1. A typical bond price/yield curve plots (maps) the bond price (the dependent variable on the y-axis) to the bond yield (independent variable on the x-axis). For a plain-vanilla bond (without embedded options), what is the proper name for the first derivative of the bond’s price with respect to yield? Can we say anything about its value as yield decreases/increases (e.g., positive increasing)?
The first derivative is the dollar duration, not duration.
Duration = -(1/P)*dP/dy such that the first derivative (dP/dy) = -Duration*Price = -DP. This is called the dollar duration.
In the case of a typical “plain-vanilla” bond (no embedded options), the price/yield curve is concave up: at low yields, the slope of the tangent line (dollar duration) is typically large and negative. As the yield increases, the slope increases; i.e., the dollar duration remains negative but becomes approaches zero as it increases.
2. A callable bond exhibits negative convexity at low yields (why?). We visually recognize negative convexity by its arching back; but how is negative convexity characterized in mathematical (derivative) terms?
Duration is the slope of the tangent line; Convexity is a function of the rate of change in the slope. Specifically, convexity is the second derivative of the bond price with respect to the yield of the bond, divided by bond price. In the case of a typical “plain-vanilla” bond (no embedded options), convexity is positive; i.e., the price-yield curve is concave upward such that dollar duration is increasing (the rate of change of dollar duration = positive = positive convexity).
Mathematically, a bond with negative convexity is characterized by, at its segment’s beginning, an inflection point; i.e., the point at which the second derivative (acceleration) is zero. This is where the price/yield curve (e.g., a mortgage-backed security) changes from concave upward to concave downward. Further, the negative convexity is reflected in dollar duration that is decreasing with yield instead of increasing.
3. Under continuous discounting, the price of a bond with face value (F) and maturity of (n) years is a function of yield (y): Price (P) = F*EXP[-y*n]. What is the first partial derivative of this bond’s price with respect to yield? If we divide this quantity itself by price(y), what do we get?
If P = F*EXP[-y*n],
then
dP/dy = F*EXP[-y*n]*(-n) = -(F)(n)*EXP[-y*n]
This is the dollar duration of a zero-coupon bond under continuous discounting.
Since P = F*EXP[-y*n], if we divide this dollar duration by price, we get:
-(F)(n)*EXP[-y*n] / (F*EXP[-y*n]) = -n
Which is the negative of the modified/Macaulay duration!
For example, if n = 30, then this is a 30-year zero-coupon bond and, under continuous compounding, the modified/Macaulay duration is 30.
4. Under the Black-Scholes-Merton, the price of a European call option (c) is a function of several risk factors (note I called them ‘risk factors’) including stock price (S), strike price (K), volatility (sigma), interest rate (r), and term to maturity (t). What is called the first partial derivative of the call price with respect to stock price? with respect to volatility? with respect to interest rate? with respect to term?
The first partial derivative of the call price with respect to stock price is delta; delta = dc/dS
The first partial derivative of the call price with respect to volatility is vega; vega = dc/dSigma
The first partial derivative of the call price with respect to interest rate is rho; rho = dc/drate
The first partial derivative of the call price with respect to term (maturity) is theta; theta = dc/dterm
5. The bond plot (above) maps price (the dependent variable) to yield (the independent variable). What are the units of the first derivative, in the case of the bond? Under the assumption of continuous discounting, what are the units of a modified/Macaulay duration?
The units of the first derivative (dollar duration) are $/%.
The units of the modified/Macaulay duration, only under this continuous discounting assumption are $/$ and therefore are unitless (they are elasticities).
See Alex’s comments at the bottom of this post for great elaboration on the units implied by various duration metrics.
6. In the case of the European call option delta, if calculate option delta = 0.7, what are the units? What does it mean to assert that the delta of a call option is 0.7?
In the webinar, I suggested that it is helpful to be mindful of the axes; e.g., in the case of a bond, the y-axis is bond price and the x-axis is yield, such that the units of dollar duration are price($)/yield(%).
In the case of option delta, the axes are both dollars ($). The first derivative (delta) is therefore unitless as the dollars cancel.
7. How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7? [Hull 14.2]
You may need the following background:
1. position delta = delta multiplied by number of options; e.g., 1,000 options with a delta of 0.4 each implies a position delta of 400.
2. A share has a delta of 1.0, and
3. to make the position delta neutral is to make the total position delta equal to zero.
A short position in 1,000 options (where the per option delta = 0.7) has a position delta = - 700.
Because an outright share a delta of 1.0, the net position can be delta neutralized by purchasing (going long) 700 shares.
8. Why does a share have a delta of 1.0? Can you visualize that with a graph (even if it’s a little silly)?
Becase dS/dS = 1.0. The visualization could be a 45 degree line plotting change in share price (y-axis) against change in share price (x-axis).
9. If we succeed in making the short position above delta neutral, what does that imply about the position?
The delta hedge is not a complete hedge. The delta neutral position only applies for a relatively short period of time. And for small changes in the stock price, since the delta hedge is based only a linear approximation,
10. Is the delta neutral position perfectly hedged? If not, why not? Can you say that in mathematical (derivative) terms?
See above, the delta hedge is not nearly perfectly hedged. In mathematical terms, the hedge ineffectiveness is illustrated by the difference (the gap) between the tangent line and the non-linear call option price function.
11. The CAPM says Expected [return] = Riskless rate + (beta)*(equity risk premium). What is the first derivative of the expected return with respect to to a change in the equity risk premium (ERP)? With respect to a change in beta?
If E[r] = rf + (beta)(ERP), then
dE[r]/dERP = beta, and
dE[r]/dbeta = ERP
12. In the bond example above, dollar/Macaulay/modified durations are all first derivatives (or functions of the first derivative). What does it mean when we insist that a limitation of duration is that it is only locally accurate?
That duration only gives us the bond price change (for a given yield change) when the change is very small.
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