P1 Focus Review: 7th of 8 (Valuation)

[David Harper CFA FRM]

P1 Focus Review 7th of 8: Valuation (continued)

• The 7th (of 8) Part 1 Focus Review video (Valuation) is located here at https://learn.bionicturtle.com/frm-...valuation-and-risk-models-focus-review-2-of-2

Concepts:

• Option Valuation (Binomial Trees)
• Option Valuation (BSM)
• Option Greeks (Delta, Gamma, Vega …)
• Option Greeks (Hedge)
• Other

Option Valuation (Binomial Trees)
The FRM exam does like to test a basic understanding of the binomial tree used to price an option. Please make sure you understand the mechanics:

1. The asset price is mapped out into the future nodes, and
2. The nodes are weighted and discounted back to the present value (so-called backward induction)

If you are thinking that a full binomial tree is too tedious for an exam question, you would be correct. Instead, expect to be asked about a single parameter (e.g., p) or to compute a single tree node.

I have noticed that GARP easily switches back and forth--and therefore can assume either--between either version of the asset process reviewed in Hull's binomial chapter:

• Where (u) and (d) are symmetrical (akin to a normal price distribution); e.g., asset can jump up or down 20%, such that u = 1.2 and d = 0.80; or
• Where u = exp[volatility*SQRT[T]) and d = 1/u, which assumes the price is lognormal (i.e., log returns are normal) and converges to BSM. In this model, if u = 1.2, then d = 1/1.2 = 0.833

Be prepared for either, the question needs to specify. Whichever is the price distribution assumption, the risk-neutral probability of an up move (p) is the same and computation of risk-neutral (p) is the most frequently asked question in our historical question database: p = (exp[r-q]*T - d)/(u-d).

Also, be prepared to apply the MAX() rule in the case of an American (can be early exercised) option: at interim nodes, the value of the node is the MAX(intrinsic value, discounted expected value).

Option Valuation (BSM)
A very common BSM-related question in our forum is, do I need to memorize Black-Scholes? Well, here are the AIMs, so according to GARP you should be able to ...

• "Compute the value of a European option using the Black-Scholes-Merton model on a dividend paying stock."
• "Use Black's Approximation to compute the value of an American call option on a dividend-paying stock."

Obviously, the safest thing is to know the formula in its entirety. But if you do not have time for that, my mere opinion is:

• Just memorize the outer BSM: c = S*exp(-qT)*N(d1) - K*exp(-rT)*N(d2) and p = K*exp(-rT)*N(-d2) - S*exp(-qT)*N(-d1). For P1, the exam probably will not go deeper than this outer BSM. As neither the TI BAII+ nor the HP 12c can calculate N(.), you will be given N(d1) and N(d2); or, worst case, a lookup table from which to retrieve them.
• If you have a more time, explore d2 due to its essential similarity to distance to default (DD) in Merton credit risk (P2)

Although not necessarily squarely in the BSM chapter, other closely related concepts you should know:

• Put-call parity (as previously discussed) is highly testable. Practice it, please
• BSM assumes log (continuous) returns are normal such that the asset price, or ratio of S(t)/S(0), is lognormal
• N(d2) is the risk-neutral probability that a call option will be struck; i.e., expire in-the-money. So, a good test question would be: what is the same for a put option?
• The lower and upper bounds; e.g., lower bound of Euro call is the BSM with the N(.)s removed, or equivalently, assuming zero volatility

Finally, it is not below GARP to query the theoretical optimality of early exercise, unfortunately (I find these rules irritating):

• American call option: if no dividends, never optimal; if dividends, might be optimal immediately before the ex-dividend date
• American put option: often desirable

Option Greeks (Delta, Gamma, Vega …)
Our historical sample includes questions for all the Greeks, in some form or another. Most questions are about: delta, gamma or vega. At a minimum, in my opinion, you should memorize:

