yield-/smile-curves are not directly related to greeks

The yield curve and smile curve (or surface) are the output of curve fitting engines, calibrated using a lot of commercial market data, discarding outliers. These are the most valuable soft market data objects. Directly used by decision makers, including trade pricing. Each trading desk guards these curves as highly proprietary trading secrets.

However, these curves are imprecise. You shouldn’t compute the gradient at each point on these curves. The most you do with such gradient is computing the gradient at the anchor vol point, which gives you the skew of the entire curve.

In contrast, the price-vs-impliedYield curve and the valuation-vs-spot curve are a different class of curves. (The valuation-vs-impliedVol curve is another example.) I call these “range of possibilities” curves. They are mathematically precise enough to let us compute gradient at every point. You get duration and delta. These are known as sensitivities, essential soft market data for risk management.

quantitative feel of bond duration – map absolute 1% -> relative x%

In the simplest illustration of modified duration, if a bond has modified duration == 5 years, then a 100bps yield change translates to 5% dollar price (valuation) change.

Note that 100 bps is an Absolute 1% change in yield, whereas the 5% is a Relative 5% change in valuation. If original valuation == $90 [1], then 100 bps =>> $4.5 change.

After we clear this little confusion, we can look at dv01. Simply set the absolute yield change to 1 bp. The valuation change would be a Relative 0.05% i.e. $0.045. The pattern is

Duration == 5 years => dv01 == 0.05% Relative change
Duration == 6 years => dv01 == 0.06% Relative change
Duration == 7 years => dv01 == 0.07% Relative change

Note 0.05% Relative change means 0.05% times Original price, not Par price.  Original price can be very different from par price, esp. for zero bonds.

[1] 90 in bond price quote means 90% of par value. For simplicity we would assume par is $100, though smallest unit is $1000 in practice.

(See P10 of YieldBook publication on Duration.)

Q: when is rho important

A: long-dated options.

I was told for both equity options and currency options, rho is usually the least significant price sensitivity greek.

For a customized “structured” option, maturity could be 10 years or longer. I feel the BS assumption of a constant risk-free rate breaks down — Over 10 years, interest rate could quadruple. Stock price could be directly affected even if all other factors remain constant. Implied vol could also be affected. However, to compute theoretical rho, we need to hold all of them constant and simulate a small bump in the time-invariant interest rate. In that context, over 10 years the effect on option valuation is non-negligible.

greeks(sensitivity) – theoretical !realistic

All the option/CDS/IRS … pricing sensitivities (known as greeks) are always defined within a math model. These greeks are by definition theoretical, and not reliably verifiable by market data. It’s illogical and misleading to ask the question —
Q1: “if this observable factor moves by 1 bp (i.e. 0.01%) in the market, how much change in this instrument’s market quote?”

There are many interdependent factors in a dynamic market. Eg FX options – If IR moves, underlier prices often move. It’s impossible to isolate the effect of just one input variable, while holding all other inputs constant.

In fact, to compute a greek you absolutely need a math model. Without a model, you can say instrument valuation will appreciate but not sure by how much.

2 math models can give different answers to Q1.

theta = a rent to own gamma #my take

* large positive gamma ~ large theta decay. Note theta is always negative since option valuation always decays with time.
* small positive gamma ~ small theta decay.

The extreme cases often help us simplify and better remember the basics —
– Large gamma is characteristic of ATM options
– Small gamma is for deep ITM/OTM options.

Some say “theta is a rent to own gamma”. Imagine you delta hedged your ATM long position — long gamma. Long gamma gives you upside profit potential whether underlier moves up or down. That’s an enviable position, but comes at a “rent” — With every day passing, you position loses value thanks to decay. The loss amount is the daily theta value (always negative). The larger the “upside” (gamma), the higher the daily rent (theta).

Negative gamma is for short option positions {large Negative gamma ~ large Positive theta}. In this blog we focus on long call/put positions either European or American style, so all gammas are positive.

vega roll-up makes no sense #my take

We know dv01, duration, delta (and probably gamma) … can roll up across positions as weighted average. I think theta too, but how about vega?

Specifically, suppose you have option positions on SPX at different strikes and maturities. Can we compute weighted average of vega? If we simulate a 100bps change in sigma_i (implied vol), from 20% pa to 21% pa, can we estimate net change to portfolio MV?

I doubt it. I feel a 100 bps change in the ATM 1-month option will not happen in tandem with a 100 bps change across the vol surface.

– Along the time-dimension, the long-tenor options will have much __lower__ vol changes.
– Along the strikes, the snapshot vol smile curve already exhibit a significant skew. It’s unrealistic to imagine a uniform 100 bps shift of the entire smile (though many computer system still simulates such a parallel shift.)

Therefore, we can’t simulate a 100 bps bump to sigma_i across a portfolio of options and compute a portfolio MV change. Therefore vega roll-up can’t be computed this way.

What CAN we do then? I guess we might bucket our positions by tenor and aggregate vega. Imperfect but slightly better.

2nd differential is the highest differential we usually need

When I first encountered the concept of the 2nd derivative, I thought maybe people will be equally interested in the 3rd derivative or 4th. Now I feel outside physics (+ math itself), folks mostly use first derivative and 2nd derivative. In classical physics, 2nd derivative is useful — acceleration. Higher derivatives are less used.

Note on notation. f” is (inherently) a function in the black-box sense that for each input value, there’s an output value. This function derives from the original function f. We write f”(x) in that context. However, f” can be usefully treated as a independent variable just like x,y and t, so we write f” without the (x). In this context, we aren’t concerned about how f” depends on x. That dependency might be instrumental in our domain, but at least for the time being we endeavor to ignore that and concentrate on how f” as an independent variable affects “downstream”

Graphically, 2nd derivative describes concavity or convexity —
^ When f” is positive, curve is concave upwards, regardless whether f(x) is positive or negative, whether f(x) is rising or falling, whether f'(x) is positive or negative. However, f’ is rising for sure.
^ When f” is Negative, curve is concave Downwards, regardless.

This observation is relevant to portfolio gamma. Take a short put for example. This position’s delta is always Positive but Falling with S, towards 0. The PnL graph is concave downward, so this gamma is always Negative. (See https://www.thinkorswim.com/tos/displayPage.tos?webpage=lessonGreeks) It’s important to clarify a few points and assumptions implicit in the above context —

* This PnL graph is purely theoretical. Underlier (S) has just one price of $88 right now, and it won’t become $1 even though the graph includes that price on the S axis.
* PnL graph is about the Current valuation (and PnL) but with Imaginary S prices. It shows “what-if” underlier price S becomes $1 in the next moment — that’s the meaning of the $1 on the x-axis.
** However, the most useful part of the PnL curve is the region around the current S — $88. This region reveals our sensitivity to underlier moves. It shows how much our short put valuation (and PnL) would gain or suffer When (not “if”) underlier moves a tiny bit in the next moment.

* The delta curve is purely theoretical. At the current S = $88, our delta is, say 0.51 or 0.47 or whatever. It won’t suddenly become 0.01 even though you may see this delta value at a high S. That 0.01 delta means “if S becomes so high tomorrow, our delta would be 0.01”

* there’s no “evolution over time” depicted in any of these graphs. Time is not an axis. These curves are pure mathematical functions describing “what if S is at this level”. In this sense the delta curve is very similar to the price/yield curve. Even the standard yield curve and forward curve are similarly Unrelated to so-called time-series graphs.

If you are confused about “on the far right put is OTM, but on a smile curve OTM puts are on the far Left”, read my other blog posts about OTM put.