Basic Fixed Income Derivative Hedging Author: Financial-edu.com
This article introduces you to the basics of hedging fixed income derivatives trades. It is meant to be a practical guide to understanding basic hedging. Three common trade examples are presented with examples of different potential hedges. Volatility smiles are touched on, as are the Greeks (delta, gamma, theta, vega, rho) and other risk measures (PV01/DV01, duration, convexity). The language is written in layman’s terms as much as possible to facilitate understanding of hedging concepts without getting too deeply into complex mathematics. Real life hedging activities are typically more in-depth than the level of this article.
Hedging Examples for Fixed Income Derivative Trades
EXAMPLE #1: Plain Vanilla Swap
The first example uses a $100 million notional vanilla swap paying USD Fixed and receiving USD LIBOR.
How to hedge this trade – 5 STEPS TO HEDGING:
First, choose a point on the curve to hedge. This can be all points on the curve, all points in a segment of the curve (MM, futures, AIC swaps, etc.), or just a single point. To keep things simple, let’s select the 2Y point. The As of Date of this trade is 9/1/05, so we are looking to hedge the risk associated with 9/1/07, which is conveniently on the 2Y point of the curve.
Second, select the direction (up or down) you are hedging. Let’s hedge a +1bp change in the curve.
Third, identify the type of risk you want to hedge. Here we only have the USD LIBOR rate risk to hedge.
Fourth, identify the total dollar value of risk to hedge. For a +1bp change on the 2Y point of the curve, our model says we need to hedge -$0.48 x $1M notional = -$480,000 (this is the DV01). To determine how much is required to hedge the entire trade, simply add up the dollar risk for each point in the money market segment, futures segment, and AIC swap segment separately. Here we have risk by segment of -$386.44 money market, -$6,266.54 futures, and $76,853.69 AIC swaps. These should add up roughly to the value change of the swap if you perturb the entire curve up +1bp in a parallel shift.
Fifth, calculate the required number of units, contracts, or notional of the hedging instrument you are going to use.
Since we are looking to hedge a point 2Y out on the curve, we cannot use futures. We need a longer term instrument such as a 2Y AIC swap to hedge the 2Y point. Here, we choose a 2Y RECEIVER Swap paying 3M LIBOR and receiving FIXED with an expiration date on 9/2/07 and notional of approximately $100 million. This trade’s risk profile is the mirror image of the existing PAYER Swap paying FIXED and receiving 3M LIBOR. This is known as a "back to back" trade, and is commonly used by sell side broker-dealers and trading desks of mutual fund companies looking to hedge their fund portfolios.
For most trades, hedging a single point on the curve does not make sense. Rather, each segment of the curve, or the entire trade, would be hedged. By segment, the risks to hedge are: -$386.44 money market, -$6,266.54 EuroDollar futures, and $76,853.69 AIC swaps.
The -$386.44 money market risk is minimal, so it probably would not make sense to hedge this part of the curve.
The -$6,266.54 futures risk can be hedged by using the Required = risk / BPV formula. EuroDollar futures trade at -$25 per +1bp, so we need -$6,266.54 / -$25 = 250.66 = +/- 251 contracts.
To determine the notional of a USD LIBOR Receiver Swap needed to hedge the $76,853.69 AIC swap segment risk, use the following formula: Notional of hedge swap = $76,853.69/DV01 of hedge swap.
EXAMPLE #2: USD LIBOR cap
The second example uses a $5 million notional cap on USD LIBOR. A cap is essentially a series of European call options on an interest rate. A cap is made up of multiple caplets, each with a reset rate (strike) and its own expiration date. The value of a cap reflects the probability of an interest rate being in or out of the money at the expirations of all caplets. This probability is non-linear for a cap so there is delta, gamma (curvature), volatility (vega), and time (theta) risk.
How to hedge this trade – 5 STEPS TO HEDGING:
First, choose a point on the curve to hedge. This can be all points on the curve, all points in a segment of the curve (MM, futures, AIC swaps, etc.), or just a single point. Let’s select the 6M point here. The As of Date of this trade is 9/1/05, so we are looking to hedge the risk associated with 3/1/06, approximately the MAR06 point on the curve.
