Binomial valuation for dividend paying stocks.Please help!

[monsieuruzairo3]

Dear David,

Binomial pricing model is quite commonplace in the syllabus for non-dividend paying stocks. However if we were to have dividend paying stock we will use
P(u) (Probability of up movt) = (exp^(r-q)*t - D)/(U-D)
This is pretty clear, however when we calculate payoffs(multiplied by probability) and discount them to today do we use disounting factor of exp^(-r)*t or exp^(-(r-q)*t)

I haven't found any question asking to calculate price for dividend paying stock using binomial method. Do you think there is a possibility of it springing up in the exam?

KR
Uzi

 

[ShaktiRathore]

Hi,
We need to discount the expected payoffs at the risk free rate r only and q does not enter here. q is associated with stock magnitude/value that is we reduce stock value from S0 to S0*e^-qt because the part of value of stock goes to paying dividends and this effectively reduces stocks value. So up move probability reduce to (exp^(r-q)*t - D)/(U-D) from (exp^(r)*t - D)/(U-D). I think it spurs into the exam. Just temember that the dicount factors do not change they remains the same the dividends only affects the stock price only.
thanks

 

[monsieuruzairo3]

Hi ShaktiRathore

Would you mind anaylzing this question too

A bronze producer will sell 1,000 mt (metric tons) of bronze in three months at the prevailing market price at that time. The standard deviation of the price of bronze over a three-month period is 2.6%. The company decides to use three-month futures on copper to hedge. The copper futures contract is for 25 mt of copper.
The standard deviation of the futures price is 3.2%. The correlation between three-month changes in the futures price and the price of bronze is 0.77. To hedge its price exposure, how many futures contracts should the company buy/sell?
Choose one answer.


1. Buy 25 futures
2. Sell 25 futures

I calculated the answer to be 25 futures and since he is long physical copper, he should short 25 futures. But the answer given is buy 25 futures. I think it is incorrect. What do you think?

KR
Uzair

 

[ShaktiRathore]

Hi
Since the producer is short the bronze(he is selling the bronze in three months so he is short the forward/future) the underlying commodity, he should naturally long the futures of this commodity. He is not long the physical copper but is short as he is selling and not buying the copper in three months.Please read the question carefully.Its not incorrect.
thanks

 

[Daniel26]

I think the answer should sell 25 Futures. This is because he will sell the bronze at the market price in 3m (this is the way i understand the question). So if the market price will decrease the underyling postion will lose. Therefore short the future should be the correct position...

Buy 25 Futures should be the case if you will sell the underlying at a fixed price in 3m. If the price will increase you have to sell at the lower price agreed on 3m ago. Long the Future will offset this 'negative' payoff.

plz correct me if I'm wrong...

Daniel

 

[David Harper CFA FRM]

Daniel26 is exactly correct (this question gives problems every year, it is a rare occasion when I disagree with the knowledgeable ShaktiRathore). I cannot say it better than Daniel: the producer here is a textbook case of a producer who requires a "short hedge" because he plans to sell in the future at "the prevailing market price at that time" (this is the key phrase, except for its incorrect answer, ha ha, the original question is good for including this specific phrase). Due to this, the producer's underlying exposure is actually long the future spot price (i.e., the producer gains on the future spot price appreciation), which requires a short futures to hedge (i.e., as he loses on future spot price depreciation, he requires his derivative payoff to profit from the same depreciation). If the producer, instead, we committed to selling in the future at a price of $X, then suddenly he has an underlying, short (forward) on the spot, and would require a long futures to hedge.

What Daniel says is more straightforward

Thanks,

 

[ShaktiRathore]

@David I misread the phrase "the prevailing market price at that time", F0=25 futures 3m price and S=20
Two cases are possible either Spot price goes up or down for the commodity after 3m,
i) If price goes down: S'=15 and i short the futures to sell at 25 then profit would be 10 and the loss from decreased price that is 20-15=-5 would offset the profit but overall net profit position on the other side if i am long then loss on future position is -10 and this will add to loss already incurred due to decreased price which is -5 thus long position does not hedge but short position do hedge effectively
ii)If price goes up: S'=30 and i short the futures to sell at 25 then loss would be -5 and the profit from increased price that is 30-20=10 would offset the loss but overall net profit position on the other side if i am long then gain on future position is 5 and this will add to gain already incurred due to increased price which is 10 thus long position does not hedge but short position do hedge effectively
From above we see that short position do hedge effectively but the long does not do it rather long position has asymmetric payoff while short has symmetric payoff that's the short position hedges against both price increases and decreases but the long position is beneficial only on upside. SO short is an effective hedge position
So answer should be sell futures, not long futures
thanks

 

[David Harper CFA FRM]

Thanks ShaktiRathore for the follow-up, appreciated!

 

[monsieuruzairo3]

Hi David Harper, CFA, FRM, CIPM ShaktiRathore Daniel26

Following up on the binomial problem, How about this one?

You are using the Merton model (Black-Scholes model for options on a stock paying a dividend yield) to price a European option on foreign exchange. The underlying is the AUD/CAD spot exchange rate quoted as 1.35 AUD per 1.00 CAD(1.35 AUD/CAD). If the AUD and CAD risk free rates are 2.4% and 2%, respectively, what would the rate inputs be in the Merton model for the risk free rate and dividend yield?
Choose one answer.
a. Risk free rate = 2%, Dividend yield = 2% Incorrect
b. Risk free rate = 2.4%, Dividend yield = 2.4% Incorrect
c. Risk free rate = 2%, Dividend yield = 2.4% Correct

d. Risk free rate = 2.4%, Dividend yield = 2% Incorrect

Now I calculated the answer to be D Rf =2.4%, Q= 2% since usually spot rates on currencies are described as Domestic/foreign. However answer is B which I fail to understand. Is the question wording open for interpretation for which currency is Domestic/Foreign.

