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Measuring Delta in Derivative Portfolios

By Jasvin Josen

This article appeared in TheEdge Malaysia on June 23, 2012

In the previous article, I explained what delta is in the market risk context and illustrated the case of a bond dealer measuring the delta in a bond portfolio. After knowing his delta exposure, he next decides how to manage the delta.
In this article, I expand the previous example by introducing interest rate swaps to hedge the delta. Next I illustrate the delta measurement for options.

Bond and Swap Portfolio

Let us assume that the bond dealer mentioned in the previous article decides to hedge his simple 3-type bond portfolio with interest rate swaps. He decides only to hedge the delta in the 5-year maturity bucket. He enters into a 5-year bullet interest rate swap (IRS) with the following terms:

5 year bullet IRS:
Pay Fixed Rate:                       4.4%
Receive Floating Rate:             6m LIBOR + margin
Notional:                                  $50,000

The cash flows from the portfolio of bonds and the IRS will look like in Chart 1. To keep it simple, say the 6m LIBOR (+ margin) rate at the 5Y point (obtained from the forward interest rate curve) is also at 4.4%.

Chart 1: Cash flows for Bonds and Swap Next, the dealer measures his delta exposure for the expanded portfolio. The delta profile will look like in Chart 2.

Chart 2: Delta Exposure for Bonds and Swap We can see that his delta exposure at the 5-year point has almost eliminated. This was possible with the bullet IRS where the cash flow on the floating leg also increases when the interest rate is bumped upwards. Specifically, the cash flow increased by $5, which is obtained by multiplying the notional value of the swap ($50,000) by 1 basis point (or 0.01%).

Macro Hedging
The practical way to hedge at trading desks of investment banks however is to conduct macro hedges rather than specific hedges. Here, the bond desks position will be combined with the interest rate swaps desk and the currency swaps desks portfolios to observe the net interest rate risk (or delta) exposure. Based on this net exposure, hedging of the interest rate risk is done at a macro level at the bank.

Other market risk factors affecting the bond price
In measuring the delta, a price engine is involved which theoretically prices the bond. However this assumes that the theoretical price is similar to the observed market price. Most probability this will not be so. This is because there are factors other than interest rates that cause the bond price to change. These factors are like liquidity and creditworthiness of the bond, which can be measured as well.
Equity Options Portfolio

In this example, we assume that an options trader has three equity options on the same share, shown in Chart 3.

Chart 3: Option Portfolio and Delta Measurement The trader defines his delta in options as the difference in the options price when the underlying share price moves by $0.10. He uses the Black Scholes model to compute the option price when the underlying share price moves from $3 to $3.10. We assume that the implied volatility is 12% and the risk-free interest rate is 3%.

He will see that the call option delta has a positive value whilst the put options delta has negative values. This is simply because when the underlying share price increases, the call option increases in value whereas the put options loses value.

The delta of the in-the-money call seem be set off mostly by the delta of the in-the-money put. On a net position, his delta position is quite small. He does not see the need to further hedge this portfolio at the moment.

Market risk other than the Delta

Delta is often the most significant risk that the market player must be aware of. Other risk factors that many option-like portfolios are also sensitive to are volatility (vega), interest rates used in models (rho) and gamma, the risk that the delta does not change in a linear manner.


We can appreciate that measurement of delta is important for any market practitioner, be he a trader, market maker, fund manager or the Treasury department in a corporation. Delta measurement enables the market practitioner to know his risk and hedge the risk accordingly. If he can do this proficiently, he will be more confident in investing, trading and dealing in a wider range of financial instruments and derivatives in the market.
Posted by Jasvinder Josen Labels: delta, derivatives, options bonds, swaps