Second, select the direction (up or down) you are hedging. Since this is a long cap, we will profit if USD LIBOR goes up and incur losses if it drops, so we need to hedge a -1bp change in USD LIBOR. Also, because we are long a cap (call option), we will profit if USD LIBOR volatility rises and lose if volatility falls, so we need to hedge a -100bp change in forward USD LIBOR volatility. (NOTE: For convenience, we are going to assume perfect symmetry in sensitivities between USD LIBOR caps and floors with the same strike (3% here). This assumption is highly simplified and does not take into account the natural volatility skew built into Cap vs. Floor prices. Here we assume the dollar effect of a -1bp move in rate or volatility is simply the inverse of a +1bp move.)
Third, identify the type of risk you want to hedge. Let’s hedge both the USD LIBOR rate risk and the USD LIBOR forward volatility risk.
Fourth, identify the total dollar value of risk to hedge.
Our model shows a +$108.15 move in cap value for a +1bp move in USD LIBOR rate at the 6M point on the curve. Inverting this we get -$108.15 move in value for a -1bp shift at 6M.
Our model also shows a +$130.67 move in cap value for a +100bp move in USD LIBOR volatility at the 6M point (remember the vol sensitivity is set to a 100bp shift.) Inverting this, we get -$130.67 move in value for a -100bp move in USD LIBOR volatility at 6M.
Fifth, calculate the required number of units, contracts, or notional of the hedging instrument you are going to use.
To hedge the USD LIBOR rate risk, we need to sell (go short) the following number of futures contracts. This is an imperfect hedge. Required number of EuroDollar contracts = -$108.15 dollar risk / $25 per bp per contract = -4.33 contracts.
The 6M USD LIBOR forward volatility risk is more difficult to hedge. To cleanly hedge the -$130.67 volatility risk associated with the 6M curve point, we need to find an instrument strictly designed for USD LIBOR volatility. For example, we could buy an OTC USD LIBOR volatility put or short an OTC USD LIBOR volatility future. We could also write (go short) a similar USD LIBOR cap or floor, although this would also have delta effects that would interact with the futures delta hedge above. Buying another USD LIBOR cap or floor would be incorrect, as this would make us further exposed to USD LIBOR volatility.
EXAMPLE #3: Bermudan Swaption
The third example uses a Bermudan Swaption exercisable into a $100 million notional Fixed/Floating USD LIBOR swap, with the option to pay fixed rate payments and receive LIBOR floating payments (a Payer Swaption). The value of a swaption reflects the probability of the swap rate being in or out of the money on the exercise date. A Bermudan Swaption has multiple exercise dates, with each normally aligned with a reset date for the floating rate (here USD LIBOR). Valuation of a Bermudan Swaption is therefore very similar to valuing a Cap. The probability of the Bermudan Swaption being in or out of the money is non-linear so there is delta, gamma (curvature), and volatility (vega) risk.
How to hedge this trade – 5 STEPS TO HEDGING:
First, choose a point on the curve to hedge. This can be all points on the curve, all points in a segment of the curve (MM, futures, AIC swaps, etc.), or just a single point. Let’s select the 10Y point here, since our model says this is where the greatest LIBOR rate sensitivity lies.
Second, select the direction (up or down) you are hedging. Since this is a long Payer Swaption, we will profit if USD LIBOR rises since we will be able to exercise and enter into a swap where we pay a fixed rate of 3% and receive a higher LIBOR rate. Let’s assume we are taking a directional position (long) on USD LIBOR and hedge a -1bp change in USD LIBOR. Also, because we are long an option on an interest rate (call option), we are long volatility. We will profit if USD LIBOR volatility rises, and lose if it falls. So we could hedge a -100bp change in USD LIBOR volatility.
Third, identify the type of risk you want to hedge. Let’s hedge both the USD LIBOR rate risk and the USD LIBOR volatility risk.
