Lesson 6.2: Black-Scholes and the Greeks
Learn the Black-Scholes formula and the Greeks: delta, gamma, vega, theta and rho, how to hedge with delta, and why gamma and theta are two sides of one trade.
By the end of this lesson you will be able to price a European call and put with the Black–Scholes formula, say what each input does, compute and interpret delta, gamma, vega, theta and rho, work out the hedge that makes a position insensitive to small price moves, and explain why a position that gains from large moves must pay for it through time decay. You will show that a six-month at-the-money call on a share at 100 with 20 per cent volatility and 6 per cent interest is worth 7.16, that its delta is 0.611 so that a trader who sold 100 of them must hold about 61 shares, that its time decay is 2.4 paise a day rising to 5.7 paise when a tenth of the time remains, and that its Greeks satisfy a single equation that links theta and gamma. You should have read Lesson 6.1 and Lessons 1.3 and 2.2 on volatility. The mathematics is logarithms, exponentials, the normal distribution function and first and second derivatives. This course is education about how markets and research work, not investment advice.
Lesson 6.1 told you what an option pays at expiry. This lesson answers the question every trader asks first: what is it worth now? In 1973 Fischer Black and Myron Scholes, and independently Robert Merton, gave a formula for the price of a European option. It is the most famous result in the field, and it earned a Nobel prize in 1997. Its main value today is not as a literal truth about markets, which it is not, but as a common language for quoting prices and measuring risk. This lesson shows the formula, what it needs and what its derivatives, the Greeks, tell a trader about the risk of a position. Lesson 7.2 derives it from first principles.
The assumptions
The Black–Scholes model values an option on an asset under these conditions:
1. The asset's price follows a geometric Brownian motion: its log return over any short period is normally distributed with a constant volatility σ. (Lesson 2.2 showed that real returns have fatter tails, and Lesson 7.2 defines this process.)
2. The risk-free interest rate r is constant, and you can borrow and lend at it.
3. There are no transaction costs, no taxes and no restriction on short selling.
4. Trading is continuous, and prices do not jump.
5. The option is European, and the asset pays no dividends. A continuous dividend yield q is easy to add.
None of these is exactly true. The model's usefulness comes from the fact that, under them, an option can be replicated by trading the underlying asset and cash, which fixes its price.
The formula
For a European call and put on an asset at price S, with strike K, time to expiry T in years, risk-free rate r and volatility σ:
Call = S × N(d₁) − K e^(−rT) × N(d₂)
Put = K e^(−rT) × N(−d₂) − S × N(−d₁)
where
d₁ = [ln(S ÷ K) + (r + σ² ÷ 2) × T] ÷ (σ × √T)
d₂ = d₁ − σ × √T
and N is the cumulative normal distribution function: N(x) is the probability that a standard normal variable is below x. The call and put satisfy put-call parity of Lesson 6.1, so one gives the other.
Worked example 1. Take S = 100, K = 100, r = 6 per cent, σ = 20 per cent and T = 0.5 years.
ln(S ÷ K) = ln 1 = 0. The numerator of d₁ is (0.06 + 0.02) × 0.5 = 0.04, and σ√T = 0.2 × 0.7071 = 0.1414. So d₁ = 0.2828 and d₂ = 0.2828 − 0.1414 = 0.1414.
N(d₁) = 0.6114 and N(d₂) = 0.5562.
The present value of the strike is 100 × e^(−0.03) = 97.045.
Call = 100 × 0.6114 − 97.045 × 0.5562 = 61.14 − 53.98 = 7.16.
Put = call − S + PV(K) = 7.16 − 100 + 97.045 = 4.20.
Compare with the intrinsic value: both options are at the money, so the whole 7.16 and 4.20 is time value.
