T
Tobechi
Guest
Say you run a small airline. Fuel is your biggest expense, and you have very little control over what it costs. A war disrupts oil supplies, a refinery shuts down, or the dollar takes another hit. Any of these could push jet fuel prices up by 30% in a single quarter. The problem is that you have already sold the tickets. Your revenue is largely set, but your costs can still change. And if fuel prices rise sharply enough, the difference comes straight out of your margins. You could take your chances with the market, and hope prices come down. Or you could agree on a price today for fuel you will need six months from now.
This is what the futures market allows businesses to do, and when a company uses a financial contract to protect itself against a risk it already faces, that is hedging. The goal is to make future costs more predictable and protect the business from price movements that could eat into its margins. But locking in a price is only one way to manage that risk. What if the airline wants protection against rising fuel prices while still benefiting if they fall? A futures contract cannot offer that flexibility; an option can.
An option gives you a right, not an obligation. A call option lets you buy an asset at a fixed price, known as the strike price, on or before a set date, while a put option lets you sell at that price. You pay a premium upfront for that right, and that premium is the most you can lose as the buyer, regardless of how far the market moves against you.
Take a stock trading at $100. You buy a call with a strike price of $100 and pay a $4 premium. If the stock falls to $60, you can let the option expire. You lose the $4, but you avoid buying a stock at $100 when it is worth only $60. If the stock rises to $130, your right to buy at $100 is worth $30 per share at expiration.
After deducting the $4 premium, your profit is $26 per share. The seller, known as the writer, takes the other side of that trade. They collect your $4 upfront, but in return, they take on an obligation. If you exercise your right to buy, they must honor the contract. That obligation can become expensive.
A seller who writes a call without owning the underlying shares, for instance, faces potentially unlimited losses as the share price rises. This is why option sellers may be required to maintain margin. The buyer pays upfront for the right; the seller collects that payment and assumes the obligation.
A futures contract works differently. Both parties are obligated to fulfil the contract. The buyer and seller agree on a price today for a transaction at a future date. Neither pays an option premium, but both must post margin, which is collateral held to cover potential losses.
The exchange does not wait until the contract expires to settle those losses. Futures are marked to market daily, meaning gains and losses are credited to or deducted from each party's account as prices change. If your balance falls below the required level, you may have to deposit more money. Margin helps ensure both parties can meet their obligations.
For the airline, the choice comes down to the kind of protection it wants. Futures allow it to lock in a price, but it also gives up the benefit of favorable price movements on the hedged position. Options offer more flexibility. The airline pays a premium for protection against rising prices while retaining the benefit if prices fall. The real flexibility comes from combining these instruments. You can build positions around a particular view of the market, with a defined cost and a specific set of possible outcomes.
Example: A stock is trading at $100, the day before it reports earnings. You buy a call struck at $100 for $4 and a put struck at $100 for another $4. You have spent $8 in total, and that $8 is the hurdle. The stock has to move far enough to cover that cost before you make a profit.
At expiration, only one of the two options can have intrinsic value. If the stock finishes above $100, the call has value, and the put expires worthless. If it finishes below $100, the put has value, and the call expires worthless. Whichever side pays out has to recover the cost of both options.
That puts your break-even points at $108 and $92. Above $108, you make a profit. Below $92, you also make a profit. Between those prices, you lose some or all of the premium. If the stock closes at exactly $100, both options expire worthless, and the full $8 is gone. Outside the break-even points, the two sides behave differently.
The call has no upper limit to its potential profit, because the stock price can keep rising. The put, on the other hand, can only gain as much as the stock falls towards zero. If the stock reaches $115, the call is worth $15, and you are up $7 after the $8 you paid. If it drops to $85, the put is worth $15, and you are up the same $7.
There is another complication with earnings trades. The $8 premium reflects the market's expectations about volatility ahead of the announcement. Options tend to become more expensive when traders expect a large move, and the combined cost of the straddle can give you a rough sense of the move being priced in. Here, the $8 premium is 8% of the stock price, although that is only a rough guide to the expected move, not a precise forecast.
Once the announcement is out, much of that uncertainty disappears. Implied volatility can fall sharply, reducing the value of the options. Traders call this volatility crush. The stock can move in the direction you expected, and the trade can still lose money if the move is too small or the fall in implied volatility outweighs the gains from the price movement. What matters is whether the stock moves enough to justify what you paid for the position.