What are your views on it?

Best
Uzi

 

[ShaktiRathore]

hi
AUD/CAD is the indirect quote(is not direct quote where DC is quoted in terms of FC)where CAD is the underlying in terms of AUD. The CAD is the domestic and the AUD is the foreign, its Foreign/Domestic so that its reverse values Rf=2 and Q=2.4%.
thanks

 

[k.simpson]

I'm a bit confused. I thought the generic form of a pair is "x A/B" meaning it takes x units of B to buy 1 unit of A, regardless of if it is an indirect quote or not. Then, how are the two statements in the question equivalent?

(1) 1.35 AUD/CAD ==> 1.35 CAD per 1.00 AUD
(2) 1.35 AUD per 1.00 CAD ==> 1.35 CAD/AUD

Maybe the writer confused the wording, and it should have been 1.35 CAD per AUD? But assuming he did that, and we just have the quote, without knowing if it is indirect or direct, how could we find the answer as to which is the domestic?

Thanks!

 

[ShaktiRathore]

Hi
Indirect quote is FC/DC is the foreign currency per unit of domestic currency.1.35 AUD/CAD ==> 1.35 CAD per 1.00 AUD is right wording i think. Quotes using a country's home currency as the unit currency (e.g., USD 1.35991 = EUR 1.00 in the Eurozone) are known as indirect quotationhttp://en.wikipedia.org/wiki/Exchange_rate, FC/DC~AUD/CAD ,So In above quote the CAD is the domestic currency and AUD is the foreign currency since CAD is unit currency and FC that is AUD is price currency. You are wrongly taking the quote as DC/FC in the above question as this is not a direct quote but an indirect one.
thanks

 

[k.simpson]

So ShaktiRathore, the implied convention then is that we generally assume indirect quotation, which is where the answer would then fall out.

Is there a precedence/convention listed somewhere that you can share? I know of the somewhat vague "Commonwealth country" convention (there are alot of Commonwealth countries), but is there anything more concrete? Thanks!

 

[acebhavik]

GARP 2007 Practice Exam question and its solution. Something seems fishy about the question
116. You are using the Merton model (Black-Scholes model for options on a stock paying a
dividend yield) to price a European option on foreign exchange. The underlying is the
AUD/CAD spot exchange rate quoted as 1.35 AUD per 1.00 CAD(1.35 AUD/CAD). If the
AUD and CAD risk free rates are 2.4% and 2%, respectively, what would the rate inputs
be in the Merton model for the risk free rate and dividend yield?
a. Risk free rate = 2%, Dividend yield = 2.4%
b. Risk free rate = 2%, Dividend yield = 2%
c. Risk free rate = 2.4%, Dividend yield = 2.4%
d. Risk free rate = 2.4%, Dividend yield = 2%
CORRECT: A
A foreign currency is analogous to a stock providing a known dividend yield because the
owner of the foreign currency receives a “dividend yield” equal to the risk free rate in the
foreign currency. We defined the spot exchange rate as the value of one of foreign
currency measured in the domestic currency, thus making the AUD the foreign currency.
Reference: John Hull, Options, Futures, and Other Derivatives, 6th ed. Chapter 14.
INCORRECT: B
This answer would be correct if the exchange rate was quoted CAD/AUD and was a futures rate.
INCORRECT: C
This answer would be correct if the exchange rate was a futures rate.
INCORRECT: D
This answer would be correct if the exchange rate was quoted CAD/AUD.

 

[Daniel26]

this is really confusing me...i think if you quote a currency like CAD/AUD = 1.35 AUD then rf = 2.4% and q = 2%

Answer 'a' makes only sense if you quote the currency like 1 divided by 1.35 so that 1 AUD = 0.74074 CAD

Think about this: If interest rates in AUD are higher than interest rates in CAD then the currency should trade with a report on the Forward Market if you quote it like CAD/AUD = 1.35 AUD. --> F=S*exp(r-q) --> r>q --> report

I guess the point in the question is this sentence: " The underlying is the AUD/CAD spot exchange rate quoted as 1.35 AUD per 1.00 CAD(1.35 AUD/CAD)." So they really mean AUD/CAD = 1/1.35 CAD but the quote ist CAD/AUD = 1.35 AUD...

 

[acebhavik]

I feel the explanation to the answer and the answer itself has some mismatches.

 

[Daniel26]

Underlying = AUD/CAD but quote is CAD/AUD =1,35 AUD ...so you have to take 1/1.35 to get AUD/CAD = 1/1.35 CAD...

Then anwer a is correct.

 

[Jhoony]

Hi David,

I would just like to let you know about a possible mistake in the question & answers PDFs.
R22.P1.T4Hull_1_9_v7.01.pdf
Problem 12.21. (page 31)
The risk free probability for futures should be (1 – d )/(u – d) and not (e^rt – d)/(u – d)

 

[NNath]

That's a good question. But I believe if its an option on a futures contract then the formula is correct. if it was a just a future contract then it will be (1 – d )/(u – d)