Finite Differences Vega calculation - confirmation on proper approach

Question

I have a MC simulation that uses finite differences to calculate the Greeks. It's for baskets and calendar spreads mostly.

Now the logical (to me anyway) approach to calculate Vega is to increase the input volatility by 1% (annual vol) for each leg (leg1 Vega: leg1 vol + 1%, leg2 Vega: leg2 vol + 1%, etc.), reprice, then subtract the initial price, leg by leg. Result = $ change from a 1% volatility increase on each leg of the option.

Today, my boss told me I should be using a much smaller number than 1% (as a FD shock). I responded: but isn't Vega supposed to show you the change in option value with a 1% increase in volatility (annualized)? Did I miss something here? Please if there are other methods, or I am completely wrong in my approach, I really need to know. And if my method is satisfactory, please confirm as well.

Note on my background: I came from a market risk role, and transfered into derivatives pricing. Not the other way around, so my understanding of the meaning of Vega is the way it is understood in market risk. It may very well be looked at differently from a financial engineering perspective, but that is news to me.

Thanks for your time.

Answer 1

Did you try using your tool for vanilla options (a single underlying)? Technically, Black Scholes Greeks are for infinitesimally small changes (not 1%).

That said, making shifts too small is dangerous - especially with numerical methods because you can get into an area where your standard error will be bigger than the change in shift in vol.

This answer shows the two main ways to compute greeks with bump and reprice:

• $[P(v+d/2) - P(v-d/2)]/d$ central difference -> bump up and down

• $[P(v+d) - P(v)]/d$ forward difference -> only shifting up

The former is the most frequently used for Delta and Gamma. The latter is what most systems (I came across) use for calculating vega; mostly done with an absolute one-sided shift of 0.0050 = 0.5% = 50 bps.

In complex models, you can compute Greeks in many other ways:

• market greeks: bump market (vol surface) and reprice (you can either recalibrate or not)
• model greeks: bump model parameters (e.g. LV surface) and reprice

If you use (probably the most sensible approach) a “bump market, recalibrate and reprice technique, you should ideally get close to closed-form expressions.

Depending on your access to other prices, you could compare your model to existing tools (e.g. Bloomberg DLIB), where you can actually manually decide what the shift in your Greeks should be (with default sizes of 1% for delta and 0.5% for vega in equities).

Answer 2

When using numerical differentiation based on a Monte-Carlo estimator you encounter two sources of error.

1. Monte-Carlo Error which is of order $\mathcal{O}(1/\sqrt{N})$ for a simulation of size $N$,
2. Numerical difference error. The forward difference has error of order $\mathcal{O}(h)$. The central difference is better with order $\mathcal{O}(h^2)$ for bump size $h>0$.

You should check which bump size $h>0$ gives you the best trade-off between those two error sources. Depending on the method you are using and the number of simulated scenarios 1% can be just right, two small or to big.

Additional i would recommend checking it would be beneficial to compute directional derivatives based on identifiable principle components. In your case that could be for example as follows:

1. Parallel shift: increase volatility of leg1 and leg2 the same amount up.
2. Spread shift: increase volatility of leg1 and decrease vola of leg2 the same amount.