8.9 The Greeks · M08 Option pricing

Motivation

The Greeks are the partial derivatives of the option price. They tell a desk how the book's value changes when spot, time, volatility and rates move, and they are how risk limits are written. Analytic forms make them cheap and exact.

Closed-form Greeks#

Setup

Write $V$ for the BSM price (8.23)–(8.24) as a function of $(S,\tau,\sigma,r)$ with $q$ fixed. Define delta $\Delta=\partial V/\partial S$ , gamma $\Gamma=\partial^2V/\partial S^2$ , vega $\mathcal V=\partial V/\partial\sigma$ , theta $\Theta=\partial V/\partial t=-\partial V/\partial\tau$ , rho $\rho=\partial V/\partial r$ . We need three derivatives of $d_1,d_2$ , all immediate from (8.22) and $d_1-d_2=\sigma\sqrt\tau$ :

\frac{\partial d_1}{\partial S}=\frac{\partial d_2}{\partial S}=\frac{1}{S\sigma\sqrt\tau},\qquad \frac{\partial d_1}{\partial\sigma}-\frac{\partial d_2}{\partial\sigma}=\sqrt\tau,\qquad \frac{\partial d_1}{\partial\tau}-\frac{\partial d_2}{\partial\tau}=\frac{\sigma}{2\sqrt\tau},\qquad \frac{\partial d_1}{\partial r}=\frac{\partial d_2}{\partial r}.

Lemma 8.9.1the key identity

Se^{-q\tau}\varphi(d_1)=Ke^{-r\tau}\varphi(d_2). \tag{8.26}

Proof

$\varphi(d_2)=\varphi(d_1-\sigma\sqrt\tau)=\varphi(d_1)\exp\big(d_1\sigma\sqrt\tau-\frac12\sigma^2\tau\big)$ . By (8.22), $d_1\sigma\sqrt\tau=\ln(S/K)+(r-q+\frac12\sigma^2)\tau$ , so the exponent equals $\ln(S/K)+(r-q)\tau$ and $\varphi(d_2)=\varphi(d_1)\,\frac SK e^{(r-q)\tau}$ . Multiply by $Ke^{-r\tau}$ .∎

Every Greek below uses (8.26) to cancel the terms that come from differentiating $d_1$ and $d_2$ .

Proposition 8.9.2the Greeks of a European call

$\Delta_c=e^{-q\tau}N(d_1)$ .
$\Gamma=\dfrac{e^{-q\tau}\varphi(d_1)}{S\sigma\sqrt\tau}$ .
$\mathcal V=Se^{-q\tau}\varphi(d_1)\sqrt\tau$ .
$\Theta_c=-\dfrac{Se^{-q\tau}\varphi(d_1)\sigma}{2\sqrt\tau}-rKe^{-r\tau}N(d_2)+qSe^{-q\tau}N(d_1)$ .
$\rho_c=K\tau e^{-r\tau}N(d_2)$ .

Proof

(1) Differentiate (8.23): $\partial c/\partial S=e^{-q\tau}N(d_1)+Se^{-q\tau}\varphi(d_1)\partial_Sd_1-Ke^{-r\tau}\varphi(d_2)\partial_Sd_2$ . Since $\partial_Sd_1=\partial_Sd_2$ , the last two terms are $(\partial_Sd_1)[Se^{-q\tau}\varphi(d_1)-Ke^{-r\tau}\varphi(d_2)]=0$ by (8.26). (2) Differentiate (1): $\Gamma=e^{-q\tau}\varphi(d_1)\partial_Sd_1$ . (3) $\partial c/\partial\sigma=Se^{-q\tau}\varphi(d_1)\partial_\sigma d_1-Ke^{-r\tau}\varphi(d_2)\partial_\sigma d_2=Se^{-q\tau}\varphi(d_1)(\partial_\sigma d_1-\partial_\sigma d_2)=Se^{-q\tau}\varphi(d_1)\sqrt\tau$ , by (8.26). (4) $\partial c/\partial\tau=-qSe^{-q\tau}N(d_1)+rKe^{-r\tau}N(d_2)+Se^{-q\tau}\varphi(d_1)(\partial_\tau d_1-\partial_\tau d_2)$ , again by (8.26), and the last factor is $\sigma/(2\sqrt\tau)$ . Then $\Theta=-\partial c/\partial\tau$ . (5) $\partial c/\partial r=K\tau e^{-r\tau}N(d_2)+(\partial_rd_1)[Se^{-q\tau}\varphi(d_1)-Ke^{-r\tau}\varphi(d_2)]=K\tau e^{-r\tau}N(d_2)$ .∎