A strangle follows the same principle, but the strikes are further apart. Instead of buying a call and a put both struck at $100, you buy a call at $105 and a put at $95. Both options are out of the money when the stock is trading at $100, which generally makes them cheaper. Suppose the pair costs $5, compared with $8 for the straddle; The lower cost comes with a wider distance to profitability. The call starts to have intrinsic value above $105, and the put below $95.
But those are the points where the options start paying out, not where the position breaks even. You still have to recover the $5 premium. That puts the break-even points at $110 and $90. If the stock closes at $107, the call is worth $2, leaving you $3 down on the position. The stock has moved in the right direction, but not far enough to cover what you paid.
The maximum loss on this strangle is $5, compared with $8 for the straddle. But the strangle loses its full premium whenever the stock finishes between $95 and $105, whereas the straddle loses its full premium only when the stock finishes at exactly $100. The strangle costs less to enter, but it needs a larger move to become profitable.
Beyond the break-even points, both positions gain an additional dollar for every dollar the stock moves further in the profitable direction, assuming expiration and ignoring transaction costs. The difference lies in the cost of entry and how far the stock has to move before those gains begin to outweigh it.
Here are the two side by side at expiration, per share:
With these numbers, the strangle only comes out ahead if the stock finishes close to where it started, within about 3 dollars of 100. Anywhere else, the straddle does the same or better. You saved 3 dollars on the ticket, but the options start paying 5 dollars away from the price, so you give back 2 dollars on every big move. A strangle is cheaper to buy and has a smaller worst case. In return, it asks for a bigger move and gives you a wider range where you lose everything you paid.
Not every option strategy starts with a view on how much a stock will move. Sometimes, the shares are already in your portfolio, and the question is what you can do with them.
Take the covered call: you own shares trading at $100 and are in no hurry to sell. You write a call with a strike price of $110 and collect a $3 premium per share. If the stock finishes below $110 at expiration, the option expires worthless. You keep the premium and your shares. If the stock rises above $110, your shares can be called away at that price. Including the premium, you receive $113 per share, even if the stock climbs to $130.
You have collected income today in exchange for giving up any gains above $110, apart from the premium. That can make sense if you are comfortable selling at that price. But the premium offers only limited protection if the stock falls. It cushions the first $3 of losses per share; beyond that, you continue to bear the downside.
A protective put takes the opposite approach. You still own shares trading at $100, but this time you want to protect yourself against a sharp fall without selling them. You buy a put with a strike price of $95 for a $2 premium per share. If the stock falls to $70, your shares lose $30 in value, but the put is worth $25.
Together, those losses leave you $7 down per share: the $5 between the share price and the strike, plus the $2 premium. That is the maximum loss on the combined position at expiration, regardless of how far the stock falls. The protection comes at a cost.
You have spent $2 per share on the put, so the stock needs to rise above $102 for the combined position to show a profit, assuming you measure it against the original $100 share price and ignore other costs. If the stock holds up or rises only slightly, the put may expire worthless, and the premium is gone. Much like insurance, you pay for protection whether or not you end up needing it.
A bull call spread borrows the covered call's trick of giving up upside, but applies it to a brand new position instead of shares you already hold. You buy a call struck at 100 for 6 dollars and sell a call struck at 110 for 2 dollars, so you are out of pocket 4. Compared with buying the 100 call on its own, you have cut your cost by a third, and in return you give up any profit above 110. Your best case is 6 dollars: the 10-dollar gap between the strikes, minus the 4 you paid.
Your worst case is losing the 4, which happens if the stock finishes at 100 or lower. The break-even sits at 104. A bear put spread is the same structure turned over for a bearish view. You buy a put at the higher strike and sell one at a lower strike, so you pay less and cap your gain.
Go one step further, and you reach the iron condor. It is a bear call spread and a bull put spread on the same stock, opened together, so you collect premium on both sides. Take a stock at 100. You sell a 105 call and buy a 110 call to cap it, then sell a 95 put and buy a 90 put for the same reason. Say the whole thing pays you 2 dollars. If the stock stays between 95 and 105 at expiration, everything expires worthless, and you keep the 2.
