Options Risk and Value at Risk: Managing the Greeks and Beyond
This is Part 2 of our retelling of Chapter 3 from “Trading the Fixed Income, Inflation and Credit Markets: A Relative Value Guide” by Neil C. Schofield and Troy Bowler (Wiley, 2011, ISBN: 978-0-470-74229-7).
Previous: What Risk Really Means in Fixed Income
Swap Market Risk
Spot Swap Risk
Remember from Chapter 2: a swap where you receive fixed and pay LIBOR is like buying a fixed-rate bond and issuing a floating-rate note. So swap risk follows the same principles as bond risk. Traders talk about the DV01 of a swap (sometimes called the “delta”) and the convexity (the “gamma”).
For a 5-year swap on a 10 million notional, a one basis point parallel shift in the curve changes the value by about 3,177. That’s the DV01.
Something interesting happens with non-parallel shifts. If only the 3-year rate moves by one basis point while everything else stays put, the discount factors and forward curve change from year 3 onwards. But the overall NPV of the swap doesn’t move. Only a shift in the swap rate of the same maturity as the swap itself affects the NPV. That’s a subtle but important detail.
Swaps also display convexity. A 100 basis point move up loses 307,737, while a 100 basis point move down gains 328,117. Losses decelerate as yields rise. Profits accelerate as yields fall. Positive convexity, just like a bond.
DV01 in Action: A Curve Trade
Say a trader expects the swap curve to steepen between 2 and 5 years. The 2-year DV01 (on 10m) is 2,764 and the 5-year is 3,177. The hedge ratio is 3,177/2,764 = 1.15.
To profit from steepening: receive fixed on a 2-year swap (notional 11.5m) and pay fixed on a 5-year swap (notional 10m). This makes the position delta-neutral against parallel moves. The trader only makes or loses money on changes in the slope.
The carry on this trade depends on LIBOR. If LIBOR stays below about 5.23%, the position carries positively. Free money while you wait for the curve to move.
Forward-Starting Swap Risk
Forward-starting swaps are popular with traders because until LIBOR actually fixes, floating cash flows carry no interest rate risk. A bank that finances at LIBOR is immune to LIBOR movements on unfixed payments. This is a key insight.
For a 3-year swap starting 1 year forward, the DV01 is purely a function of forward rates during the swap’s life. No LIBOR setting before the start date can change the risk profile. Once the forward period ends and the swap goes live, it behaves like a regular spot-starting swap.
This stability is why traders prefer forward-starters for curve trades. With spot-starting swaps, as soon as LIBOR fixes, the DV01 of each leg shifts by different amounts and the hedge gets messy. Forward-starters keep things clean until the effective date.
Option Risk: The Greeks
Option models aren’t really for pricing. The market price is whatever someone will pay. Models are for understanding how the option’s value will respond to changes in market conditions. That’s where the Greeks come in.
Delta: Your Directional Exposure
Delta tells you how much the option premium moves when the underlying price moves. A call with a delta of 0.50 means the premium moves by 50 cents for every dollar the underlying moves.
Key properties:
- Out-of-the-money options have deltas near zero
- In-the-money options have deltas near one
- At-the-money options sit around 0.50
- Long calls and short puts have positive delta
- Short calls and long puts have negative delta
Delta is additive. A +30 delta option combined with a -30 delta option gives zero net delta. No directional exposure.
Think of delta as a few things at once: your directional bet, the hedge ratio you need to neutralize the position, and (roughly) the probability of exercise.
One thing to watch: delta changes with time even if the price stays flat. As expiry approaches, delta becomes more extreme. Near-ATM options snap toward 0.50 while OTM options bleed toward zero. This “delta bleed” means a hedged position gradually drifts unless you adjust.
Gamma: The Rate of Change
Gamma is how fast delta changes when the price moves. It’s the second-order effect. Think of it as the fixed income equivalent of convexity.
Gamma matters most when you’re delta hedging. High gamma means delta shifts rapidly, forcing frequent rebalancing. This creates real costs.
