 8 month American Put option

The stock price and the options value at maturity is shown below

		68.9976		0.0000
	62.16	55.9440		2.0560
56.00	50.40	45.3600		12.6400
t=0	t=1	t=2		Value of Put option at t=2

Risk neutral probability

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

Value of American put option (P1h) at t=1 when stock price is $62.16 is given by

= max ((p*value of option when stock price is $68.9976 at t=2 + (1-p)*value of option when stock price is $55.944 at t=2)/exp(0.06*4/12) , 58-62.16)

=max ((0.5724*0+0.4276*2.056)/exp(0.06*4/12), 58-62.16)

= 0.8618

Value of American put option (P1l) at t=1 when stock price is $50.4 is given by

= max ((p*value of option when stock price is $55.944 at t=2 + (1-p)*value of option when stock price is $45.36 at t=2)/exp(0.06*4/12) , 58-50.4)

=max ((0.5724*2.056+0.4276*12.64)/exp(0.06*4/12), 58-50.40)

= $7.6

So, Value of option(P) at t=0 when stock price is $56 is given by

= max ((p*value of option when stock price is $62.16 at t=1 + (1-p)*value of option when stock price is $50.40 at t=1)/exp(0.06*4/12) , 58-56)

=max ((0.5724*0.8618+0.4276*7.6)/exp(0.06*4/12), 58-55)

= $3.67

So, the value of the American Put option is $3.67