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Delta-Gamma Value at Risk (VaR) as Taylor Series

Alberto asked a good question here about using the delta-gamma formula to estimate the VaR of an option position. Lu Shu (lushukai) gave an excellent reply and he itemized the four possible long/short call/put scenarios. This refers to one of the most fundamental quantitative applications in risk finance, which is to say the **Taylor Series**** expansion**. It is a pattern that applies to options (delta-gamma), bonds (duration-convexity), and even portfolio VaR (marginal VaR). Why do we do this? Rather than re-price complex positions, we use the first (and second, if needed) partial derivatives to *approximate *the potential loss given a shock to primitive risk factor(s). Our go-to example would be a vanilla bond. Rather than reprice the bond, we can approximate the price change given an assumption of shock to the yield, where negated dollar duration, ∂P/∂y, is the first derivative with respect to yield. For myself, I am a visual learner and prefer to visualize the slope’s tangents (like I did here), but it is not necessary. My only advice–and you can see this coming–is that you try to get underneath (or beyond) mere memorization, especially as it pertains to +/- signs. From a merely memorized perspective, I think Taylor Series can seem daunting, but it’s really not.

**Position Greeks versus Per-option (aka, Percentage) Greeks**

My focus here is to illustrate the four scenarios that Lu Shu itemized, and we will pay particular attention to the distinction between a percentage Greek and a position Greek (borrowing from my previous note here). Let’s recap that now. A call option must have a positive percentage (i.e., per option) delta between zero and 1.0; for an ATM call option, we might expect Δ(call) = N(d1) = 0.60. For the associated put option, we can expect Δ(put) = N(d1) – 1 = 0.60 – 1.0 = -0.40. Percentage delta arguably is a slight misnomer given they are unitless, but I am going to maintain this terminology given it already has a history. The percentage (per) option put delta is always negative. The percentage (per) option call delta is always positive. Now let’s refer to an option contract for which the size is 100 options. We can be long or short the option contract such that we have four possibilities (and illustrated with our examples) :

- Long call option contract is long 100 call options; e.g., the
*position*delta =**+100*** 0.60 =**+60** - Short call option contract is short 100 call options; e.g., the
*position*delta =**-100*** 0.60 =**-60** - Long put option contract is long 100 put options; e.g., the
*position*delta =**+100*** -0.40 =**-40** - Short put option contract is short 100 put options; e.g., the
*position*delta =**-100*** -0.40 =**+40**

The final signs (+/-) should be appealing. If we have a *positive position delta* (which is the case for a long call or short put), our position **gains **when the underlying risk factor (i.e., the stock price) increases. If we have a *negative position delta* (which is the case for a short call or a long put), our position **loses** when the underlying risk factor (i.e., the stock price) increases. Is the percentage/position semantic distinction really vital? No, not really. It’s just that a common question from new learners is, put delta was always negative, how can it be positive? And, in reality, we don’t need the percentage/position labels because the context will tell us. But, in the meantime, until you are comfortable, I prefer to stress the explicit multiplication wherein the short (long) is represented by negative (positive) quantity as a multiplier.

**Delta-Gamma approximation for Long And Short Option Positions**

The question pertains to the delta-gamma version (i.e., the version for the option asset class) of the truncated Taylor Series. Where δ is the delta and Γ is the gamma, the approximated price change is given by **Δprice = df = δ*ΔS + 0.5*Γ*ΔS^2. **In Lu Shu’s reply to the question, he itemized the four scenarios ...

… and then I illustrated them numerically. See below. My **deltas are 0.60 and -0.40**, while my **gammas are +0.10**; that’s a bit unrealistically high, for illustrative convenience (otherwise identical calls and puts will have the same gamma, and the percentage gamma is always positive). In yellow, you can see the risk factor assumptions: if we are long calls or short puts, the exposure is to a stock price drop; if we are long puts or short calls, the exposure is to a stock price increase. For simplicity, I assume a $1.00 shock to the risk factor. The delta term is a simple linear approximation; e.g., the short put has per-option delta of -0.40; the short put contract has a position delta of -100*-0.40 = +40; and if the stock drops by $1.00, the linear approximated *loss *on a single short put is $0.40 because the approximated price change is $0.40. The risk factor, ΔS, enters in its natural mathematical form. Under the Taylor Series columns, the gamma term is *(positive) +0.05 under all scenarios* because the ΔS gets squared, ΔS^2. Due to this, *the gamma term always adds* to the delta term: for the long call, the -0.60 linear price drop gets mitigated to -0.55; for the long put, the -0.40 linear price drop gets mitigated to -0.35; for the short call, the +0.60 linear price increase (which is a loss to the short position!) gets exacerbated to +0.65; and for the short put, the +0.40 linear price increase (again, a loss to the short position) gets exacerbated to +0.45. I hope that is a helpful illustration!