Outside that range you start losing, up to a worst case of 3 dollars, which is the 5 dollar width of either spread minus the 2 you took in. Only one side can be breached at a time, so you never lose on both. Your break-evens are 93 and 107. You can think of it as a straddle or strangle turned inside out. Instead of paying to bet that something big happens, you get paid to bet that it does not, with a hard limit on what going wrong can cost.
Knowing what a strategy pays at expiration is useful, but someone holding a live position wants to know how its value is moving today. That is what the Greeks measure. Each one isolates how an option's price responds when a single input changes and everything else stays put.
Delta is how much the option's price moves for a 1 dollar move in the stock. A call with a delta of 0.60 gains about 60 cents for each dollar the stock rises. It is also a rough read on the odds of finishing in the money. Traders lean on it most for hedging. A call on 100 shares with a delta of 0.60 is offset by selling 60 shares, and as delta shifts, you adjust that hedge. This is the same replication argument the Black-Scholes model was built on.
Gamma is how quickly delta itself changes as the stock moves. A high gamma position needs its hedge adjusted often, because its sensitivity keeps shifting under you. It is highest for options near the strike that are close to expiry. A low gamma position stays hedged for longer.
Theta is the value an option loses just from time passing, with everything else held still. If you own options, it is a steady drag, and it accelerates as expiry gets close. If you sell them, the same number is income. That is a big part of why a covered call makes money simply by sitting there.
Vega is sensitivity to volatility. The straddle and strangle from earlier are, at heart, long vega positions. They gain when the market expects bigger moves ahead, whichever way the stock ends up going.
Rho is sensitivity to interest rates. It is usually the smallest of the five for short-dated options. It matters more for long-dated ones, or when rates are moving sharply.
Here is a small piece of code that calculates all five for a European option, building directly on the Black-Scholes pricing function.
Then run it on the same call, a stock at 100, struck at 100, six months out, 20 percent volatility and a 5 percent risk free rate:
A delta of about 0.60 means that, for each dollar the stock moves, this option moves roughly 60 cents. Options are quoted per share, but one standard contract covers 100 shares, so a single contract behaves a lot like holding about 60 shares. A theta of about negative 0.022 means the option loses roughly 2.2 cents a day just from time passing, with the stock price and everything else held still. On a full contract, that is about 2 dollars and 22 cents every calendar day.
Understanding how a strategy works is one thing. Deciding how much of your capital to put behind it is another. Position sizing is where risk management becomes practical, because even a strategy with favorable odds will experience losing streaks. If you take on too much exposure, a run of ordinary losses can do serious damage to your account before the strategy has a chance to recover.
Futures add another layer to this problem because gains and losses are settled daily through marking to market. Your account is credited or debited as the futures price moves, and you must maintain the required margin balance. If your account falls below the maintenance requirement, your broker may issue a margin call, requiring you to deposit more funds. If you cannot meet it, your position may be closed, even if you still believe the trade will eventually work in your favor.
Consider a crude oil futures contract covering 1,000 barrels. A $1 fall in the futures price translates into a $1,000 loss on a long position. That is the effect of leverage: you control a large position with a comparatively small deposit, so relatively small price movements can produce substantial gains or losses against the margin you posted.
The same mechanism can create a cash-flow problem for a hedger. An airline that has taken a futures position to protect itself against rising fuel prices may face margin calls when futures prices fall. The position loses money, but the airline may benefit from lower prices when it buys physical fuel later. Those two effects can offset each other economically, yet the timing may not match. The margin payment is due now; the cheaper fuel may be purchased weeks or months later.
Position size is one part of the picture; you also need to understand what risks your positions share. The Greeks help make those exposures visible. Imagine a portfolio holding straddles across several companies, sectors, and earnings announcements. On paper, the positions appear diversified. But if they are all long vega, they are all exposed to changes in implied volatility. A broad fall in volatility could hurt several positions at once, regardless of how different the underlying companies are.
This is why risk management has to account for the portfolio as a whole. The number of positions you hold tells you very little if those positions respond to the same market event in the same way.
Return to the airline. Its problem was exposure to fuel prices, and the instruments we've covered offer different ways to manage that exposure or shape other market risks.
