Questions from Hull: Options, Futures and Other Derivatives

[rekrap]

This would be a great place to post particular questions about examples and exercises from the Hull textbook.

An example of a question can be found here (http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?t=100165&page=2).

[campbell]

Thanks for doing this. I was thinking of starting a thread like this, as I am working Hull problems now.

[Kenny]

I probably have an older version than you (4th ed) since mine discusses Asion options in Ch 17. I'm also unsure of your exact question (again since I don't have the same edition and I don't see a similar question in mine), but perhaps this quote will help
When, as is nearly always the case, Asian options are defined in terms of arithmetic averages, exact analytic pricing formulas are not available. This is because the distribution of the arithmetic average of a set of lognormal distributions does not have analytically tractable properties.

[campbell]

I think I know what he's asking

The reason you can't "use" binomial trees for Asian options is that you lose that "recombinant" quality in regular binomial trees. You can't simply go back a node to value. All the 2^n paths are totally separate, so it's not exactly a binomial tree any more.

[seattleIslander]

The binomial tree constructed is based on the underlying stock. it can be used to value derivatives. The fact we are using it to value asian options does not make it recombine or not recombine. it is still the same tree.

In fact, Hull has an example for valuing asia options. The difficulty lies in the fact that there are many possible derivative values in one node. that's why he proposed to get the max and min, and a points in the middle equally spaced. Then interpolate to get the derivative price. Refer to his example in the chapter that talks about lookback options.

[campbell]

sI: I was referring to the option value not recombining on the tree. The whole point of using binomial trees as a method of calculation is to prevent having to calculate every possible path. Hull notes there's a way to approximate this in the manner you suggest, and I remember working out something like this with a lookback option.

More seriously, in an exam situation, we could be asked to value an asian option using info from a given binomial tree (or construct our own). I doubt they would give us anything beyond 3 time steps, and it is still doable by hand, as there's only 8 paths in that case.

[campbell]

With regards to arbitrage, it can be either side. It depends where the price disparities lie.

Just as one would short (or sell) a stock that's overvalued, or buy a stock that's undervalued.

[carzymathematician]

Could someone pls explain the difference between the LIBOR zero rates, LIBOR fwd rates and the LIBOR fwd swap rates?? If anyone has Goldfarb's manual refer to pg 429, Q5

[coca tea]

Could someone pls explain the difference between the LIBOR zero rates, LIBOR fwd rates and the LIBOR fwd swap rates?? If anyone has Goldfarb's manual refer to pg 429, Q5

LIBOR zero rates are the LIBOR rates, because there is no intermediate coupon assumed.
LIBOR forward rates are LIBOR rates expected in the future. They can be derived from the zero rates.
LIBOR forward swap rates are the swap rates that would apply between the maturity of a swaption (t) and the tenor of the underlying swap (T) such that the swap, at time t, has an "NPV" of zero.

[campbell]

Let me make sure I understand that third: the swaption would be entered into at time t, but the actual cashflow swap occurs at time T. The "value" of the swaption at time t would be 0.

You would have to determine the forward swap rates (other than market pricing on it) by Monte Carlo modeling of the LIBOR yield curves from now til time T, right?

[carzymathematician]

LIBOR zero rates are the LIBOR rates, because there is no intermediate coupon assumed.
LIBOR forward rates are LIBOR rates expected in the future. They can be derived from the zero rates.
LIBOR forward swap rates are the swap rates that would apply between the maturity of a swaption (t) and the tenor of the underlying swap (T) such that the swap, at time t, has an "NPV" of zero.

Tx coca tea! Ok, help me walk through this. If you have a swaption that matures at time t this means that you have an option to enter into a T-yr swap at time t. Now, a swap at its inception ( paying fixed + receiving floating) can be thought of as borrowing the notional amount at a fixed rate and investing in a floating rate bond. Since the NPV of swap at the inception must be equal to 0 then you can value the swap as the PV(fixed)- PV(floating) = 0 but note that the PV(floating) = notional amount. I get this but the part I don't understand is that Goldfarb in his question values the pv of the fixed side using the LIBOR fwd swap rates as the fixed coupon rates. But why is this so? Why not use the fixed rate?

