Estimating Swaption Delta and Gamma from P&L Shocks

# Swaptions Gamma Interview Questions


# Swaptions Gamma Interview Questions












A while ago, I interviewed for a trader role and was given the below assignment (I didn't get the job). I wanted to revisit the questions to learn from my mistakes and be better prepared next time. Generally speaking, I have a good understanding of options and greeks but for some reason I'm having a hard time applying it here.

> Question 1 Based on the P&L profile below please calculate a 25bps gamma profile for the up and down shocks.

I'm not sure what's meant by gamma profile or why the +/-50bp values are highlighted. I see that the changes in P&L are non-linear and asymmetrical, similar to the payoff diagram of a long put.

I know that

$\\P\&L$ = $\delta$ * $\Delta$S + $\frac{1}{2}$ * $\Gamma$ * ($\Delta S^2$)

but I'm not sure how to isolate the delta and gamma. Any pointers would be appreciated!

> Question 2 The desk would like to purchase 5k of 25bps gamma with a goal of remaining delta and vega neutral. Utilizing the trades below develop the most cost efficient (cheapest 1m carry) way to achieve this goal.

To calculate 25bps Gamma, I used the following formula:

$\Gamma_{25}$ = $\frac{Dn_{25} + Up_{25} - 2 * Base}{2 * Base * 0.0025^2}$

which gives me the following values:

1m/10y = 102,204 3m/10y = 32,932 6m/10y = 16,095 1y/10y = 7,515 2y/10y = 3,671

I thought I should be able to re-calculate the Delta (10bp) as $\\Up_{10} - Base$ but the results don't make sense.

Can you confirm that my delta/gamma calculations are correct or tell me where I went wrong?

Finally, I tried to find a combination of straddles where the Gamma ~= 5k and Delta and Vega are close to 0 but I'm having a hard time finding a feasible solution, which makes me think my Gamma calculations might be incorrect.

## Answer by MaPy (score 5)

https://quant.stackexchange.com/a/42322

Using Taylor polynomials of 2nd order:$$V(r+h)\approx V(r) + \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$ $$V(r-h)\approx V(r) - \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$

The sum of the previous 2 equation will give us gamma as: $$Gamma = \frac{\partial^2{V}}{\partial{r}^2} \approx \frac{V(r+h) -2V(r) + V(r-h)}{h^2}$$ whereas the difference of the two equations will give us delta as: $$Delta = \frac{\partial{V}}{\partial{r}} \approx \frac{V(r+h) -V(r-h)}{2h}$$

if you substitute Up10 for $V(r+h)$, Dn10 for $V(r-h)$ and 0.001 for $h$ in the Delta equation and multiply by 0.0001 (to get the 1 bp Delta) you will re-calculate the Delta (as presented in your table).

You can calculate Gamma in the same fashion (Up25 for $V(r+h)$, Dn25 for $V(r-h)$ and 0.0025 in gamma equation and multiply by 0.0001)

## Answer by user35980 (score 3)

https://quant.stackexchange.com/a/75249

Refreshing to see this type of question in this forum. Regarding the first part of the question (let's leave the detailed calculation aside for a moment), I think what the interview is trying to get at is your ability at being able to spot a book's 'general' delta and gamma in the first instance (i.e. by just eyeballing) before actually getting to any computations (e.g. is the book long gamma?).

Normally traders would look at their delta and gamma profiles (these just mean your 1st and 2nd order risks profiles rather than the P/L profile shown in the question) - and these lead to various P/L scenarios (which a trader will be able to 'predict' based on her knowledge of these risk profiles). Here things seem to be the other way around and asks for some reverse engineering. P/L profiles of the sort shown in the question, though revealing and important for management, are not especially useful for traders for exactly this reason.

So in the first instance, it's quite clear this book is long gamma. Why? Well the first thing to answer is what's the delta? As user MaPy has indicated above, this is just -3,957 per 25bps ((-2745-5168)/2). This is your delta now i.e. for 0 shift in rates when your P/L is 0. Now if gamma was zero, you'd see +3,957 in the -25bps column and -3,957 in the +25bps column of the P/L profile. But what you actually see in these columns is +5,168 (=3,957+1,211) and -2,745 (=-3,957+1,211), respectively. In other words your delta changes by +1,211. So the -25bps/+25bps gamma profile is +1,211/+1,211. Hence the book is long gamma both on the up and downside (gamma profiles can be skewed too - not the case here).