Reading the formula
The terms have a meaning that is worth keeping. Under the model's pricing measure, which Lesson 7.3 defines, N(d₂) is the probability that the option finishes in the money, here 55.6 per cent. The second term, K e^(−rT) N(d₂), is then the present value of the strike you expect to pay. The first term, S N(d₁), is the present value of the share you expect to receive, which weights the share price by a slightly higher number, 61.1 per cent, because when the call finishes in the money the share is more than average. The call is the difference.
Four things raise a call's price: a higher share price, a higher volatility, a longer time to expiry and a higher interest rate. A higher strike lowers it. The effect of volatility, which the Greek vega captures, is the one that makes options a way to trade volatility.
The Greeks
A Greek is the sensitivity of the option's value to one input, holding the others fixed. Together they describe how the position will behave. The standard five, with their values for the call of worked example 1:
Greek · Measures · Formula for a call · Value in the exampleDelta (Δ) · Change in value for a 1-unit rise in the price · N(d₁) · 0.611
Gamma (Γ) · Change in delta for a 1-unit rise in the price · φ(d₁) ÷ (S σ √T) · 0.0271
Vega (ν) · Change in value for a 1 percentage-point rise in volatility · S φ(d₁) √T ÷ 100 · 0.271
Theta (Θ) · Change in value as one day passes · see below · −0.0237 a day
Rho (ρ) · Change in value for a 1 percentage-point rise in the interest rate · K T e^(−rT) N(d₂) ÷ 100 · 0.270
Here φ is the normal density, φ(x) = e^(−x²/2) ÷ √(2π). Theta for a call is −S φ(d₁) σ ÷ (2√T) − r K e^(−rT) N(d₂), divided by 365 for a daily figure. For a put, delta is N(d₁) − 1, which is −0.389 here, and gamma and vega are the same as for the call.
Delta: the hedge ratio
Delta is the slope of the option value against the price, and the most important Greek, because it is the hedge ratio. A call with a delta of 0.611 gains about 0.611 for each 1 the share rises, so selling 100 calls and buying 61.1 shares leaves a position that is, for a small move, unaffected. That is a delta hedge. Delta is also close to the probability of finishing in the money, though not equal to it. It moves from 0 for a call far out of the money to 1 deep in the money, and from −1 to 0 for a put.
Gamma: the curvature, and the risk of a hedge
A delta hedge is exact only for tiny moves. As the price moves, delta changes, and gamma measures by how much: 0.0271 per 1 rupee in the example. A trader who is long an option has positive gamma: as the price rises their delta rises, so they hold more of the rising asset, and as it falls their delta falls, so they hold less. The delta-hedged position gains from a move in either direction. The seller is short gamma, and loses from a large move either way, because the hedge must be rebalanced at worse and worse prices: buying after the rise, selling after the fall.
For a delta-hedged position the profit from a move of size ΔS is approximately ½ × Γ × (ΔS)². For 100 options with Γ = 0.0271, a move of 1 gives 0.5 × 100 × 0.0271 × 1 = 1.36, and a move of 3 gives 12.2, nine times as much, since it grows with the square. Gamma is largest for options at the money and close to expiry: our option has a gamma of 0.027 with six months to go, 0.089 with 0.05 years (about 18 days) left and 0.199 with 0.01 years left.
Theta: the price of being long gamma
An option that gains from big moves must cost something while it waits. That cost is theta, the loss of value as time passes with the price unchanged. The call in the example loses 0.0237 a day, about 0.33 per cent of its value each day. Time decay is not constant: it accelerates as expiry approaches, because the time left is what gives an at-the-money option its value, and it falls with the square root of the time.
Time to expiry · Call value · Theta per day · Gamma · Vega per volatility point0.50 years · 7.16 · −0.0237 · 0.0271 · 0.271
0.25 years · 4.75 · −0.0302 · 0.0391 · 0.196
0.05 years · 1.94 · −0.0572 · 0.0889 · 0.089
0.01 years · 0.83 · −0.1176 · 0.1993 · 0.040
When a tenth of the time remains the daily decay is 2.4 times larger, and the call has lost more than three quarters of its value. Gamma rises and vega falls: near expiry the option is sensitive to price and nearly insensitive to volatility.