Fourth, identify the total dollar value of risk to hedge. For a -1bp change at the 10Y point, we need to hedge -$49,841.44 of USD LIBOR rate risk and -$9,678.80 of USD LIBOR volatility risk on this swaption.
Fifth, calculate the required number of units, contracts, or notional of the hedging instrument you are going to use.
The problem with hedging such a long-dated swaption is finding an appropriate instrument to hedge it with. There are no liquid exchange traded futures or options with a 10Y tenor. Therefore any hedge must be done in the OTC market. There are several potential choices available:
• A combination of a 3M LIBOR volatility put and a floor on 3M LIBOR. • A forward-starting Receiver Swap receiving fixed and paying 3M LIBOR. • Long a 10Y Treasury Bond receiving LIBOR+ • A Total Return Swap using this Bermudan Swaption as the reference asset – you would give up the total returns of the Swaption for a set period in exchange for receiving payments of LIBOR +/- a spread and an offsetting payment (insurance) on any capital losses of the Swaption. • Others…
Each of these positions has its own unique delta, gamma, vega, theta, and rho. Thus the selection and sizing of any hedge would require a fair amount of “what if” analysis and computing power.
The Volatility Smile
Many hedging models for instruments with embedded optionality assume a normal distribution of prices or rates centered around the current price or rate. This is a simple and CPU-efficient approach to calculating basic hedges.
However, real world option valuations do not reflect normal distributions of prices or rates. Rather, option prices reflect a volatility “smile” which presumes the option’s value behaves differently depending on whether the direction of price or rate volatility is up or down and the level of price or rate. For example, the market assumes that the long term probability of interest rates being very high in the future (say 15%+) is greater than the probability they will be extremely low (say 0.75%, 0%, or negative), thus for interest rates the volatility “smile” is skewed to the upside.
Volatility “smile” also exists in foreign currency rates since governments tend to modify their economic policies to manage currency rates in a desirable “band” or “range” (say 80-120 EUROs per USD), making probabilities of currency rates being outside the band much lower than expected with a normal distribution.
Volatility “smile” is less of a factor for equity prices, although the dynamic exists to a greater extent at very low stock prices than higher stock prices. The value curve for each type of option has an ever-changing shape with a volatility “smile” related to the natural price paths of their underlying assets, and is not typically bell-shaped as a normal distribution would be. Normal distribution models are therefore only useful for basic hedging.
The Greeks
WARNING! The “rough” formulas used below are NOT the correct formulas. They are simple “ballpark” formulas that allow a quick calculation and are easy to understand. The actual formulas are more complex.
The term “Greeks” refers to 5 basic measures of risk. They are called “Greeks” because the risk figures are denoted by Greek letters Delta (Δ), Gamma (Γ), Theta (Θ), Vega (V) and Rho (Ρ). Each Greek symbolizes a single component of risk as follows:
• DELTA (Δ): The change in price or value of a derivative in relation to the change in price or value of the underlying security, index, or rate. For an equity option, this would be the dollar option price change per underlying dollar stock price change. For an interest rate cap, this would be the dollar cap price change per basis point move in the reference interest rate (e.g. LIBOR). For call options or long positions in other derivatives, delta is positive. For put options or short positions in other derivatives, delta is negative. Hedging Delta requires matching the trade with another position with the same total Delta in the opposite direction, for example a long call option hedged with a short future, both with equal but opposite Deltas (and possibly some convexity adjustment to the future contract hedge).
Rough Formula: Delta (Δ) = (DP0 – DP1) / (UP0 – UP1)
Where: DP0 = derivative price now DP1 = derivative price previous UP0 = underlying price or value now UP1 = underlying price or value previous
• GAMMA (Γ): The change in Delta in relation to the change in price or value of the underlying security, index, or rate. Gamma is only applicable to non-linear option-type derivatives like basic vanilla options, swaptions, caps/floors, convertible bonds, callable bonds, etc. Gamma represents the estimated “slope” or “curve” of the Delta profile over a range of underlying prices or values. Calculating Gamma requires two Delta calculations as inputs. The Gamma formula is the same for both calls and puts. Hedging Gamma requires going long or short a security with the same or similar delta and delta slope values, but in the opposite direction. Hedging Gamma is an imperfect science, and must be done on a continual basis as prices change.
