## FRM Doubts

s.roy
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### FRM Doubts

1. What are the property of Principal Component Analysis.

2. When we apply effective duration and duration formula. Pl distinguish between two with suitable example.

3. What is full revaluation approach. How it is used.

4. When call option are sold delta becomes negative. When put option are sole delta becomes positive. Why ? I don't understand the concept for this

5. If gamma exposure is positive how to make gamma neutral. If gamma is negative how to make gamma neutral.

6. If delta exposure is positive, how to make delta neutral. If delta exposure is negative how to make delta neutral. Please explain clearly to understand the concept.

edupristine
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### Re: FRM Doubts

1.Principal Components Analysis (PCA) is a multivariate statistical technique that is used further as a dimension reduction technique. You could use PCA to identify the drivers of the response, which are called principal components. The principal components are not actual variables but rather hypothetical constructs (or factors similar to factors in factor analysis) that are ordered in terms of most explanatory to least explanatory. Typically, the first 2-3 components explain a significant amount of the information in the data, so the statistician makes a decision with respect to to how many components to utilize. PCA is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is the simplest of the true eigenvector-based multivariate analyses.

2. Effective Duration
Duration is the approximate percentage change in price for a 100 basis point change in rates. To compute duration, you can apply the following equation that was presented earlier in the guide.
Price if yield decline - price if yield rise / 2(initial price)(change in yield in decimal)
Effective duration takes into account the way in which changes in yield will affect the expected cash flows. It takes into account both the discounting that occurs at different interest rates as well as changes in cash flows. This is a more appropriate measure for any bond with an option embedded in it.

3."Full revaluation method calls for revaluation of the derivative at Var of the underlying" while Jorion says"Full valuation methods measure risk by full re-pricing the portfolio over a range of scenarios". This repricing could be done using risk factors generated by historical stimulation or Monte Carlo methods or Boot Strapping.

4. The delta is a ratio comparing the change in the price of an asset, usually a marketable security, to the corresponding change in the price of its derivative. Delta values can be positive or negative depending on the type of option.
For example, the delta for a call option always ranges from 0 to 1, because as the underlying asset increases in price, call options increase in price. Put option deltas always range from -1 to 0 because as the underlying security increases, the value of put options decrease. For example, if a put option has a delta of -0.33, if the price of the underlying asset increases by \$1, the price of the put option will decrease by \$0.33. Technically, the value of the option's delta is the first derivative of the value of the option with respect to the underlying security's price.

5.To effectively neutralize the gamma, we first need to find the ratio at which we will buy and write. Instead of going through a system of equation models to find the ratio, we can quickly figure out the gamma neutral ratio by doing the following:

Find the gamma of each option.
To find the number you will buy, take the gamma of the option you are selling, round it to three decimal places and multiply it by 100.
To find the number you will sell, take the gamma of the option you are buying, round it to three decimal places and multiply it by 100.

For example, if we have our \$30 call with a gamma of 0.126 and our \$35 call with a gamma of 0.095, we would buy 95 \$30 calls and sell 126 \$35 calls. Remember this is per share, and each option represents 100 shares.

Buying 95 calls with a gamma of 0.126 is a gamma of 1,197 (9,500*0.126).
Selling 126 calls with a gamma of -0.095 (negative because we're selling them) is a gamma of -1,197 [12,600*(-0.095)].

6. Position delta can be understood by reference to the idea of a hedge ratio. Essentially, delta is a hedge ratio because it tells us how many options contracts are needed to hedge a long or short position in the underlying. By changing the ratio of calls to number of positions in the underlying, we can turn this position delta either positive or negative.
For example, if an at-the-money call option has a delta value of approximately 0.5 - which means that there is a 50% chance the option will end in the money and a 50% chance it will end out of the money - then this delta tells us that it would take two at-the-money call options to hedge one short contract of the underlying. In other words, you need two long call options to hedge one short futures contract. (Two long call options x delta of 0.5 = position delta of 1.0, which equals one short futures position). This means that a one-point rise in the S&P 500 futures (a loss of \$250), which you are short, will be offset by a one-point (2 x \$125 = \$250) gain in the value of the two long call options. In this example we would say that we are position-delta neutral.

s.roy
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Joined: Wed Jul 26, 2017 7:32 am

### Re: FRM Doubts

In connection with question No. 5. you have explained as under.

For example, if we have our \$30 call with a gamma of 0.126 and our \$35 call with a gamma of 0.095, we would buy 95 \$30 calls and sell 126 \$35 calls. Remember this is per share, and each option represents 100 shares.