Vega: the sensitivity to volatility
The model's one free input is σ, and vega measures what happens when it changes. A call with a vega of 0.271 gains 0.271 for each percentage point of volatility: if the market's implied volatility rises from 20 to 25 per cent, the call gains about 5 × 0.271 = 1.36, nearly 19 per cent of its value. Vega is largest at the money and for long-dated options, and zero for an option that is certain to expire in or out of the money. A trader who wants to hold volatility and not direction hedges the delta and keeps the vega.
Rho
Rho, the effect of the interest rate, is small for short-dated options: 0.270 per percentage point here. It matters for long-dated options and for interest-rate products.
The link between theta and gamma
The Greeks are not independent. The Black–Scholes price satisfies a partial differential equation, derived in Lesson 7.2, which in terms of the Greeks reads:
Θ + ½ σ² S² Γ + r S Δ − r V = 0
where Θ is theta per year and V is the option value. Check it with the numbers of the example: Θ = −8.659 per year, ½ σ² S² Γ = 0.5 × 0.04 × 10,000 × 0.0271 = 5.42, r S Δ = 0.06 × 100 × 0.6114 = 3.668 and r V = 0.06 × 7.156 = 0.429. The sum is −8.659 + 5.42 + 3.668 − 0.429 = 0, to within rounding. In words, the daily loss from the passage of time is paid for, one for one, by the gain from gamma over the move that the model's volatility implies, and the interest terms balance the financing of the hedge. Gamma and theta are two sides of one trade. A long-gamma position makes money if the price moves more than the volatility in the price of the option implies, and loses the time decay if it moves less.
That tells you a break-even. A delta-hedged long option breaks even over one day if ½ Γ (ΔS)² equals the day's cost of carrying the position. Ignoring the financing terms, the daily theta of 0.0237 and gamma of 0.0271 give a break-even move of √(2 × 0.0237 ÷ 0.0271) = 1.32. Including the financing of the hedge, the break-even is σ S ÷ √365 = 0.2 × 100 ÷ 19.1 = 1.05, which is simply the daily move that matches 20 per cent annual volatility. The practical statement is the important one: a long delta-hedged option position breaks even when the market's realised volatility equals the volatility that was paid for. Lesson 6.4 tests this by simulation.
A pricing calculator
Use the calculator below to see how the price and the Greeks move with the inputs. It uses a close approximation to the normal distribution function, accurate to about 0.0002.
What Black–Scholes gets wrong, and why it still matters
The model's assumptions fail in ways every trader knows. Volatility is not constant: it moves, clusters and jumps, as Lesson 2.2 showed. Returns have fat tails and prices can jump, so the probability of a large fall is higher than the model says. Hedging is discrete and costs money. Yet the formula remains the standard language of the market, for two reasons. First, it maps a price into one number, the implied volatility, that can be compared across options, days and markets: traders quote volatility, not price. Second, the Greeks computed from it are good first approximations of how a position behaves, even when the model is imperfect. The next lesson studies what the market's own implied volatilities look like, and the pattern, the smile, is where the model's failure shows.
The code below computes the price and the Greeks, checks them against finite differences and verifies the link between theta and gamma.