Rough Formula: Gamma (Γ) = (Δ 0 – Δ1) / (UP0 – UP1)
Where: Δ 0 = Delta now Δ 1 = Delta previous UP0 = underlying price or value now UP1 = underlying price or value previous
• THETA (Θ): The change in price or value of a derivative in relation to units of time (usually 1 day). Theta primarily applies to option-type derivatives like basic vanilla options, swaptions, caps/floors, convertible bonds, callable bonds, etc. Theta represents the change in time value of an option as time progresses to an expiration date. Theta is almost always negative, because the probability of an option being in the money decreases as time to expiration decreases since there is less time for large price swings to occur. Theta typically has an exponentially decreasing value, with the slope of Theta increasingly negative (probability of going in the money increasingly less possible) as expiration nears. Hedging Theta of a long call option position requires shorting/selling/writing a same or similar call, or buying a put with the same strike and expiration.
Rough Formula: Theta = (DP0 – DP1) / (T0 – T1)
Where: DP0 = derivative price now DP1 = derivative price previous T0 = time now T1 = time previous
• Vega (V): The change in price or value of a derivative in relation to the unit volatility of the underlying security price, index value, or rate. Vega applies to non-linear option-type derivatives like basic vanilla options, swaptions, caps/floors, convertible bonds, callable bonds, etc. Vega also applies to options and futures on volatility indexes such as the VIX. As volatility of the underlying price, value, or rate increases, an option’s theoretical value also increases because there is greater probability that it will go in the money. The opposite is also true; as underlying volatility decreases, an option’s price should also decrease. Volatility of an underlying price, index, or rate is measured in standard deviations over a chosen look back period. For example, estimating the volatility of an underlying stock price would typically require calculating the standard deviation of price over the previous year’s trading days. Likewise, estimating the volatility of an underlying interest rate (e.g. Fed Funds) would require calculating the standard deviation of interest rates over the previous year. The Vega calculation is the same for both calls and puts. Hedging Vega requires going long or short volatility in the opposite direction as the derivative position. For example, a long call or put option is “long volatility” – either a long call or long put option increases in value if the underlying reference volatility increases. To hedge this, you would go “short volatility” by shorting/selling/writing equivalent call or put options.
Rough Formula: Vega (V) = (DP0 – DP1) / (V0 – V1)
Where: DP0 = derivative price now DP1 = derivative price previous V0 = underlying volatility now V1 = underlying volatility previous
• RHO (P): The change in price or value of a derivative in relation to the risk-free discount rate used to NPV the future value of the derivative to present value. Lower risk-free discount rates imply a higher present value, and vice-versa. Short-term derivatives like listed equity options (<1 year) are less sensitive to discount rates (i.e. have lower rho) than longer-term derivatives like 10-year swaps. Rho can be hedged through the use of futures, options, or swaps on the risk-free discount rate (e.g. LIBOR or Fed Funds), or the 90-day T-Bill rate.
Other Risk Measures
• PV01 (a.k.a. DV01): The change in the dollar present value of a bond or interest rate trade in relation to a 1 basis point (0.01%) move in the underlying interest rate. PV01 is equivalent to Delta, where the underlying rate is an interest rate.
Rough Formula: PV01 = PVr0 – PVr1
Where: PVr0 = present value of bond or trade using rate now PVr1 = present value of bond or trade using rate previous
• Duration: The approximate percentage of value the bond or trade will lose for a 1% increase in interest rates. Duration is roughly equivalent to the time in which the purchase price of a bond will be repaid by its internal cash flows. Duration is considered an alternative to PV01 or Delta.
• Convexity: The curvature of how the price of a bond changes as the interest rate changes. Convexity is essentially the same as Gamma, and measures changes in Duration over different interest rates. |