[carzymathematician]

Can anyone say how good Actex Problem Supplement is? I would appreciate any feedback ASAP

[coca tea]

Tx coca tea! Ok, help me walk through this. If you have a swaption that matures at time t this means that you have an option to enter into a T-yr swap at time t. Now, a swap at its inception ( paying fixed + receiving floating) can be thought of as borrowing the notional amount at a fixed rate and investing in a floating rate bond. Since the NPV of swap at the inception must be equal to 0 then you can value the swap as the PV(fixed)- PV(floating) = 0 but note that the PV(floating) = notional amount. I get this but the part I don't understand is that Goldfarb in his question values the pv of the fixed side using the LIBOR fwd swap rates as the fixed coupon rates. But why is this so? Why not use the fixed rate?

The forward swap rate 7.63% is the expected fixed rate (given the LIBOR zero rates) you would have to pay if you enter a swap at t=1.
The fixed rate given 6% is the fixed rate you can lock in today for the swap at t=1 (the strike rate).

[carzymathematician]

The forward swap rate 7.63% is the expected fixed rate (given the LIBOR zero rates) you would have to pay if you enter a swap at t=1.
The fixed rate given 6% is the fixed rate you can lock in today for the swap at t=1 (the strike rate).

This would make sense since the only way the optionholder would enter into a swap is if the rate they are to pay is less than the rate that the market is expected to offer at that time.

Well my next question is, is the floating rate pd in a swap locked in for the term of the swap?

[snoo]

I have a question about problem 20.11. In the solutions manual they say M(-.84162,-1.03643;0.3) = 0.0522. I believe M is a bivariate normal distribution. How do you go about calculating M?

Thanks in advance.

[carzymathematician]

Re: Q13.7, Hull - The question asks for the prob distribution of the rate of return earned over a 2-yr period. Is this the average rate of return or is it the total rate of return that the question is referring to? I interepreted it as the total, the book uses the distn for the average. Any thoughts?

[rekrap]

I have a question about problem 20.11. In the solutions manual they say M(-.84162,-1.03643;0.3) = 0.0522. I believe M is a bivariate normal distribution. How do you go about calculating M?

Thanks in advance.
M is the bivariate normal.

Hull tells you how himself here (http://www.rotman.utoronto.ca/%7Ehull/Technical%20Notes/TechnicalNote5.pdf) [warning: PDF]

I wouldn't anticipate a question like this on the exam, if that calculation is needed. Moving along, nothing to see here.

[rekrap]

Re: Q13.7, Hull - The question asks for the prob distribution of the rate of return earned over a 2-yr period. Is this the average rate of return or is it the total rate of return that the question is referring to? I interepreted it as the total, the book uses the distn for the average. Any thoughts?

See example 13.3 on page 284 for a similar problem (3-year period). Regardless of what you might think, that result is intended, and it will be the expected result if a similar question is asked on the exam.

I would consider this to be a simpler problem, maybe part of a larger question on the exam, where they would ask to do a confidence interval, like the example. Know how the formula works... it's probably on the formula sheet (though I don't have mine handy to verify).

[Jy88]

Anyone understand how this works? I couldn't get the example illustrated.

I understand we trying to replicate up-and-out option payoff and that at
t=0.75, payoff is a call option with srike price 50. And at 0<=t<=0.75, payoff is call option with stock price 60. Did I get this right?

For Option A with strike price 50, t=0.75, r=10%, sigma=30% we get 6.99 i.e. the initial value.
For Option B position, onwards, I don't understand how we get -2.66 and initial value of -8.21.
The same applies for Option C and Option D.

[rekrap]

For Option A with strike price 50, t=0.75, r=10%, sigma=30% we get 6.99 i.e. the initial value.
For Option B position, onwards, I don't understand how we get -2.66 and initial value of -8.21.
The same applies for Option C and Option D.

We can do this as follows:
Option A is the baseline boundary (standard call on S=50 with 9-mo maturity and strike of 50): Position is taken to be +1.00, price is 6.99

Then choose time steps (the more steps the better the price approx). Hull chose 3 month steps to make the example simple/quick.