A futures contract can lock in a price, subject to the details of the hedge. A protective put establishes a floor for shares you own, in exchange for a premium. A covered call generates premium income but caps the gains you can make above the strike price. Spreads can define a narrower range of potential outcomes. A straddle or strangle gives you exposure to a sufficiently large move in either direction, while an iron condor takes the opposite view, earning a premium when the underlying asset stays within a range, but accepting losses if it moves too far.
Each position has a different payoff, cost and set of risks. Choosing between them means understanding which exposure you want and which trade-offs you are prepared to accept.
The Greeks help you track how that exposure changes as prices, time and implied volatility move. Delta measures sensitivity to the underlying price, gamma measures how delta changes, theta captures the effect of time passing, vega measures sensitivity to implied volatility, and rho measures sensitivity to interest rates. A payoff diagram shows the possible outcomes at expiration; the Greeks help explain how the position behaves before then.
But knowing what you hold is only half the job. You also need to know how much you can afford to lose, how much liquidity you need to maintain the position, and whether several apparently different trades are exposing you to the same underlying risk. A sound strategy can still lose money, and leverage or insufficient margin can force you out of a position before your original thesis plays out.
Pricing tells you what a contract is worth. Strategy determines the exposure you take on. Risk management determines how much of that exposure your capital can support.
The aim is to understand the risk before you take it, size the position accordingly, and avoid carrying exposures you never intended to hold.
This is what the futures market allows businesses to do, and when a company uses a financial contract to protect itself against a risk it already faces, that is hedging. The goal is to make future costs more predictable and protect the business from price movements that could eat into its margins. But locking in a price is only one way to manage that risk. What if the airline wants protection against rising fuel prices while still benefiting if they fall? A futures contract cannot offer that flexibility; an option can.
An option gives you a right, not an obligation. A call option lets you buy an asset at a fixed price, known as the strike price, on or before a set date, while a put option lets you sell at that price. You pay a premium upfront for that right, and that premium is the most you can lose as the buyer, regardless of how far the market moves against you.
Take a stock trading at $100. You buy a call with a strike price of $100 and pay a $4 premium. If the stock falls to $60, you can let the option expire. You lose the $4, but you avoid buying a stock at $100 when it is worth only $60. If the stock rises to $130, your right to buy at $100 is worth $30 per share at expiration.
After deducting the $4 premium, your profit is $26 per share. The seller, known as the writer, takes the other side of that trade. They collect your $4 upfront, but in return, they take on an obligation. If you exercise your right to buy, they must honor the contract. That obligation can become expensive.
A seller who writes a call without owning the underlying shares, for instance, faces potentially unlimited losses as the share price rises. This is why option sellers may be required to maintain margin. The buyer pays upfront for the right; the seller collects that payment and assumes the obligation.
A futures contract works differently. Both parties are obligated to fulfil the contract. The buyer and seller agree on a price today for a transaction at a future date. Neither pays an option premium, but both must post margin, which is collateral held to cover potential losses.
The exchange does not wait until the contract expires to settle those losses. Futures are marked to market daily, meaning gains and losses are credited to or deducted from each party's account as prices change. If your balance falls below the required level, you may have to deposit more money. Margin helps ensure both parties can meet their obligations.
For the airline, the choice comes down to the kind of protection it wants. Futures allow it to lock in a price, but it also gives up the benefit of favorable price movements on the hedged position. Options offer more flexibility. The airline pays a premium for protection against rising prices while retaining the benefit if prices fall. The real flexibility comes from combining these instruments. You can build positions around a particular view of the market, with a defined cost and a specific set of possible outcomes.
Example: A stock is trading at $100, the day before it reports earnings. You buy a call struck at $100 for $4 and a put struck at $100 for another $4. You have spent $8 in total, and that $8 is the hurdle. The stock has to move far enough to cover that cost before you make a profit.
At expiration, only one of the two options can have intrinsic value. If the stock finishes above $100, the call has value, and the put expires worthless. If it finishes below $100, the put has value, and the call expires worthless. Whichever side pays out has to recover the cost of both options.
That puts your break-even points at $108 and $92. Above $108, you make a profit. Below $92, you also make a profit. Between those prices, you lose some or all of the premium. If the stock closes at exactly $100, both options expire worthless, and the full $8 is gone. Outside the break-even points, the two sides behave differently.