python
import numpy as np
from math import log, sqrt, exp, pi
from scipy.stats import norm
def bs(S, K, r, sigma, T):
d1 = (log(S / K) + (r + 0.5 sigma 2) T) / (sigma sqrt(T))
d2 = d1 - sigma sqrt(T)
call = S norm.cdf(d1) - K exp(-r T) norm.cdf(d2)
put = K exp(-r T) norm.cdf(-d2) - S norm.cdf(-d1)
return call, put, d1, d2
def greeks(S, K, r, sigma, T):
call, put, d1, d2 = bs(S, K, r, sigma, T)
pdf = norm.pdf(d1)
delta = norm.cdf(d1)
gamma = pdf / (S sigma sqrt(T))
vega = S pdf sqrt(T) / 100 # per volatility point
theta = (-S pdf sigma / (2 sqrt(T)) - r K exp(-r T) norm.cdf(d2)) / 365 # per day
rho = K T exp(-r T) norm.cdf(d2) / 100 # per interest-rate point
return call, delta, gamma, vega, theta, rho
S, K, r, sigma, T = 100, 100, 0.06, 0.20, 0.5
call, put, d1, d2 = bs(S, K, r, sigma, T)
print("d1", round(d1, 4), "d2", round(d2, 4), "N(d1)", round(norm.cdf(d1), 4), "N(d2)", round(norm.cdf(d2), 4))
print("call", round(call, 3), "put", round(put, 3), "| parity check", round(call - put, 4), round(S - K exp(-r T), 4))
call, delta, gamma, vega, theta, rho = greeks(S, K, r, sigma, T)
print("delta", round(delta, 4), "put delta", round(delta - 1, 4), "gamma", round(gamma, 4), "vega", round(vega, 3),
"theta/day", round(theta, 4), "rho", round(rho, 3))
finite-difference checks
h = 0.01
up, down = bs(S + h, K, r, sigma, T)[0], bs(S - h, K, r, sigma, T)[0]
print("delta by differences", round((up - down) / (2 h), 4), "| gamma by differences", round((up - 2 call + down) / h 2, 4),
"| vega by differences", round((bs(S, K, r, sigma + 1e-4, T)[0] - bs(S, K, r, sigma - 1e-4, T)[0]) / 2e-4 / 100, 3))
the link between theta and gamma: theta + 0.5 sigma^2 S^2 gamma + r S delta - r V = 0
thetayear = theta 365
print("theta + gamma term + carry terms =", round(thetayear + 0.5 sigma 2 S 2 gamma + r S delta - r call, 10))
print("break-even move ignoring carry", round(sqrt(2 abs(theta) / gamma), 2), "| with carry (sigmaS/sqrt(365))", round(sigma S / sqrt(365), 2))
time decay and gamma by maturity
for T in (0.5, 0.25, 0.05, 0.01):
c, d, g, v, th, = greeks(S, K, r, sigma, T)
print("T", T, "value", round(c, 3), "theta/day", round(th, 4), "gamma", round(g, 4), "vega", round(v, 3))
delta hedge of 100 short calls and the gamma effect of a move
print("shares to hold against 100 short calls:", round(100 delta, 1), "| gamma gain on a move of 1:", round(0.5 100 gamma 1, 2), "and of 3:", round(0.5 100 gamma 9, 2))
Common mistakes
Using Black–Scholes prices as if the model were true. It is a quoting convention and a first approximation, not a law.
Treating delta as a fixed hedge. Delta changes with the price, with time and with volatility, so the hedge must be adjusted.
Ignoring gamma when short options. A short-gamma position is safe until a large move, and then the losses come quickly.
Forgetting that theta is not free money. The seller collects time decay as compensation for being short gamma, and the compensation is fair only if the realised moves are as small as the price implies.
Mixing the units of the Greeks. Vega may be per 1.00 or per 1 percentage point of volatility, theta per year or per day, and rho per 1.00 or per 1 percentage point. State which.
Using the wrong time or rate. T is in years, and the rate must match the period and compounding.
Hedging a long-dated option with a one-off delta hedge. Over weeks, delta drifts a long way.
Ignoring vega at long maturities. A long-dated option's price depends heavily on the volatility view.
Treating the probability of finishing in the money as the real-world probability. N(d₂) is a probability under the pricing measure, not under the real world.
Key takeaways
Black–Scholes prices a European option as S N(d₁) − K e^(−rT) N(d₂), with d₁ = [ln(S ÷ K) + (r + σ² ÷ 2)T] ÷ (σ√T) and d₂ = d₁ − σ√T. For S = K = 100, r = 6 per cent, σ = 20 per cent and T = 0.5, the call is 7.16 and the put 4.20.
