12.9. A stock price is currently $50. It is known that at the end of 2 months it will be either $53 or $48.

The risk-free interest rate is 10% per annum with continuous compounding.

What is the value of a 2-month European call option with a strike price of $49?

Use no-arbitrage arguments.

If the stock price moves from $50 to $53 then the value of shares
is to be 53Δ and the value of option is (53-49)=4, the total
value of the portfolio will be 53Δ- 4.

If the stock price moves down from $50 to $48, the value
of shares is 48Δ and the value of option is 0, the total value
of the portfolio will be 48Δ.

The portfolio is riskless, if the value of Δ is chosen so that the
final value of the portfolio is the same for both alternatives
(53Δ-4= 48Δ, or Δ = 4/5).

A riskless portfolio is therefore

Long: 4/5 shares

Short: 1 option

If the stock moves up to $53, the value of the portfolio
is S_U×Δ-(S_U-K)=53×4/5-(53-49)=38.4 If the stock moves
down to $48, the value of the portfolio is S_U×Δ=48×4/5=38.4

Regardless of if the stock price moves up or down, the value
of the portfolio is always 38.4 at the end of the life of option.

This shows that Δ is the number of shares necessary to hedge
a short position in one option.

The value of the stock is $50. The value of portfolio today is

[image]

Alternatively, we can calculate

50u=53 u =53/50=1.06

50d=48 d=48/50=0.96 

Then

[image]

[image]

12.11. A stock price is currently $40. It is known that at the end of 3 months it will be either $45 or $35.

The risk-free rate of interest with quarterly compounding is 8% per annum.

Calculate the value of a 3-month European put option on the stock with an exercise price of $40.

Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

If the stock price moves from $40 to $45 then the value of shares
is be 45Δ and the value of option is 0, the total value of the
portfolio will be -45Δ.

If the stock price moves down from $40 to $35, the value
of shares is 35Δ and the value of option is 5, the total value
of the portfolio will be -35Δ+5.

The portfolio is riskless, if the value of Δ is chosen so that
the final value of the portfolio is the same for both
alternatives (-35Δ+5= -45Δ, or Δ = -0.5, for both
alternatives (-35Δ+5= -45Δ, or Δ = -0.5).

A riskless portfolio is therefore

Long: 1 shares

Short: 1 option

If the stock moves up to $45, the value of the portfolio is

-45×(-0.5) = 22.5

If the stock moves down to $35, the value of the portfolio is

-$35×(-0.5) + 5 = 22.5

Regardless of if the stock price moves up or down, the value
of the  portfolio is always 22.5 at the end of the life of option.

This shows that Δ is the number of shares necessary
to hedge a short position in one option.

The value of the stock is $40. The value of portfolio today is

[image]

(20 – f)×1.02 = 22.5

20-f=22.5/1.02

f = 2.06

If we use risk neutral option, then

[image]

The expected value of the option in the neutral world is

0 x 0.58 + 5(1-0.58) = 2.10

The present value is

[image]

The result is consistent with no-arbitrage answer.

12.13. For the situation considered in Problem 12.12, what is the value of a 6-month European put option with a strike price of $51?

Verify that the European call and European put prices satisfy put–call parity.

If the put option were American, would it ever be optimal to exercise it early at any of the nodes on the tree?

[image]

 The value of option

[image]

payoff = K-S

The value of option is

[image]

The value of put plus stock price is

1.376 + 50 = 51.376

The value of call plus stock price is

1.635 +51e^-0.05*6/12 = 51.376

This proves that call parity holds.

To test whether it worth exercising the option early we
compare the value calculated for the option at each node
with the payoff from immediate exercise.

At node C the payoff from immediate exercise is

K-(S-(1-%)=51- (50×(1-0.05)) = 3.5.

Because this is greater than 2.866, the option should be
exercised at this node.

The option should not be exercised at either node A or node B.

12.15. Calculate u, d, and p when a binomial tree is constructed to value an option on a foreign currency.

The tree step size is 1 month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.

[image]

12.16. A stock price is currently $50. It is known that at the end of 6 months it will be either $60 or $42.

The risk-free rate of interest with continuous compounding is 12% per annum.

Calculate the value of a 6-month European call option on the stock with an exercise price of $48.

Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

If the stock price moves from $50 to $60 then the value of shares
is be 45Δ and the value of option is (60-48)=12, the total
value of the portfolio will be 60Δ-12.

If the stock price moves down from $50 to $42, the value of
shares is 42Δ and the value of option is 0, the total value of
the portfolio will be 42Δ.

The portfolio is riskless, if the value of Δ is chosen so that
the final value of the portfolio is the same for both
alternatives (60Δ-12= 42Δ, or Δ = 12/18).

A riskless portfolio is therefore

Long: 1 shares

Short: 1 option

If the stock moves up to $60, the value of the portfolio is

[image]

If the stock moves down to $35, the value of the
portfolio is [image]

Regardless of if the stock price moves up or down, the value
of the portfolio is always 28 at the end of the life of option.

This shows that Δ is the number of shares necessary
to hedge a short position in one option.

The value of the stock is $50. The value of portfolio today is
[image]

If we use risk neutral option, then

[image]

The expected value of the option in the neutral world is

12 × 0.6162 + 0×(1-0.6162) = 7.3944

The present value is

7.3944/ e^0.12*6/12 = 6.96

The result is consistent with no-arbitrage answer.

12.17. A stock price is currently $40.

Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%.

The risk-free interest rate is 12% per annum with continuous compounding.

(a) What is the value of a 6-month European put option with a strike price of $42?

(b) What is the value of a 6-month American put option with a strike price of $42?
[image]

[image]

[image][image]

12.19. A stock price is currently $30.

During each 2-month period for the next 4 months it will increase by 8% or reduce by 10%.

The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off [max(30-ST, 0)]², where ST is the stock price in 4 months.

If the derivative is American-style, should it be exercised early?

[image]

This type of option is known as a power option. The risk-neutral
probability of an up move, p, is given by

[image]

The value of the European option is 5.394. The second number
at each node is the value of the European option.  Early exercise
at node C would give 9.0, which is less than 13.2449.

The option should therefore not be exercised early if it is American.