The call has no upper limit to its potential profit, because the stock price can keep rising. The put, on the other hand, can only gain as much as the stock falls towards zero. If the stock reaches $115, the call is worth $15, and you are up $7 after the $8 you paid. If it drops to $85, the put is worth $15, and you are up the same $7.
There is another complication with earnings trades. The $8 premium reflects the market's expectations about volatility ahead of the announcement. Options tend to become more expensive when traders expect a large move, and the combined cost of the straddle can give you a rough sense of the move being priced in. Here, the $8 premium is 8% of the stock price, although that is only a rough guide to the expected move, not a precise forecast.
Once the announcement is out, much of that uncertainty disappears. Implied volatility can fall sharply, reducing the value of the options. Traders call this volatility crush. The stock can move in the direction you expected, and the trade can still lose money if the move is too small or the fall in implied volatility outweighs the gains from the price movement. What matters is whether the stock moves enough to justify what you paid for the position.
A strangle follows the same principle, but the strikes are further apart. Instead of buying a call and a put both struck at $100, you buy a call at $105 and a put at $95. Both options are out of the money when the stock is trading at $100, which generally makes them cheaper. Suppose the pair costs $5, compared with $8 for the straddle; The lower cost comes with a wider distance to profitability. The call starts to have intrinsic value above $105, and the put below $95.
But those are the points where the options start paying out, not where the position breaks even. You still have to recover the $5 premium. That puts the break-even points at $110 and $90. If the stock closes at $107, the call is worth $2, leaving you $3 down on the position. The stock has moved in the right direction, but not far enough to cover what you paid.
The maximum loss on this strangle is $5, compared with $8 for the straddle. But the strangle loses its full premium whenever the stock finishes between $95 and $105, whereas the straddle loses its full premium only when the stock finishes at exactly $100. The strangle costs less to enter, but it needs a larger move to become profitable.
Beyond the break-even points, both positions gain an additional dollar for every dollar the stock moves further in the profitable direction, assuming expiration and ignoring transaction costs. The difference lies in the cost of entry and how far the stock has to move before those gains begin to outweigh it.
Here are the two side by side at expiration, per share:
| Stock price | Straddle (cost 8) | Strangle (cost 5) |
|---|---|---|
| 70 | +22 | +20 |
| 85 | +7 | +5 |
| 90 | +2 | 0 |
| 95 | −3 | −5 |
| 100 | −8 | −5 |
| 105 | −3 | −5 |
| 110 | +2 | 0 |
| 115 | +7 | +5 |
| 130 | +22 | +20 |
With these numbers, the strangle only comes out ahead if the stock finishes close to where it started, within about 3 dollars of 100. Anywhere else, the straddle does the same or better. You saved 3 dollars on the ticket, but the options start paying 5 dollars away from the price, so you give back 2 dollars on every big move. A strangle is cheaper to buy and has a smaller worst case. In return, it asks for a bigger move and gives you a wider range where you lose everything you paid.
Trading the stock you already hold.
Not every option strategy starts with a view on how much a stock will move. Sometimes, the shares are already in your portfolio, and the question is what you can do with them.
Take the covered call: you own shares trading at $100 and are in no hurry to sell. You write a call with a strike price of $110 and collect a $3 premium per share. If the stock finishes below $110 at expiration, the option expires worthless. You keep the premium and your shares. If the stock rises above $110, your shares can be called away at that price. Including the premium, you receive $113 per share, even if the stock climbs to $130.
You have collected income today in exchange for giving up any gains above $110, apart from the premium. That can make sense if you are comfortable selling at that price. But the premium offers only limited protection if the stock falls. It cushions the first $3 of losses per share; beyond that, you continue to bear the downside.
A protective put takes the opposite approach. You still own shares trading at $100, but this time you want to protect yourself against a sharp fall without selling them. You buy a put with a strike price of $95 for a $2 premium per share. If the stock falls to $70, your shares lose $30 in value, but the put is worth $25.
Together, those losses leave you $7 down per share: the $5 between the share price and the strike, plus the $2 premium. That is the maximum loss on the combined position at expiration, regardless of how far the stock falls. The protection comes at a cost.
You have spent $2 per share on the put, so the stock needs to rise above $102 for the combined position to show a profit, assuming you measure it against the original $100 share price and ignore other costs. If the stock holds up or rises only slightly, the put may expire worthless, and the premium is gone. Much like insurance, you pay for protection whether or not you end up needing it.
