Introduction to risk management

Table of Contents

What is Delta?

Delta measures how much the price of a derivative—such as an option—changes in response to a $1 move in the price of the underlying asset (e.g., a stock). It is one of the key “Greeks” used by traders to understand and manage risk.

For a call option, delta ranges between 0 and +1. For a put option, it ranges between -1 and 0. A positive delta means the option's value moves in the same direction as the underlying asset, while a negative delta means it moves in the opposite direction.

A call option gives the buyer the right (but not the obligation) to buy the asset at a set price. A put option gives the buyer the right (but not the obligation) to sell it.

Example – Call Option: Imagine a company called BigCorp with shares trading at $20. A BigCorp call option has a delta of 0.35. This means a $1 increase in the share price raises the option's price by $0.35. If the call is currently $2.00 and the stock rises to $21, the call would be worth about $2.35.

Example – Put Option: If a BigCorp put option has a delta of -0.65, a $1 rise in the share price causes the option to lose $0.65 in value. So if the put is $2.00 when the stock is $20, and the stock rises to $21, the put would drop to around $1.35.

Delta is also essential for hedging. Traders often aim for a delta-neutral portfolio—meaning the total delta adds up to zero—so that small movements in the underlying asset's price have little or no effect on profit and loss.

Hedging Example: If you buy 100 call options with a delta of +0.40, your total delta is +4,000 (since each option controls 100 shares). To become delta-neutral, you could short (sell shares you do not currently own, with the aim of buying them back later at a lower price) 4,000 shares of the stock. Selling stock creates a delta of -1 per share, which offsets positive delta from calls. Similarly, buying 100 puts with a delta of -0.30 would give you a total delta of -3,000, which could be neutralized by buying 3,000 shares of the stock, each adding +1 delta to your position.

Another Simple Hedge: Suppose you buy 1 call option with a delta of +0.50 and short 50 shares of the stock (delta = -1 per share). The total delta is +0.50 × 100 − 50 = 0. If the stock rises $1, the call gains $50 while the short stock loses $50, resulting in a net P&L of $0.

Understanding delta helps traders fine-tune risk exposure, construct effective hedges, and predict how option prices will react to market movements.

What is Vega?

Vega measures how much an option's price is expected to change for a 1% change in implied volatility. It reflects an option's sensitivity to volatility in the underlying asset. Both call and put options have positive vega, meaning they generally increase in value as volatility rises, and decrease when volatility falls. Options that are at-the-money and have a longer time to expiration typically have higher vega.

Traders often use vega to hedge or speculate on volatility changes. For example, if volatility is expected to rise—perhaps due to an upcoming earnings announcement—a trader might buy high-vega options to benefit from the increase. Conversely, if they expect volatility to drop, they might sell such options, anticipating a decline in their value. Volatility itself measures the magnitude and speed of price movements, which can be assessed using recent price action, historical data, or expected future moves.

At the Money (ATM): The strike price is equal (or very close) to the current market price of the underlying asset. Example: If a stock is trading at $100, a $100 strike call or put is ATM. ATM options have the highest time value but no intrinsic value.

In the Money (ITM): The option has intrinsic value, meaning it would be profitable if exercised immediately. For a call option: The strike price is below the current stock price (buying below market). For a put option: The strike price is above the current stock price (selling above market). ITM options are more expensive but have immediate value.

Out of the Money (OTM): The option has no intrinsic value, only time value. For a call: The strike price is above the stock price (not worth buying above market). For a put: The strike price is below the stock price (not worth selling below market). OTM options are cheaper but riskier.

Time Value: The extra amount an investor is willing to pay because there's still time until expiration, representing the potential for the option to gain value. Even if an option is OTM, it can still be worth something because there's time for the underlying to move favorably. Higher volatility increases time value.

Time Decay (Theta): Time value erodes as expiration approaches, and this decay accelerates near expiry. Example: A stock is trading at $105 and you buy a call option with a $100 strike for $8. Intrinsic Value = $105 − $100 = $5. Time Value = $8 − $5 = $3. The $3 reflects what you’re paying for the possibility that the stock rises further before expiration.

Notional Value

Notional value refers to the total value of the underlying asset represented by a derivative contract. It represents how much value a position controls — not necessarily how much was paid for it. This concept is used across various derivative markets, including options, futures, forwards, and currency swaps.

Traders and institutions use notional value to assess portfolio exposure and determine hedge ratios. It provides a way to understand how much of the underlying market a position influences, even if the actual investment (or market value) is much smaller due to leverage.

The notional value can generally be calculated as:

Notional Value (NV) = Contract Size × Underlying Price

Because derivatives use leverage, the notional value is often much higher than the amount of money actually invested. The degree of leverage can be calculated as:

Leverage (L) = Notional Value ÷ Market Value

Interest Rate Swap Example

In an interest rate swap, the notional value is the reference amount upon which interest payments are calculated. No actual exchange of the notional amount occurs — it's simply a base used for computation.

For instance, two banks enter a swap with a notional value of $10 million. One agrees to pay a fixed interest rate of 3%, while the other pays a floating rate based on LIBOR. The $10 million is never exchanged — it only serves as a benchmark for calculating interest payments.