Option B is the first call 3 months earlier (so only T=.25 until 9 months maturity): price (S=K=60, T=0.25) is 4.33
The price of Option A (S=60, K=50) at this point (T=.25) is 11.54, and since we must have a total value of 0 we must get enough of B to negate the +1.00 position of Option A, and so position of Option B is -11.54/4.33 = -2.66

Option C is a 6-mo call at 3 months (so T = 0.25 until Maturity) and price (S=K=60, T = 0.5) is 4.33 still. Value of Otion A (S=60, K=50, T=.5) and Option B (S=K=60, T=0.5) at this point is 1*(13.22)-2.66*(6.54) = -4.21
So the position of C is 4.21/4.33 = 0.97

Option D is a 3-mon call at time 0, so price still 4.33. Price of Option A (S=60, K=50, T=0.75) is 14.77, Option B (S=K=60, T=0.75) is 8.39, and Option C (S=K=60, T=0.5) is 6.54, so the position at this point is (14.77)-2.66*(8.39)+0.97*(6.54) = -1.23, so the position in Option D is 1.23/4.33 = 0.28

Now as for the initial value at each point in the table, the original Option A (S = 50, K = 50, T = 0.75) costs 6.99; the original Option B (S = 50, K = 60, T = 0.75) costs 3.08 (so the value is -2.66*3.08 = -8.21); the original Option C (S = 50, K = 60, T = 0.5) is 1.83 (so the value is .97*1.83 = 1.78); and the original Option D (S = 50, K = 60, T = 0.25) is 0.61 (so the value is .28*.61 = .17). The total initial value of the portfolio is 6.99-8.21+1.78+.17 = 0.73

Hull tells us the analytic market price is 0.31, which means we need more time steps to get more accurate. 18 steps (a new option every half-month) gets us 0.38 (much better, but a lot more work), and 100 steps gets 0.32.

This is a good example to work through, becuase it's prime for an exam type question, so I'm glad you asked.

[campbell]

Well, they did do such an exam question last fall, so perhaps they will skip it this sitting.... of course, I totally screwed it up last fall and perhaps many other people did as well, so they'd think it a great question type to try again.

[rekrap]

Well, they did do such an exam question last fall, so perhaps they will skip it this sitting.... of course, I totally screwed it up last fall and perhaps many other people did as well, so they'd think it a great question type to try again.

That's right! How could I forget that question. Stupid barriers... :swear:

Of course, it makes much more sense the second time around, so why would they ask us to do it again? :dsmile:

[campbell]

Oh, I know how to do that problem =now=.

[Jy88]

Really...thank you so much rekrap for explaining that. Now is clearer.

I got stucked on another question on the study notes. Would you have any clue on that? (pasted from the study notes link)

Exhibit 4 in the study notes V-C107-07 chapter 21?

I don't understand how accretion, rolldown, shift, twist, butterfly component works? Anyone can shed some light?

The example says the Alabama Power 6.85% of 2002 was priced for settlement on 1/1/96 at 101.72, for a yield to maturity of 6.525%. Repricing at this yield for settlement on 2/1/96 obtain a price of 101.7, which give a price return of -1bp. How do you get -1bp? And 101.712+2.854 = 104.566. Where does 2.854 come from?

It follows on the say the OAS 21.6bp. Where is that from? and so on. At the end it says the OAS of 22.2bp which is intepreted as a -1bp return due to change in spread?? Why??

Thank you so so much. I am totally confused.

[rekrap]

:-? in goldfarb manual, section 1, hull chapt13,
question1
if y=ln(St), St follow gemetric brownian motion, dS=mean*S*dt+vol. * S* dz

y will follow normal distribution,
mean of that normal distribution is ln(S0)+(mean-vol^2/2)*T

but the mean of St is S0*exp(mean*T),

why mean of St is not S0*(exp(mean-vol^2/2)*T)? Thanks.

I believe Goldfarb makes it clear at the very beginning of his summary. He explains that your final equation simplifies (the exercise is left to the reader?) to the penultimate equation you anticipated.

So, it's both! :tfh:

[rekrap]

is asian option path dependent as look back option?

if use binomial tree, for convenience , look back option 's tree is not recombined, but asia option's tree is still recombined...