The price rises with the share price, volatility, time and rate for a call, and the time value is largest at the money.
Delta, N(d₁), is the hedge ratio: 0.611 here, so selling 100 calls means buying about 61 shares. Delta runs from 0 to 1 for a call and becomes a switch near expiry at the money.
Gamma, 0.0271, is the curvature: a delta-hedged position gains about ½ΓΔS² from a move, so long-gamma positions profit from large moves and short-gamma positions lose.
Theta, −0.0237 a day, is the cost of being long gamma. It accelerates near expiry, to −0.0572 with 0.05 years left. Vega, 0.271 per volatility point, measures the sensitivity to volatility.
The Greeks obey Θ + ½σ²S²Γ + rSΔ − rV = 0, so gamma and theta are two sides of one trade: a delta-hedged long option breaks even when realised volatility equals the volatility paid.
The model's constant volatility, thin tails and continuous hedging are false, but it remains the language for quoting volatility and a first guide to risk.
Check yourself
1. A call has a delta of 0.35. You are short 400 of them. How many shares do you buy to be delta-neutral?
400 × 0.35 = 140 shares.
2. An at-the-money call has a vega of 0.30 per volatility point. Implied volatility falls from 22 to 18 per cent. What is the approximate change in its value?
A fall of 4 points: 4 × 0.30 = −1.20. The call loses about 1.20.
3. A delta-hedged position is long 200 options, each with a gamma of 0.04. The price moves by 2. What is the approximate gamma profit, ignoring theta?
½ × 200 × 0.04 × 2² = 16.
4. Why is a short at-the-money option position in its last week more dangerous than one with three months left?
Gamma is much larger near expiry, so a small price move changes delta a lot and the hedge needs frequent large adjustments. The theta income is high, but a single large move can wipe out many days of it.
5. For the example option, why is the break-even daily move that ignores financing (1.32) larger than the one that includes it (1.05)?
Theta includes the interest cost of the option's value, and the delta hedge earns interest on the cash it holds. When both are counted, what remains is the pure trade of gamma against volatility, which breaks even at σS ÷ √365 = 1.05.
Try this
Set aside forty-five minutes. You need Python and the code in this lesson.
1. Run the code. Check the price, the Greeks, the finite-difference checks and the theta-gamma identity.
2. Plot the price of the call, and its delta, gamma and vega, against the share price from 60 to 140, for a maturity of six months. Which are largest at the money?
3. Repeat for a maturity of one week. How do the shapes change?
4. Use the calculator simulator to find the volatility at which the call is worth 10 when the share is at 100 with six months to go. This is a rough implied volatility, the topic of the next lesson.
5. Compute the delta, gamma and theta of a position that is short 100 calls with a strike of 105 and long 100 calls with a strike of 95, with the same expiry. What kind of position is it?
6. Use the break-even simulator to find the daily move needed to cover the time decay of your option, and compare it with the average daily move of a real stock or index.
7. Write a short note: the Greeks of one position, the hedge you would make against its delta, and the market move that would hurt it most.
A good result is a set of Greek curves you can sketch from memory, a worked hedge in numbers and an understanding that the gamma you earn is the theta you pay. The next lesson inverts the formula: given the market's price, what volatility does it imply, and what does the pattern across strikes say?
Read more
Fischer Black and Myron Scholes's 1973 paper "The Pricing of Options and Corporate Liabilities" in the Journal of Political Economy and Robert Merton's 1973 paper "Theory of Rational Option Pricing" are the sources. John Hull's Options, Futures and Other Derivatives gives the formulas for all the Greeks with examples, and Sheldon Natenberg's Option Volatility and Pricing explains them from a trader's point of view. Nassim Taleb's Dynamic Hedging is a practitioner's account of managing the Greeks. For a detailed look at how the Greeks interact, see Peter Carr's work on the link between gamma and theta.
Tags: Quant Trader Black-Scholes Option Greeks Delta hedging Gamma and theta