Equity Option Example

In an option, notional value represents the total value of the shares the option controls. For example, assume stock ABC trades at $20, and one call option (which controls 100 shares) costs $1.50 per share.

Cost of the option: $1.50 × 100 = $150
Notional value: $20 × 100 = $2,000

In this case, the trader controls $2,000 worth of stock for only $150 — a demonstration of how leverage amplifies exposure relative to investment.

Notional vs. Market Value

Think of notional value as the reference amount a contract is based on. It's usually fixed when the contract begins and doesn't fluctuate with market movements. It's used for calculating potential gains or losses.

Market value, in contrast, reflects the current worth of the contract itself — that is, how much the contract could be sold for today. Market value changes as underlying prices move.

For example, if you enter into a futures contract for 1,000 barrels of oil at $80 per barrel:
Notional Value = $80 × 1,000 = $80,000

Market value refers to the current fair value (or profit/loss) of the contract after market prices change. If oil prices rise to $85, your contract gains value because you can buy oil below market price. The market value (your profit) would increase by:($85 − $80) × 1,000 = $5,000

Yield Curve

The yield curve is a graphical representation that shows the relationship between the yield (interest rate) and the time to maturity of bonds with similar credit quality, typically government securities. It plots bond yields on the vertical axis and maturities on the horizontal axis.

In essence, the yield curve illustrates how much investors expect to earn from lending money over different time horizons. It serves as a key indicator of market expectations about future interest rates, inflation, and overall economic growth.

Shapes of the Yield Curve

  • Normal (Upward Sloping): Long-term bonds have higher yields than short-term bonds, reflecting expectations of future economic growth and potential inflation.
  • Inverted: Short-term yields exceed long-term yields — often seen as a predictor of an upcoming economic slowdown or recession.
  • Flat: Short- and long-term yields are similar, suggesting uncertainty about future economic conditions or a transition period in the economy.

The slope of the yield curve is especially important. A steep curve generally signals strong future growth expectations, while a flattening or inverted curve can indicate tightening monetary policy or reduced confidence in economic expansion.

Volatility

Volatility (σ) measures the degree to which the price of an asset fluctuates over time. In simple terms, it captures how “noisy” or “stable” an asset's price is. A high volatility means that the price tends to move sharply — both upward and downward — over short periods, while low volatility suggests that prices move gradually or remain relatively stable. Volatility is a key input in risk management, option pricing, and portfolio optimization, as it quantifies the uncertainty or risk associated with an investment.

Historical volatility measures how much the asset price has fluctuated in the past. It is computed from a series of past price returns — usually daily returns — and then annualized using the formulaσ annual= √252 × StdDev(daily returns), where 252 represents the approximate number of trading days in a year. For example, if the standard deviation of daily returns is 1%, the annualized volatility would be √252 × 0.01 ≈ 15.9%. This provides a statistical measure of how volatile the asset has been historically.

Implied volatility, in contrast, is forward-looking. Instead of relying on past data, it is inferred from the current market prices of options using models likeBlack–Scholes. The logic is: given the observed option price, what volatility must the market be assuming for the model price to match the market price? Implied volatility therefore reflects market expectations about how much the underlying asset is likely to move in the future.

In practice, traders often monitor changes in implied volatility to gauge market sentiment. When they say “vol is up 2 points,” they mean that the market's implied volatility has risen by 2 percentage points, indicating higher expected uncertainty. During periods of market stress or major economic announcements, implied volatility usually increases as investors anticipate larger potential price swings.

Spread

The spread is the difference between two prices, rates, or yields — most commonly the ask and the bid. In plain terms:

Spread = Ask price − Bid price.The bid is the highest price a buyer is willing to pay; the ask is the lowest price a seller will accept. When you buy, you pay the ask; when you sell, you receive the bid.

Example: if Bid = 150.00 and Ask = 150.10, the spread is 0.10. If you immediately buy at the ask and then sell at the bid, you lose the spread — that loss is the basic transaction cost of trading.

The spread exists to compensate liquidity providers (market makers, dealers, or electronic liquidity pools) for the risk and capital they commit to make two-sided markets. By quoting both a bid and an ask they enable instant execution for other participants; the spread is their fee for providing that service and for bearing the risk that prices may move while they hold inventory.

Several factors influence the size of the spread. Highly liquid instruments (major forex pairs, large-cap stocks, government bonds) tend to have very tight spreads because many participants trade them and order books are deep. Conversely, illiquid instruments, news-driven volatility, or times of market stress tend to widen spreads — fewer counterparties and greater inventory risk increase the cost of providing liquidity.

Spread can also be expressed relative to price to compare across instruments. For example, the percentage spread = (Ask − Bid) / Midprice × 100%, where midprice = (Ask + Bid) / 2. This makes it easier to see trading cost as a fraction of value.

In practice, consider the spread when sizing trades and estimating round-trip costs (buy then sell). For high-frequency or large-volume strategies, tiny differences in spread can materially affect P&L; for long-term investors, spread is usually a smaller component of total trading cost but still worth considering, especially in less liquid markets.