Question

In: Finance

Use a 2 step binomial tree to value a new exotic derivative. Draw the tree and...

Use a 2 step binomial tree to value a new exotic derivative. Draw the tree and label the stock prices and derivative values at each node.

The option expires in 6 months. The interest rate is 10% annually continuously compounded. The Strike Price (K) is 100.

The spot price is at 100. U= 1.2 and D= 0.8 for each quarterly period. The payoff of this derivative is (ST/K). By this I mean that the payoff is the price of the underlying stock divided by the Strike Price.

Solutions

Expert Solution

Given

option expiry (t) = 6 Months

Interest (r) = 10% per anum continously compounded

Strike Price (K) = 100

Spot Price (S0) = 100

Price Up by (u) = 1.2

Price goes down by (d) = 0.8

CALL OPTION: It gives an entitlement for the buyer to get a right to purchase from the option seller on the date of expiry.

A call option is excercised by the buyer when Spot Price (St) is GREATER than Strike price(K)

At Node 1 from the diagram uploaded, the spot price is greater than strike price

Hence the buyer will excercise the option. Thus Payoff when price went up wii be (fu) = St-K = 120-100 = 20.

At node 2 when the price goes down to 80, the option buyer donot excercise the option, Hence Payoff will be (fd) = 0

Call option price will be calculated as follows

={(fu-fd)+(e^-rt)*(ufd-dfu)}/u-d

Where fu = 20, fd = 0, u=1.2, d=0.8, r=10%pacc, t=3/12 or 0.25

={(20-0) + e^-(0'10*0.25)*((1.20*0-0'8*20))}/(1.20-0.8)

={ 20 + (0.9753)*(0-16)}/0.4

={20-15.6048)/0.4

=4.3952/0.4

= 10.988

Hence Call option Price when Price takes its first movement up and down = Rs. 10.988

PUT OPTION

In this case option buyer will have right to sell to the seller of the option at the expiry.

A Put option is excercised by the buyer when Spot Price (St) is LESS than Strike price(K)

At Node 1 from the diagram uploaded, the spot price is greater than strike price

Hence the buyer will not excercise the option. Thus Payoff when price went up wii be (fu) = 0

At node 2 when the price goes down to 80, the option buyer excercise the option, Hence Payoff will be (fd) = K-St = 100-80 = 20

Put option price will be calculated as follows

={(fu-fd)+(e^-rt)*(ufd-dfu)}/u-d

Where fu = 0, fd = 20, u=1.2, d=0.8, r=10%pacc, t=3/12 or 0.25

={(0-20) + e^-(0'10*0.25)*((1.20*20-0'8*0))}/(1.20-0.8)

={ (-20) + (0.9753)*(24-0)}/0.4

={(-20)+22.9752)/0.4

=2.9752/0.4

= 7.438

Hence Put option Price when Price takes its first movement up and down = Rs. 7.438

CALCULATION OF CALL OPTION PRICE AND PUT OPTION PRICE AFTER DOUBLE MOVEMENTS IN SPOT PRICE

CALL OPTION:

At node 3 from the diagram, the payoff will be (fuu) = 144-100 = 44

At node 4, the payoff will be (fud) = 0

At node 5, the payoff will be (fdd)= 0

Note that at node 4 and 5, the spot price is less than strike price, hence payoff will be Zero

The formula to find call option price in case of two steps is as follows

=e^-2rtdelta*{p^2 fuu + 2p*(1-p)*fud + (1-p)^2* (fdd)}

fuu = 44, fud = 0, fdd = 0, r=10%pacc, delta t = 6/12 or 0.5

p=(e^r(delta t) - d)/(u-d) = probability of happening

p={e^(0.10*0.50) - 0.8}/(1.20-0.80)

p= (1.0513-0.8)/0.4

p= 0.62825

1-p = 1- 0.6283 = 0.3718

e^-2rdelta t = e^ (-2*0.10*0.50) = 0.9048

Substituting everything in below formula

=e^-2rtdelta*{p^2 fuu + 2p*(1-p)*fud + (1-p)^2* (fdd)}

=0.9048*{(0.6283^2)*44 + (2*0.6283*0.3718*0) + (0.3718)^2 * 0}

= 0.9048*(17.3695 + 0 + 0)

= 15.7159

Call option price after double change in price = Rs. 15.7159

PUT OPTION:

At node 3 from the diagram, the payoff will be (fuu) = 0

At node 4, the payoff will be (fud) = K-St = 100 - 96 = 4

At node 5, the payoff will be (fdd)= K-St = 100-64 = 36

Note that at node 3, the spot price is greater than strike price, hence payoff will be Zero

The formula to find put option price in case of two steps is as follows

=e^-2rtdelta*{p^2 fuu + 2p*(1-p)*fud + (1-p)^2* (fdd)}

fuu = 0, fud = 4, fdd = 36, r=10%pacc, delta t = 6/12 or 0.5

p=(e^r(delta t) - d)/(u-d) = probability of happening

p={e^(0.10*0.50) - 0.8}/(1.20-0.80)

p= (1.0513-0.8)/0.4

p= 0.62825

1-p = 1- 0.6283 = 0.3718

e^-2rdelta t = e^ (-2*0.10*0.50) = 0.9048

Substituting everything in below formula

=e^-2rtdelta*{p^2 fuu + 2p*(1-p)*fud + (1-p)^2* (fdd)}

=0.9048*{(0.6283^2)*0 + (2*0.6283*0.3718*4) + (0.3718)^2 * 36}

= 0.9048*(0 + 1.8688 + 4.9765)

= 6.1936

Call option price after double change in price = Rs. 6.1936


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