• The shape of delta; e.g. deeply ITM calls have (percentage) delta approaching 1.0, deeply OTM calls have delta approaching zero
• How the shape of delta informs gamma (1st derivative of delta): at both OTM/ITM extremes gamma tends toward zero and therefore peaks ATM
• Why (percentage) gamma and vega are always positive such that long option positions (call or put) are long gamma and long vega
• call option delta = exp(-qT)*N(d1) and put option delta = exp(-qT)*[(d1)-1]; for non-dividend-paying stocks, call delta = N(d1) and put delta = [N(d1)-1]

Note we have a symbolic way to perceive the BSM for a European call:
c = S*exp(-qT)*[N(d1) = delta] - K*exp(-rT)*[N(d2) = risk-neutral probability of expiration in-the-money]

Option Greeks (Hedge)
I think hedging with Greeks just requires a few practice question to get the hang of it. Whe know what to expect: a question will ask you to hedge or neutralize a position with respect to one or two of the Greeks; e.g., neutralize delta, or delta and gamma with two trades.

As I mentioned here (http://forum.bionicturtle.com/threads/l1-t4-7-dynamic-delta-hedging.4839/), I think this is easier if you keep in mind that you want to neutralize the position Greek. For example,

• Consider a position in 100 call options with per-option (percentage) delta of 0.6. This Percentage Delta is 0.6. It is the unitless first partial derivative, dc/dS
• The Position Delta is 60 because Position Delta = Quantity * Percentage Delta.
• If we are long, we use (+) quantity: Position Delta (long 100 calls) = +100 * 0.6 = +60; if we are short , we use (-) quantity: Position Delta (short 100 calls) = -100 * 0.6 = -60
• To neutralize is to get the position Greek to zero; for example to delta neutralize 100 long call options with per-option (percentage) delta of +60, we need a trade that creates an offsetting -60 position delta. This can be achieved with a short potion in 60 shares: -60 * 1.0 (percentage) delta per share. Or, a long position in 150 put option with percentage delta each of -0.40 because +150 * -0.40 = -60. Or, a short position in 200 OTM call options with percentage delta per share of 0.30 because -200*0.30 = -60.

Other (Ratings, Ong, Dows' measures)
Among the remaining T4 topics, the most important for the exam are the following:

• Applying a one-year credit rating transition (aka, migration) matrix
• Definitions of investment versus non-investment (speculative) ratings
• Through-the-cycle versus at-the-point approaches
• Ong's expected loss (EL) is a popular test question
• While it is a good idea to memorize Ong's formula for unexpected loss (UL), most likely is a qualitative question; e.g., unlike EL, UL is non-linear with EDF
• Please note the EDF or PD is a Bernoulli, which is a special case of the binomial with n = 1, and therefore variance(PD) = PD*(1-PD)
• Dowd's weaknesses of VaR, especially: VaR is not sub-additive (can you given an example?) and therefore VaR is not coherent. But ES is coherent.
• Expected shortfall (ES): you should understand how ES is a conditional average and you should be able to calculate x% ES given the worst (n) losses in historical window.

 

[RiskNoob]

This is fantastic! thanks for posting reviews (as well as mock exams) in time-crunching period David, (I mean from BT's perspective, I am guessing BT team is busy preparing contents for 2013 on top of all this)

Regarding the remaining T4 topics, weights for operational risk and stress test seem to be small so I am expecting few qualitative questions (if any), do you have any opinion on these topics?

RiskNoob

 

[Bryon]

Hi David,
I have a question on part 4 under Operational Risk (Hull).
Do we need to memorize the beta for each business lines under the standardized approach? There are eight!
I've seen a question like this without betas given.

Thanks
Bryon

 

[David Harper CFA FRM]

RiskNoob thanks for for your nice feedback! In regard to T4 operational risk and T4 stress testing, you make an excellent point: my focus reviews have largely ignored them. Candidly, this is a function of my own bias, especially in regard to operational risk. My feedback to GARP, which includes my imminent feedback on the 2013 AIMs is that "operational risk" in T4 is something of a rogue topic undermined by two weak readings weak Allen reading. I think operational risk should devote entirely to T7 and its abridged appearance in T4 is not helpful (e.g. Hull Chapter 18 is an abrupt dive into Basel, without the top-down intro to Basel of T7. Basel ... how does that make sense?). Okay but that is just my bias ...