Spreads narrow the bet, and narrow the cost with it
A bull call spread borrows the covered call's trick of giving up upside, but applies it to a brand new position instead of shares you already hold. You buy a call struck at 100 for 6 dollars and sell a call struck at 110 for 2 dollars, so you are out of pocket 4. Compared with buying the 100 call on its own, you have cut your cost by a third, and in return you give up any profit above 110. Your best case is 6 dollars: the 10-dollar gap between the strikes, minus the 4 you paid.
Your worst case is losing the 4, which happens if the stock finishes at 100 or lower. The break-even sits at 104. A bear put spread is the same structure turned over for a bearish view. You buy a put at the higher strike and sell one at a lower strike, so you pay less and cap your gain.
Go one step further, and you reach the iron condor. It is a bear call spread and a bull put spread on the same stock, opened together, so you collect premium on both sides. Take a stock at 100. You sell a 105 call and buy a 110 call to cap it, then sell a 95 put and buy a 90 put for the same reason. Say the whole thing pays you 2 dollars. If the stock stays between 95 and 105 at expiration, everything expires worthless, and you keep the 2.
Outside that range you start losing, up to a worst case of 3 dollars, which is the 5 dollar width of either spread minus the 2 you took in. Only one side can be breached at a time, so you never lose on both. Your break-evens are 93 and 107. You can think of it as a straddle or strangle turned inside out. Instead of paying to bet that something big happens, you get paid to bet that it does not, with a hard limit on what going wrong can cost.
The Greeks
Knowing what a strategy pays at expiration is useful, but someone holding a live position wants to know how its value is moving today. That is what the Greeks measure. Each one isolates how an option's price responds when a single input changes and everything else stays put.
Delta is how much the option's price moves for a 1 dollar move in the stock. A call with a delta of 0.60 gains about 60 cents for each dollar the stock rises. It is also a rough read on the odds of finishing in the money. Traders lean on it most for hedging. A call on 100 shares with a delta of 0.60 is offset by selling 60 shares, and as delta shifts, you adjust that hedge. This is the same replication argument the Black-Scholes model was built on.
Gamma is how quickly delta itself changes as the stock moves. A high gamma position needs its hedge adjusted often, because its sensitivity keeps shifting under you. It is highest for options near the strike that are close to expiry. A low gamma position stays hedged for longer.
Theta is the value an option loses just from time passing, with everything else held still. If you own options, it is a steady drag, and it accelerates as expiry gets close. If you sell them, the same number is income. That is a big part of why a covered call makes money simply by sitting there.
Vega is sensitivity to volatility. The straddle and strangle from earlier are, at heart, long vega positions. They gain when the market expects bigger moves ahead, whichever way the stock ends up going.
Rho is sensitivity to interest rates. It is usually the smallest of the five for short-dated options. It matters more for long-dated ones, or when rates are moving sharply.
Here is a small piece of code that calculates all five for a European option, building directly on the Black-Scholes pricing function.
Code:
import math
from statistics import NormalDist
def option_greeks(S0, K, r, T, sigma, option_type="call"):
"""Calculate the five main Greeks for a European option under Black-Scholes.
S0 : current price of the underlying asset
K : strike price
r : risk free interest rate, annualized, as a decimal
T : time to expiration, in years
sigma : volatility of the underlying, annualized, as a decimal
option_type : "call" or "put"
"""
d1 = (math.log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))
d2 = d1 - sigma * math.sqrt(T)
N = NormalDist().cdf
n = NormalDist().pdf
gamma = n(d1) / (S0 * sigma * math.sqrt(T))
vega = S0 * n(d1) * math.sqrt(T) / 100 # per 1% change in volatility
if option_type == "call":
delta = N(d1)
theta = (
-(S0 * n(d1) * sigma) / (2 * math.sqrt(T))
- r * K * math.exp(-r * T) * N(d2)
) / 365 # per calendar day
rho = (K * T * math.exp(-r * T) * N(d2)) / 100 # per 1% change in rates
else:
delta = N(d1) - 1
theta = (
-(S0 * n(d1) * sigma) / (2 * math.sqrt(T))
+ r * K * math.exp(-r * T) * N(-d2)
) / 365
rho = (-K * T * math.exp(-r * T) * N(-d2)) / 100
return {"delta": delta, "gamma": gamma, "theta": theta, "vega": vega, "rho": rho}Then run it on the same call, a stock at 100, struck at 100, six months out, 20 percent volatility and a 5 percent risk free rate:
Code:
greeks = option_greeks(S0=100, K=100, r=0.05, T=0.5, sigma=0.20, option_type="call")
for name, value in greeks.items():
print(f"{name}: {value:.4f}")
Code:
delta: 0.5977
gamma: 0.0274
theta: -0.0222
vega: 0.2736
rho: 0.2644A delta of about 0.60 means that, for each dollar the stock moves, this option moves roughly 60 cents. Options are quoted per share, but one standard contract covers 100 shares, so a single contract behaves a lot like holding about 60 shares. A theta of about negative 0.022 means the option loses roughly 2.2 cents a day just from time passing, with the stock price and everything else held still. On a full contract, that is about 2 dollars and 22 cents every calendar day.
