Question about Hull's interest rate swap as FRAs

[Karthik & Vaidy]

Dear Harper

I am unable to understand the logic for the following question- source Market Risk question 9
Hull makes the distinction between investment and consumption assets- Answer Difference to the forward price-

Please explain the logic.

With REGARDS
m.kARTHIK

 

[David Harper CFA FRM]

M Karthik,

The answer was mistakenly truncated in the quiz application, I just replaced with the full answer:

"Hull wants to introduce the concept of a convenience yield, which the investment commodity does not have. Hull wants to show that consumption assets are not under the same no-arbitrage law as an investment asset due to the convenience yield. So that: the forward price (F0) can be less than the compounded spot price, (S0)*EXP[(rate+storage)(T)]."

Let me know if that isn't clear....Thanks, David

 

[Karthik & Vaidy]

Dear Harper
Question

Gamma Industries, Inc. issues an inverse floater with a face value of USD 50,000,000 that pays a semi-annual coupon of 11.50% minus LIBOR. Gamma Industries intends to execute an arbitrage strategy and earn a profit by selling the notes, using the proceeds to purchase a bond with a fixed semi-annual coupon rate of 6.75% a year, and hedging the risk by entering into an appropriate swap. Gamma Industries receives a quote from a swap dealer with a fixed rate of 5.75% and a floating rate of LIBOR. What would be the most appropriate type of swap Gamma Industries, Inc. should enter into to hedge their risk?

Answer is Pay floating and receive fixed swap.

I feel that the answer should be pay floating and receive fixed. Please explain the rationale of the solution .

With Regards
M.Karthik

 

[Karthik & Vaidy]

Dear Harper

Please help on the following question

An option on a stock has a payoff equal to the square of the positive excess of the stock price over the exercise price at expiration only if the stock exhibits an annual growth rate of 15% or more every year. Given the following assumptions and using a three-step binomial model and rounding to the nearest USD, which of the following would be the option’s price?

Time to expiration is 3 years, Stock price is USD 10, Standard Deviation is 15%, Risk free rate of return is 5%, Strike Price is USD 10. No dividend is paid.

Answer comes to USD 7.

I have understood the calculation of a, u d and P. However I am unable to understand the final step in the answer. Where the formula is given as

Option price = exp(-r * t) * p 3 * (current_price * u - strike_price) .

I shall be thankful to you if you can explain. I know this is a three step trinomial.

With Regards
M.Karthik

 

[Karthik & Vaidy]

Dear Harper

What is the likely date by which the formulas will be uploaded. Also please reply for the two questions posted by me yesterday.

With Regards
M.Karthik

 

[David Harper CFA FRM]

M.Karthik,

In regard to the formulas: as soon as i can.

Regarding the inverse floater, I illustrated for you here.

I think it is easy to make a mistake with this question. What ultimately matters is Gamma issues an (inverse) floater; therefore, they are pay-floating. Thye will receive fixed coupons to offset, but their exposure is the pay floating. To neutralize the net floating payments (i.e., pay inverse floater minus receive 6.75% fixed), they translate this (net) floating obligation into a fixed by entering a pay-floating, receive fixed.

you can also solve via:

sell inverse = pay = -(11.5% - L)
bond = receive= +6.75%
combined = +6.75% - 11.5% + L = L - 4.75%

So, the investor (before hedge) is receiving = L - 4.75%
So the investor hedges by paying LIBOR. If pay libor, recieve fixed (F), then investor net-net is:
(L-4.75%) - (L + F) = F-4.75% which is fixed (hedged)

David

 

[David Harper CFA FRM]

Hi M. Karthik,

In regard to the binomial option tree:

u = exp(15%) = 1.16. The condition the option has payoff "only if the stock exhibits an annual growth rate of 15% or more every year" is a way to limit our need for the binomial tree to only the upper path; i.e., the path that takes three steps up. Any other pay does not qualify and gives a zero. So that Stock(up, up, up) = ($10)*exp(15%)^3 = $10*exp(15%*3) = $15.68 at end of third year. So that payoff = ($15.68 - 10)^2 = $32.30.

That future payoff is discounted back to present value (and simplified because only the on path is needed):
= EXP(5%)*p*EXP(5%)*p*EXP(5%)*p*$32.30

Where each EXP(5%)*p is discounting for a single year. And, again this is because the binomial is typically:
= EXP(5%)*[p*Option_up + (1-p)*Option_down] but the Option_down value = 0 b/c it does not meet threshold.

Above simplifies to:

= EXP(-5%*3)*p^3*[$32.30]
= EXP(-5%*3)*p^3*[(10*exp(15%*3)-10)^2] = $7.05

I am not quite sure i get the formula you are showing but i don't feel the need to try and simplify further as this formula above seems to match the intuition of the tree...

Hope that helps!

David

 

[Karthik & Vaidy]

Dear Harper

I have a doubt on the folloiwing question

A currency swap has remaining life of 15 months. The swap exchanges 5% on 50 million pound sterlings for interest at 8% on $80 million US dollars once a year. Both interest rate term structures are flat: US dollar at 6% and UK pound sterling at 8%. All interest rates are quoted with annual compounding. The current exchange rate is 1.750 dollars per pound sterling. What is the value of the swap to the party paying pound sterling?

In the excel page valuation under the FORWARD RATE AGREEMENT (FRA) after calculating (Pound 2.50*1.741841 for 3 months why you are directly showing the calculation for 15 months (1+6%)^1.25/(1+8%)^1.25*1.75*2.50 and not showing for 9 months.

I am asking this question as the swap has a got remaining life of 15 months i.e. (3+6+6).

Please explain.

wITH Regards
M.Karthik

 

[kgolf20]

Hi David, just a quick question about your binomial tree response. I understand how we arrive at the stock value of 15.68 at the end of 3 years. But I would think the payoff would be equal to just 15.68 - 10 (Stock Px - Strike). Why is the payoff equal to (15.68 - 10)^2 ?

Thanks

Kyle

 

[David Harper CFA FRM]

@ M. Kathik:

My currency swap mimic Hull in assuming ANNUAL not semi-annual swap payments (coupons). Note in question, "The swap exchanges 5% on 50 million pound sterlings for interest at 8% on $80 million US dollars once a year." I realize we go from semi-annual interest rate swaps to annual currency, but I frankly try not to be original and I rather follow the author (Hull, here). Please note: It can certainly, of course, be done with semi-annual payments. That would be a fine solution, too, if the question were worded slightly differently.

@ Kyle: please note the question i was replying to. It was an atypical question, the payoff = the SQUARE of the intrinsic value. That's only why, otherwise, just as you say.

David