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A second source of convexity gap is atonality. The relationship between the market value of an option and the underlying parameter is highly convex when the option is at-the-money and the payoff of an option, as a function of the underlying asset, has a "kink." Options create "kinks" in duration and create convexity risk. When the option is out-of-the money, the sensitivity, or delta, is very low, and conversely, when the option is in-the-money, the delta is close to one. Hence, all options, including embedded options in the balance sheet of banks, have a high convexity effect when close to at-the-money.

Options are a major source of convexity in a balance sheet, and the main source of errors when using local measures of sensitivity to measure risk. Embedded options cap the value of assets and impose floors to the value of liabilities.

FIGURE 27.8 "Scissors Effect" on EV due to embedded options

For instance, the value of a loan with a renegotiation option cannot go above some maximum value, since the borrower can renegotiate the loan interest rate. A decline in interest rate normally increases the value of the asset, until we reach the point when renegotiation occurs. When we reach the cap, the loan behaves as if it were variable, at least at the time of renegotiation. The difference between the maximum loan value and the value of the straight loan is the value of the option.

On the liability side, options generate a floor for the value of some liabilities when the interest rates rise. Beyond some upper level of market rates, depositors shift their funds from deposits that earn no return at all, or earn a low fixed rate, towards interest-bearing assets. The value of the liability hits a floor instead of decreasing with the rise of interest rates. Even if resources stay within the bank, they become variable rate liabilities and have a constant minimum value when rates increase.

Beyond some variations of interest rates, fixed-rate assets and liabilities tend to behave as variable rates assets and liabilities. The shape of the "value/interest rate profile" of assets and liabilities flattens and becomes horizontal when hitting some upper bounds when interest rates decline and lower bounds when interest rates increase, as shown in Figure 27.8.

Figure 27.9 shows that, whatever the direction of the interest rate movement, the effect on economic value is always adverse to the bank. This is sometimes called "the scissors effect" on economic value. When interest rates remain in a narrow range, the EV remains positive. However, if they deviate beyond upper and lower bounds that triggers the embedded caps or floors, options gain values and generate a negative EV. This is the general case. Remember that EV is the present value of all Nils. When interest rate volatility grows higher, it has always an adverse effect on both EV and NIL

Adverse effect of convexity on EV and Nil

FIGURE 27.9 Adverse effect of convexity on EV and Nil

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