This seems strangely familiar to your first question (http://www.actuarialoutpost.com/actuarial_discussion_forum/showpost.php?p=1934942&postcount=3) in this thread.

[rekrap]

when value the credit swap, they use a formual set the prob. of default conditional on M, what is M refer to? Thanks.

M is a macro/market factor that affects defaults for all companies.

[bagheera]

This reply is with reference to hw0799's question about the expected stock price.
In my opinion, E(S_T)=S_0*exp(mu*T) where mu is the arithmetic mean of the percentage changes over small, successive time segments in the stock price over T. So the first equation is correct. This is because if y follows the lognormal distribution E(y)=exp(alpha+beta^2/2), where alpha and beta are the parameters of the lognormal. Here S_T is lognormal and so alpha=ln S_0 + (mu-sigma^2/2)T, beta=sigma*T^0.5. (See 13.3 in Hull to get alpha and beta.) After substitution you can get the above result.
The second equation is incorrect. The implicit reasoning you probably used to obtain the second result is this:
S_T=S_0*exp(xT) (here x is the geometric mean of the percentage changes in the stock price over small, successive time segments in time T or the continuously compounded rate of return over T)
=>E(S_T)=E(S_0*exp(xT)) ...1
=>E(S_T)=S_0*E(exp(xT)) ...2
=>E(S_T)=S_0*exp(E(xT)) ...3 (problematic..see below))
=>E(S_T)=S_0*exp((mu-sigma^2/2)*T)...4

Now step 2=>3 in the above is wrong because E(exp(xT)) > exp(E(xT)). See Hull page 286, first line. The above inequality can be derived from the inequality mentioned on the first line by exponentiating both sides. In effect, 4 above is wrong.

[campbell]

That last bit bagheera explained -- pulling the expected value operator through the exponential -- does cause problems. The expected value operator is linear, so you can do that only with linear functions on the random variable.

One of the more important things to learn with regards to this is something called Jensen's Inequality. It helps explain why an annuity factor is not the same as an annuity-certain to life expectancy (unless the interest rate is zero, in which case, they're the same).

I've had to explain that more than once to annuitants, and it's something that actuaries should know.

[bagheera]

when to value of default risk for currency forward or swap, when to treat it as call option , when put?

in goldfarm manual, page 474, q16, it is a currency swap, the default risk is calculated as a put option.

in 2005, q 13, a currency forward contract, the currency forward's value of default is treated as a call option.

it is kind of confusing. Thanks. :popcorn:

If you are selling a foreign currency (which is nothing but an asset like a stock with respect to option valuation), a put will be involved, and if you are buying a foreign currency, a call will be involved. In the example on page 474, party A is paying (selling) sterling and receiving dollars. So the claim amount will be of the form "dollars receivable - sterling payable*exchange rate in dollars/sterling" . An informal way to think of this is to note that the future exchange rate (the "unknown") appears after the minus sign, which again validates the presence of a put option.
In the 2005 example, the company is buying Euro. Therefore using a similar logic, a call is involved. Loss, when the counterparty defaults, is of the form
"exchange rate (dollars/euro) at the time of forward delivery - current exchange rate". The future, unknown exchange rate appears before the minus sign indicating that this is a call option.
What I did not understand in the 2005 example is where r_f (the foreign risk free rate) comes from. Can anybody point that out?

[hershey220]

There are two optimal hedge ratios. In ch 3 of Hull, h=(Corr)(vol of S)/(vol of F). Then in IMI ch 26, h=-(beta of S)/(beta of F). It seems like they are both using futures. How do you know which one to use? Are they just variations on each other?

[rekrap]

There are two optimal hedge ratios. In ch 3 of Hull, h=(Corr)(vol of S)/(vol of F). Then in IMI ch 26, h=-(beta of S)/(beta of F). It seems like they are both using futures. How do you know which one to use? Are they just variations on each other?

IMI is a bunch of Babbel... :wink:

Seriously, you should be able to tell which formula to used based on the information and context of the question (i.e., do they give you Betas or volatities?).