• In regard to T4. Operational Risk, clearly the two readings contain a large number of AIMs; i.e., I should revise the T4 focus review to include something on Operational Risk, candidly, assuming GARP retains OpRisk in T4 2013. GARP has previously quizzed some of the Linda Allen topics (e.g., top-down versus bottom-up) so her Chapter 5 should be reviewed. But it is also qualitative ("Describe," "Compare"), right?, the T4 OpRisk are almost entirely qualitative
  ... With the exception of Hull's "Calculate the regulatory capital using the basic indicator approach and the standardised approach"
  
• In regard to T4. Stress Testing, this sub-topic challenges me, in the Focus Review, because there simply is not much of an historical exam sample; e.g., only two questions in the FRM handbook. I just don't find much in our historical database, so for purpose of Focus Review, I'm actually okay with excluding it

Byron Technically, according to the AIMs ("Calculate the regulatory capital using the basic indicator approach and the standardised approach"), you do need to know the betas. However, pragmatically, I do not think they will ever be tested in Part 1 (is just my opinion). Can you identify or point me to the question you are thinking of, I am not aware of it, I am happy to be wrong?

I hope that helps, thanks,

 

[Bryon]

I was referring to a schweser practice test question. In fact, there is another question where we are asked to rank betas given business lines.

 

[David Harper CFA FRM]

Bryon, well, it's possible but I think Basel OpRisk SA betas would be a borderline "cheap shot" in the P1 exam. As I understand, some of schweser's questions are previous GARP sample questions (I can't verify) which, if it is the case that the question pre-dates 2009, then the question would pre-date the P1/P2 split. And if it is a pre-split OpRisk question, then I would especially stand by my belief that they won't be tested. But it's just my opinion, I have no inside knowledge, and the AIM does say what it says, so maybe i should not be arguing against the literal AIM ... thanks

 

[afterworkguinness]

Hi David, I'm not clear on what you said about GARP switching back and forth on the asset process... can you clarify with an example ? Thanks!

 

[David Harper CFA FRM]

Hi afterwork,

I'm referring to the two different approaches illustrated by Hull's comparison of an American put option in Figure 12.8 and 12.10. The setup is the same: a one-year American put option with strike (K) = 52 while asset (S) = 50 and riskless rate in 5.0%:

• [Hull Fig 12.8] If the question says "the stock price can go up or down by 20% in each time step" (the approach used by GARP in 2012 Practice P1.7, see here), then u = 1.2 and d = 0.8 and p = 0.6282
• [Hull Fig 12.10] If the question says "the stock price has a volatility of 30% per annum," then u = 1.350 and d = 1/u = 0.741 and p = 0.510. The probability of an up move (p) is computed with the same formula in either (normal or lognormal) process. How would you know to use this "lognormal process?" You would be given volatility: it has no other purpose in the binomial, if not to inform (u) and (d)

 

[afterworkguinness]

Sorry still don't get what you mean by "probability of an up move (p) is computed with the same formula in either (normal or lognormal) process. How would you know to use this "lognormal process?" You would be given volatility: it has no other purpose in the binomial, if not to inform (u) and (d)"

I was under the impression the only way to calculate p for a non dividend paying stock was: e^rt-d / u -d ???

 

[David Harper CFA FRM]

yes, there is only one p in the binomial; i.e., p = (exp[r-q]*T - d)/(u-d).
... but if u = 1.2, then either down (d) = 0.80 (if "symmetrical" or what i am imprecisely calling price ~ normal) or d = 1/0.80 = 0.8333 (if volatility is used to inform u and d; which is price ~ lognormal). Either is okay: binomial's virtue is flexibility, but only the lognormal converges on BSM if steps increase. Formula for (p) in same regardless. thanks,