Managing risk when the position is real money.
Understanding how a strategy works is one thing. Deciding how much of your capital to put behind it is another. Position sizing is where risk management becomes practical, because even a strategy with favorable odds will experience losing streaks. If you take on too much exposure, a run of ordinary losses can do serious damage to your account before the strategy has a chance to recover.
Futures add another layer to this problem because gains and losses are settled daily through marking to market. Your account is credited or debited as the futures price moves, and you must maintain the required margin balance. If your account falls below the maintenance requirement, your broker may issue a margin call, requiring you to deposit more funds. If you cannot meet it, your position may be closed, even if you still believe the trade will eventually work in your favor.
Consider a crude oil futures contract covering 1,000 barrels. A $1 fall in the futures price translates into a $1,000 loss on a long position. That is the effect of leverage: you control a large position with a comparatively small deposit, so relatively small price movements can produce substantial gains or losses against the margin you posted.
The same mechanism can create a cash-flow problem for a hedger. An airline that has taken a futures position to protect itself against rising fuel prices may face margin calls when futures prices fall. The position loses money, but the airline may benefit from lower prices when it buys physical fuel later. Those two effects can offset each other economically, yet the timing may not match. The margin payment is due now; the cheaper fuel may be purchased weeks or months later.
Position size is one part of the picture; you also need to understand what risks your positions share. The Greeks help make those exposures visible. Imagine a portfolio holding straddles across several companies, sectors, and earnings announcements. On paper, the positions appear diversified. But if they are all long vega, they are all exposed to changes in implied volatility. A broad fall in volatility could hurt several positions at once, regardless of how different the underlying companies are.
This is why risk management has to account for the portfolio as a whole. The number of positions you hold tells you very little if those positions respond to the same market event in the same way.
Where this leaves you.
Return to the airline. Its problem was exposure to fuel prices, and the instruments we've covered offer different ways to manage that exposure or shape other market risks.
A futures contract can lock in a price, subject to the details of the hedge. A protective put establishes a floor for shares you own, in exchange for a premium. A covered call generates premium income but caps the gains you can make above the strike price. Spreads can define a narrower range of potential outcomes. A straddle or strangle gives you exposure to a sufficiently large move in either direction, while an iron condor takes the opposite view, earning a premium when the underlying asset stays within a range, but accepting losses if it moves too far.
Each position has a different payoff, cost and set of risks. Choosing between them means understanding which exposure you want and which trade-offs you are prepared to accept.
The Greeks help you track how that exposure changes as prices, time and implied volatility move. Delta measures sensitivity to the underlying price, gamma measures how delta changes, theta captures the effect of time passing, vega measures sensitivity to implied volatility, and rho measures sensitivity to interest rates. A payoff diagram shows the possible outcomes at expiration; the Greeks help explain how the position behaves before then.
But knowing what you hold is only half the job. You also need to know how much you can afford to lose, how much liquidity you need to maintain the position, and whether several apparently different trades are exposing you to the same underlying risk. A sound strategy can still lose money, and leverage or insufficient margin can force you out of a position before your original thesis plays out.
Pricing tells you what a contract is worth. Strategy determines the exposure you take on. Risk management determines how much of that exposure your capital can support.
The aim is to understand the risk before you take it, size the position accordingly, and avoid carrying exposures you never intended to hold.