My doubts. How \$30 call with a gamma of 0.126 we would buy 95 \$30 calls and \$35 call with a gamma of 0.095 would sell 126 \$35 calls
please explain in details the calculation. In your example it is not clear gamma of 0.126 and gamma of 0.095 ( which is for buy and which is for sell).

Buying 95 calls with a gamma of 0.126 is a gamma of 1,197 (9,500*0.126).
Selling 126 calls with a gamma of -0.095 (negative because we're selling them) is a gamma of -1,197 [12,600*(-0.095)].

edupristine
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### Re: FRM Doubts

Long Gamma(+ve)
--all long options have +ve Gamma
--indicates that the position delta increases as the stock price rises, & decreases as the stock price falls.

Short Gamma
--all short options have -ve Gamma
--indicates that the position delta decreases as the stock price rises, & increases as the stock price falls.

We use to do the following in order to make gamma neutral (taking a quick fresh eg.)
Call Delta = 50, (sell 50 shares)
If Spot price up by 10, Delta = 60
So sell 10 more shares
If Spot price down by 10, Delta =50 again

Similarly, in case of Put delta
Put Delta = -50, (buy 50 shares)
If Spot price up by 10, Delta = -40
So sell 10 more shares
If Spot price down by 10, Delta =-60 again

The simple rule we should follow is that
Sell high

s.roy
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Posts: 25
Joined: Wed Jul 26, 2017 7:32 am

### Re: FRM Doubts

My doubts. How \$30 call with a gamma of 0.126 we would buy 95 \$30 calls and \$35 call with a gamma of 0.095 would sell 126 \$35 calls
please explain in details the calculation. In your example it is not clear gamma of 0.126 and gamma of 0.095 ( which is for buy and which is for sell).

The above calculation is not still clear. I understand delta /gamma concept.

My doubt is how \$30 call with a gamma of 0.126 we would buy 95 \$30 calls

edupristine
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### Re: FRM Doubts

Please refer the simpler example to understand the concept behind. Am giving you the crux in two lines, hope this would help you out:

If we have positive gamma that means one must be having all Brought options. And to make it neutral we have to sell the options (whether Call or Put),
&
If we have negative gamma that means one must be having all Sold options. And to make it neutral we have to buy the options (whether Call or Put).

s.roy
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Joined: Wed Jul 26, 2017 7:32 am

### Re: FRM Doubts

I am very much aware about the gamma and delta concept.

You are not addressing the fact which I wanted to know . In your example below please explain how one would buy 95 \$30 calls and sell 126 \$35 calls

For example, if we have our \$30 call with a gamma of 0.126 and our \$35 call with a gamma of 0.095, we would buy 95 \$30 calls and sell 126 \$35 calls. Remember this is per share, and each option represents 100 shares.

edupristine
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### Re: FRM Doubts

If we want to create a gamma neutral portfolio with two options: One of them is a call with 0.126 gamma and another is a call with 0.095 gamma. First of all, gamma of both the options is positive (in fact whenever you buy any option be it a call or put, gamma will be positive). It means we have to sell one option and buy another one. If we buy one of first option (forgetting the lot size, as it is common in both options so will not matter), then we have 0.126 and if we sell second option, we will get -0.095 gamma.

Lets say we buy x number of first options and sell y number of second options, our net gamma will be: 0.126x - 0.095y. Now, we want this portfolio to be gamma neutral then this equation is equal to zero.

So, 0.126x - 0.095y = 0
which means, 0.126x = 0.095y
so, x/y = 0.095/0.126 = 95/126
So, we can say x = 95 and y = 126 which means we are buying 95 number of first option and selling 126 second options.

We don't have to worry about the strike price as we are working solely on gamma.

Let me know if you have any further queries.

s.roy
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Posts: 25
Joined: Wed Jul 26, 2017 7:32 am

### Re: FRM Doubts

But still I have two doubts

1. If both gamma is positive then to make it gamma neutral we need to sell two call options. or as you suggested one buy one call and sell one call. In my opinion if gamma is positive then we need to sell both call option to make it neutral

2. Here call 30 and 35 is not used in the problem

I will be happy if you kindly clear my this doubts

edupristine
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### Re: FRM Doubts

1. You are right that if gamma is positive then we need to sell options. This one we just gave an example if we are creating a portfolio then buying will make it positive and selling will make it negative

2. We have used call 30 and call 35 by using their gamma