Question

A corporate treasurer is contemplating buying a five-month down-and-out put option on the Australian dollar with an exercise price equal to the current spot rate of the Australian dollar of USD0.7200 and a barrier at USD0.6000. Her treasury analyst estimates that the Australian dollar will either rise or fall by 5% during each one- month period. The term structure is flat in both Australia and the US, with risk free rates of 1.5% and 0.5% p.a. respectively, continuously compounded.

(a) Build a five-period binomial model in EXCEL to price the down-and-out put option. Make sure you include a binomial tree diagram for the exchange rate. (How is this done)

(b) Compare the price of the down-and-out put option with that of a standard European put option priced using a five-period binomial model. Account for the difference in price.

Answer 1

Exercie price = Current Spot Rate of the Australian dollar = USD0.7200

Barrier at USD0.6000

The Australian dollar will either rise or fall by 5% during each one-month period

Thus, let u (up) = 1.05 and d (down) = 0.95

(a) Five-year binomial model

Here, we start with the decision tree. We start with the starting price and iterate it with the possible outcomes of price either going up or going down.

In an down-and-out put option, if the price falls below the barrier price, then the option expires even before it reaches the expiration date.

There are 6 likely outcomes at the end of month 5. We consider the risk-free rate in the U.S. since we are buying the put option for an Australian dollar in USD.

rate = 0.5% p.a. continuous compounding.

Thus, converting it into monthly compounding rate r,

e0.005*(5/12) = (1+r)5

Thus, r = 0.04%

The risk-neutral probability for the underlying security to go up is as follows.

p = (e^r - d)/(u - d) = (e^0.0004 - 0.95)/(1.05 - 0.95) = 0.504

Thus, the risk-neutral probability for the underlying security to go down is 1-p = 0.496

The value of the put option would be as follows (note that in case of price exceeded strike price, the option will not be excercised, making the price to be 0. And in case the price falls below the barrier price, the option will expire).

The 5-year binomial model can be shown as follows to calculate the option price for a standard European option is as follows:

P = (e-2r)*[5C5.p5.P1 + 5C4.p4.(1-p).P2 + 5C3.p3.(1-p)2.P3 + 5C2.p2.(1-p)3.P4 + 5C1.p.(1-p)4.P5 + 5C0.(1-p)5.P6]

In this standard European option, there is no barrier price and thus the option does not expire before the expiry date due to any scenarios.

P1 to P6 are the six likely profits from exercising the regular European put option from price outcomes at the end of period 6.

P1	$0.7200	minus	$0.9189	=	$0.0000
P2	$0.7200	minus	$0.8314	=	$0.0000
P3	$0.7200	minus	$0.7522	=	$0.0000
P4	$0.7200	minus	$0.6806	=	$0.0394
P5	$0.7200	minus	$0.6158	=	$0.1042
P6	$0.7200	minus	$0.5571	=	$0.1629

Calculation in Excel:

	5	0.504001	0.495999
n	5Cn	p^n	(1-p)^(5-n)	Px	e^-2r	Product	Product
5	1	0.03252	1	0		0
4	5	0.064525	0.495999	0		0
3	10	0.128025	0.246015	0		0
2	10	0.254017	0.122023	0.039416		0.012217
1	5	0.504001	0.060523	0.104233		0.015898
0	1	1	0.03002	0.162878		0.00489
					0.9992	0.033004	0.032978

Thus, standard European put option Price = 0.032978 or $0.0330.

For the down-and-out put option, the probabilities of occurence of likely prices will change.

The option will expire if the price goes below $0.6000 anytime. This is possible in month 4 (if the price becomes $0.5864).

This will impact the probabilities of occurence in month 5.

The 5-year binomial model can be shown as follows to calculate the option price for the down-and-out put option is as follows:

P = (e-2r)*[5C5.p5.P1 + 5C4.p4.(1-p).P2 + 5C3.p3.(1-p)2.P3 + 5C2.p2.(1-p)3.P4 + p.4C1.p.(1-p)3.P5]

Note that the term in P6 is eliminated since the option cannot reach there, it will expire before that.

The term in P5 has been calculated from the only outcome in month 4 ($0.6482) from which it can occur with a chance of p (probability of stock moving up)

Calculation in Excel:

	5	0.504001	0.495999
n	5Cn	p^n	(1-p)^(5-n)	Pn	e^-2r	Product	Product
5	1	0.03252	1	$0.0000		0
4	5	0.064525	0.495999	$0.0000		0
3	10	0.128025	0.246015	$0.0000		0
2	10	0.254017	0.122023	$0.0394		0.012217
	4C1	p^2	(1-p)^3
1	4	0.254017	0.122023	$0.1042		0.012923
					0.9992	0.025141	0.02512

Thus, down-and-out put option Price = 0.02512 or $0.0251.

(b) As we can see, the down-and-out put option price is lower than the standard European put option price. This is because of the additional constraints and likelihood of the option expiry on hitting the barrier price even before the expiry period of 5 months. It lowers the likelihood of profit and thus the expected profit for the investor and thus, the option is valued at a lower price.