For the put, differentiate parity $p=c-Se^{-q\tau}+Ke^{-r\tau}$ : $\Delta_p=\Delta_c-e^{-q\tau}=-e^{-q\tau}N(-d_1)$ ; $\Gamma$ and $\mathcal V$ are the same as for the call; $\Theta_p=\Theta_c-qSe^{-q\tau}+rKe^{-r\tau}=-\frac{Se^{-q\tau}\varphi(d_1)\sigma}{2\sqrt\tau}+rKe^{-r\tau}N(-d_2)-qSe^{-q\tau}N(-d_1)$ ; $\rho_p=\rho_c-K\tau e^{-r\tau}=-K\tau e^{-r\tau}N(-d_2)$ .

Table 8.2. Greeks for $S=K=100$ , $\tau=1$ , $r=5\%$ , $q=0$ , $\sigma=20\%$ (units: vega per 1.00 of $\sigma$ , theta per year, rho per 1.00 of $r$ ). Tested to $10^{-6}$ .

	Delta	Gamma	Vega	Theta	Rho
Call	0.636831	0.018762	37.524035	−6.414028	53.232482
Put	−0.363169	0.018762	37.524035	−1.657880	−41.890461

Desk units: vega per vol point is $37.52/100=0.375$ ; call theta per calendar day is $-6.414/365=-0.0176$ .

The theta–gamma relation#

Proposition 8.9.3theta–gamma relation

For any price $V(t,S)$ satisfying the PDE (8.18),

\Theta+\tfrac12\sigma^2S^2\Gamma+(r-q)S\Delta-rV=0. \tag{8.27}

Proof

This is (8.18) with the Greeks substituted for the partial derivatives.∎

For the ATM reference call: $-6.414028+\frac12(0.04)(10{,}000)(0.018762)+0.05\times100\times0.636831-0.05\times10.450584=-6.4140+3.7524+3.1842-0.5225=0.0000$ . The tests check (8.27) for calls and puts at several spots to $10^{-9}$ .

For a delta-hedged position (net $\Delta=0$ ) with small $rV$ , (8.27) says $\Theta\approx-\frac12\sigma^2S^2\Gamma$ . Long gamma costs theta; short gamma earns it. Over $dt$ , the hedged long option's P&L is approximately

d\Pi\approx\tfrac12\Gamma S^2\big[(dS/S)^2-\sigma^2dt\big], \tag{8.28}

positive if the realised squared return beats the implied variance. This is the one-line summary of option trading, and M09 develops it.

Rupee Greeks for a Nifty lot#

Worked example 8.10rupee Greeks for one Nifty lot

Use Worked example 8.8: Nifty December 24,700 call, $S=24{,}536.60$ , $\tau=54/365$ , $r=5.5\%$ , $q=0.49\%$ (from the future), $\sigma=12.5\%$ .

Greek	Per unit of index	Per lot of 65	Reading
Delta (spot)	0.5156	33.52 units	₹33.52 gain per 1-point rise; equivalent to 0.516 lots of futures (0.512 using the futures delta)
Gamma	0.000338 per point	0.02195 units per point	a 100-point rise adds 2.19 units of delta per lot; convexity gain $\frac12\Gamma(\Delta S)^2\times65\approx\text{₹}110$ for a 100-point move
Vega	3,759.3 per 1.00	₹2,443.56 per vol point	a rise in implied vol from 12.5% to 13.5% adds ₹2,444 per lot
Theta	−2,195.5 per year	−₹390.97 per calendar day
Rho	1,800.8 per 1.00	₹1,170.5 per 1% of $r$	small for short-dated index options

Check (8.27) in per-unit, per-year terms: $\Theta=rV-(r-q)S\Delta-\frac12\sigma^2S^2\Gamma=26.38-633.70-1{,}588.14=-2{,}195.47$ , matching $-2{,}195.5$ (use the unrounded $\Gamma=0.00033765$ ; with the rounded $0.000338$ the gamma term comes out near 1,590). The largest part of theta here is the gamma term, as it is for near-the-money options.

Practitioner notes

Gamma and theta of an ATM option scale like $1/\sqrt\tau$ . A weekly Nifty option on its last day has enormous gamma per rupee of premium; this is the pin risk of expiry day and part of the reason for SEBI's extra 2% ELM on short index options on expiry day (circular of 1 Oct 2024).
Greeks are local. For a 3% Nifty gap, delta–gamma is a poor approximation for short-dated options; reprice the book under shocks instead (stress testing, M09).
Vega is largest for ATM options with longer maturities; theta per day is largest for short-dated ATM options. A "theta-positive" book of short weeklies is a short-gamma book, by (8.27).