Here’s the classic example. A trader sells a short-dated ATM call on 10,000 shares at 700p with a delta of -0.50. They buy 5,000 shares to hedge. The stock rallies to 720p, and delta jumps to -0.92. Now they need 9,200 shares short equivalent, but only have 5,000 long. They buy 4,200 more shares at 720p.
Then the price drops back to 700p and delta returns to -0.50. They sell those 4,200 shares at 700p. Net result: bought at 720, sold at 700. Loss of 840 on the rebalancing alone.
This is negative gamma. The option seller loses money from rebalancing in a volatile market. The buyer of the option (positive gamma) would profit from the same moves. Gamma is essentially your exposure to actual price volatility, not implied volatility.
Gamma is highest for short-dated, at-the-money options. Long-dated OTM or ITM options have low gamma. Buyers are gamma positive. Sellers are gamma negative.
Theta: Time Decay
Theta measures how much value the option loses each day just from time passing. It’s negative for buyers (time works against you) and positive for sellers (time works for you).
Time decay isn’t linear. It starts slow and accelerates as expiry approaches. ATM options lose the most in absolute terms. OTM options lose the most in percentage terms.
Theta is sometimes called “gamma rent.” Selling options gives you positive theta (you earn the passage of time), but negative gamma (you lose from rebalancing). The game is to earn more from time decay than you lose from hedging.
Vega: Implied Volatility Exposure
Vega measures sensitivity to changes in implied volatility. This is different from gamma. Gamma is about actual price movements right now. Vega is about the market’s expectation of future price range at expiry.
Higher implied volatility means higher option prices for both calls and puts. Buyers are vega positive (they want implied vol to rise), sellers are vega negative.
The relationship between vega and strike matters. For ATM options, vega is roughly constant with respect to implied vol. For OTM options at low implied vol, a doubling of implied vol can more than double the premium.
A common myth: spot price moves and implied volatility moves go hand in hand. They don’t have to. They’re independent inputs to the model. A trader might see a volatile day but not change their vol quotes if they think it’s temporary.
Smiles, Skews, and Surfaces
Black-Scholes assumes implied volatility is constant across all strikes and maturities. Reality disagrees.
When OTM and ITM options trade at higher implied vol than ATM options, that’s a volatility smile (common in FX). When the smile is lopsided, it’s a skew (equity markets tend to skew to the downside).
Why? Three explanations:
- Fat tails. Extreme events happen more than the model predicts.
- Hedging feedback. Traders who sold OTM puts face growing gamma risk as markets crash, so they charge higher vol upfront.
- Supply and demand. Traders buying OTM calls and selling OTM puts push vol higher on one side.
The smile tends to get more pronounced closer to expiry because gamma risk intensifies for short-dated ATM options.
Value at Risk
All the measures so far are instantaneous: what happens if rates move one basis point right now? Value at Risk (VAR) takes a forward-looking approach. It estimates the potential loss over a specific time period at a given confidence level.
Example from Barclays: their Daily VAR at 98% confidence means they expect losses to exceed the VAR number on only 2 out of every 100 business days. Important caveats: VAR is just an estimate, it changes daily, and losses can exceed it.
Three Methods
Variance/covariance uses statistical properties of returns. Historical simulation revalues current positions using 2 years of actual daily prices, generating about 500 hypothetical P&L scenarios, then picks the relevant percentile. Monte Carlo simulation generates random scenarios based on assumed distributions.
Barclays favored historical simulation. Their process: revalue every position using each historical day’s prices, sum the P&Ls across the portfolio for each day, then select the 98th percentile loss.
During the 2008 crisis, VAR numbers spiked dramatically as the historical data now included extreme market moves. This highlights both the strength of VAR (it adapts to changing conditions) and its weakness (it only reflects what has already happened).
That finishes Chapter 3. From the philosophical question of what risk means, through duration, convexity, and DV01, to the Greeks and Value at Risk. The thread connecting everything: risk isn’t a single number. It’s a collection of sensitivities to different market factors, and managing it means understanding how all those sensitivities interact.