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楊宗緯-對愛渴望MV

Great song

Great song...2nd attempt.

FRM questions by bionic turtle

• The partial first derivative is very common in risk measurement. It appears in various asset classes and metrics; e.g., option delta, bond duration, risk contribution, marginal value at risk (marginal VaR)
• The first derivative is an instantaneous rate of change; i.e., the limiting ratio illustrated by the convergence of a secant line to a tangent line.
• We looked at some basic differentiation rules (e.g., power rule).
• I highly recommend the excellent, free calculus resources at www.analyzemath.com. My favorite calculus texts are The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides) and Calculus Know-It-ALL: Beginner to Advanced, and Everything in Between by Stan Gibilisco. I also like the affordable Schaum's Outline of Calculus.
• We looked at a handy idea: The first derivative of the natural log of a function equals the function’s growth rate or relative rate of change: if f(x) = ln g(x), f’(x) = g’(x)/g(x).
• Given the price function of a 30-year zero-coupon bond under continuous discounting, p(y) = 100*EXP(-y*30), we showed that the first derivative is the dollar duration: p’(y) = –3000*EXP(-y*30). Further, while duration has various specific "flavors," all are variations on (functions of) this first derivative dollar duration. For example, by dividing by price [i.e., p(y)], we produce the modified/Macaulay duration of 30.
• It is helpful to be mindful of the axis units. In the case of dollar duration, our first derivative at 5% yield was about –670. What are the units? In this case (i.e., dollar duration), this refers to –670 $/%. Specifically, 670 dollars in price change for a one-unit (100 basis point) change in yield. Still confused about duration units? See the comments to this post.
• The option delta is a first derivative; i.e., the change in call price with respect to a change in stock price. What are the units in this case? Since we have dollars (option price on y axis) divided by dollars (stock price on x axis), they cancel and option delta is unitless.
• We applied the first derivative rule to confirm that (i) futures contract delta is EXP(r*t) and (ii) the Eurodollar futures contract implies a $25 dollar change for each one basis point move.
• Finally, I hope I succeeding in conveying the one big idea underlying this webinar. We may characterize portfolios as responding to (mapped to) a set of underlying risk factors (the call option analogy: the value of the call option reacts to risk factors such as volatility and interest rates). The partial first derivative returns a linear approximation of the portfolio’s sensitivity to an underlying risk factor.
• Phillip made an excellent point about the key weakness of the first derivative linear approximation: is it is only locally accurate; the larger the change in the underlying factor, the less accurate it becomes.

FRM questions by bionic turtle

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.

BT FRM

Assume the following performance for a bond cohort:

Year T	Value of bonds outstanding at the beginning of Year T	Dollar value of bonds defaulted on during Year T
1	$1,000	$45
2		$55
3		$80

Questions:

• Calculate the marginal mortality rate (MMR) in year 3 for the following class of issuers
• Calculate the cumulative mortality rate (CMR) over the three year period

Answers:

According to Caouette, Altman et al:

The individual mortality rate [MMR(t)] of bonds in a specific rating class for each year (marginal mortality rate or MMR) is given by:
MMR(t)= Total value of defaulting debt in the year(t) / Total value of the population of bonds at the start of the year(t)

The cumulative mortality rate (CMR) is measured over a specic time period (1, 2, ......, T years) by subtracting the product of the surviving population of each of the previous years from one (1.0); i.e.,
CMR (t) = 1 - [Survival rate, year t] * [Survival rate, year t-1] * ... * [Survival rate, year 1]

Here are the calculations in a simple EditGrid spreadsheet.
Note the CMR is calculated per above, but also equals: cumulative defaults ($180) divided by $1,000 = 18%

D. Harper note:
MMR/CMR is analogous to Saunder’s marginal default probability (1 - p) and cumulative default probability (Cp):
Saunders’ (1-p) is an ex ante probability of default; MMR is an (ex post) realized default (mortality)
Saunders’ Cp is an ex ante cumulative probability of default; CMR is the realized default (mortality)