[bagheera]

There are two optimal hedge ratios. In ch 3 of Hull, h=(Corr)(vol of S)/(vol of F). Then in IMI ch 26, h=-(beta of S)/(beta of F). It seems like they are both using futures. How do you know which one to use? Are they just variations on each other?


In my opinion, you use the Hull ratio when the asset underlying the futures contract is different from the asset being hedged. In this case, the best you can do is to the minimize the variance of the hedge.
But when the asset underlying the futures is also the asset being hedged, you use the hedge in IMI. This hedge is called a full hedge in which you can reduce the variance of the hedge to zero due to perfect correlation between the change in futures value and the change in asset value.
So I suppose the full hedge, the hedge in IMI, is a special case of the Hull hedge.

[Jy88]

In the example shown, to calculate the expected loss from default;

Isn't that just the difference between the PV of the bond discounted at risk free rate and the PV of the bond discounted at risky rate? Why do we need to divide that by the PV of bond at risk free rate again?

Thanks.

[campbell]

Because you want to look at LGD in percentage terms, not absolute terms.

[Jy88]

hmm...true...ok thanks :-p

[Jy88]

In Hull, he notes that a critical input to the determination of value of a CDS is the recovery rate estimate that we cannot observe. However, for plain vanilla creidt default swaps the price is generally insensitive to different assumptions about the recover rate. Why is this true?

In goldfarb, he provided the answers as follows:

Because recovery rate assumption enters into the pricing of CDS in two largely offsetting ways. It affects the risk neutral default probabilities and it affects the payoffs in event of default. Larger recovery rates imply larger default probabilities and smaller CDS payoffs and so these effects are offsetting.

Anyone can share some insight on this.

I don't really get the point. Say I based on Merton's model where the probability of default is N(-d2). And the recovery in event of default is
= V0*N(-d1)/N(-d2). So if the probability of default N(-d2) increase, then the recovery in event of default will decrease. The seond sentence looks sensitble to me.

However, the Po = [Fe^-rT - V0*N(-d1)/N(-d2)] * N(-d2). Then in this case, the price would increased as if the N(-d2) increase. Even the recovery amount is offset. How would that then become insensitive to pricing assumption?

Thanks.

[Jy88]

see his question Hull, 20, q1 in goldfarm manual.

there is a approximate formula for default intensity rate: h=s/(1-R).
I understand by using this formula, when R increase, h increase, but the expected default loss equal to (1-exp(-h*t))*(1-R), expected default loss stay roughly the same.



Hw0799,

I think we are on the wrong track actually, the question ask about Credit default swap. On page 517 of the goldfarb manual it mention that if we use the same recovery rate to calculate the default probability and for the CDS payoff, then it would not have significant impact on the estimated CDS spread. What do you think? So if our
Expected payoff = default probability * recovery rate.

If recovery rate increase, using h=s/(1-R); then we expect a decrease in default probability. Thereby the expected payoff would not change. Therefore, we have offsetting effect. Does this make sense to you?

[bagheera]

In my opinion, you use the Hull ratio when the asset underlying the futures contract is different from the asset being hedged. In this case, the best you can do is to the minimize the variance of the hedge.
But when the asset underlying the futures is also the asset being hedged, you use the hedge in IMI. This hedge is called a full hedge in which you can reduce the variance of the hedge to zero due to perfect correlation between the change in futures value and the change in asset value.
So I suppose the full hedge, the hedge in IMI, is a special case of the Hull hedge.

There is another way of looking at this which leads me to conjecture that both ratios are nearly equivalent..
Let me present an informal argument. Say we are hedging asset A with futures. I assume here that the futures contract is perfectly correlated with the market, almost a proxy for the market index. Let rho_A be the correlation of A with the market or futures.
According to IMI hedge ratio=beta_A/beta_futures=(rho_A*sigma_A/sigma_market)/(sigma_futures/sigma_market)
=rho_A*sigma_A/sigma_futures=Hull's ratio
The point is beta_A/beta_futures can be thought of informally as beta of A relative to futures. And we know that this is rho_A*sigma_A/sigma_futures in a similar way that beta_A with respect to the market index is rho_A*sigma_A/sigma_market (CAPM).