chapter 28 i'm confused about the difference between liability hedging and perfect matching. The notes describe it as choosing assets that behave in the same way as liabilities in all ways. Isn't that perfect matching? So what's the difference?

Hedging and matching are closely related. Usually if you are matched you'll also be hedged. But its possible to hedge without matching. Matching requires that the amount and timing of the cashflows for the asset and liabilities are the same. The nature and currency of the cashflows must also be the same. So liability cashflows of 10, 30, 40 at times 1, 2, 3 must be matched by identical asset cashflows at these times. The value of the matching assets should have the same value as the liabilities at all times, so we'll also be hedged. Hedging requires that the values of assets and liabilities move in exactly the same way. For example, we can hedge an option which expires in one year's time with a mixture of cash and shares. The option liability has a cashflow at time 1, but the assets produce a regular stream of income (interest and dividends), so we are not matched. I hope this helps explain the difference. Best wishes Mark

Hi Mark, Two questions: 1. In the example above, can you clarify how the assets and liabilities are moving in the same way? Which is the asset and which is the liability here? 2. Can you give an example where we are hedging and matching as well? Thanks, Phani

Hi Phani In my first example above we are hedging and matching. The liability cashflows are 10, 30 and 40 and times 1, 2 and 3. We find assets (perhaps three zero coupon bonds) that also have cashflows of 10, 30 and 40 at times 1, 2 and 3. We are matched because the cashflows are of the same amounts and happen at the same times. As these two sets of cashflows are identical, then they must surely have the same value, so that means we are hedged too. Best wishes Mark

Hi Mark, Thanks for the reply. Sorry I don't think I was clear in my question about which example I didn't understand. When you wrote "For example, we can hedge an option which expires in one year's time with a mixture of cash and shares", I didn't quite understand this. The value of the option would be what we would get on exercising it, which would depend on the exercise price agreed upon. How then would the shares have the same value? How does cash come into the picture? And finally is the option our asset or liability in this example? Thanks, Phani

Hi Phani An option can be an asset or liability. But in this thread we're thinking about the matching or hedging of liabilities. So let's assume that an investment bank has a liability of 100 call options, ie the bank has sold 100 call options to a client. Perfect matching of this liability is impossible. As you say, the liability at maturity is the excess of the value of 100 shares over the exercise price multiplied by 100, subject to a minimum of zero. There are no assets that will match this (other than the call option itself). But we can hedge this liability. CM2 describes in detail how we can hedge call options. It's covered in various parts of the course: delta hedging under the Greeks, using a portfolio of cash and shares using the binomial model, using a replicating portfolio to derive the Black-Scholes equation. I'll leave you to check out CM2 for the full discussion, but here are the basics. If the share price goes up by 1, then the value of the call option will go up, but not by the same amount. How much the call option goes up will depend on how in the money the option is. Let's say that when the share price goes up by 1, the call option goes up by 0.7 (0.7 is the delta of the option). We have 100 call options, so we could hedge with 70 shares. That way, the bank's assets go up by 70 and liabilities go up by 70 if the share price goes up by 1. So we are hedged. We aren't matched as the shares will pay dividends that don't coincide with the option expiry date. We need our asset and liabilities to have the same value. But 70 shares are likely to cost more than 100 call options, so we will borrow the difference in cash. So we have a hedging (or replicating) portfolio made up of a positive amount of shares and a negative amount of cash. We'll need to rebalance our portfolio of shares and cash over time to ensure that the hedging still works as time goes by and the share price changes - again there's more about how to make a replicating portfolio self financing in CM2. Best wishes Mark