Other

Question 1

Bond PD

Answer 1

~PD = spread/LGD

Question 2

Map of risk types

Answer 2

Insert

Question 3

How to calculate ES? (historically)

Answer 3

ES is the average of the worst returns (wich lie below the corresponding VAR)

For a standard normal, the ES is given by pdf/(1-confidence)

Question 4

If R(i) = α(i) + β(i)*R(M) + e(i);

where:

• R(i) is the excess return for security(i) and
• R(M) is the market’s excess return
• σ(M) is known
• σ(e(i)) is known

How to calculate σ(R)?

Answer 4

σ²(R) = β(i,M)² σ(M)² + σ[e(i)]²

Question 5

Basis and roll yield in contango and backwardation

Answer 5

• contango is when F(0) > S(0) such that basis, B = S(0) - F(0,T), is negative
• backwardation is when F(0) < S(0) such that basis, B = S(0) - F(0,T), is positive
• In contango and a static forward curve, the forward price will decrease as maturity approaches. This is why we say that during contango (backwardation) the roll yield is negative (positive)

Question 6

Normal backwardation

Answer 6

Normalbackwardation refers to F(0) < E[S(t)], when the futures price is less than the expected future spot price

… for example, we can have normal backwardation expectation while in contango

Question 7

Dollar duration is?

Answer 7

Dollar (value) duration = (-) modified duration * Price

! without change of interest rate (0.01)

Question 8

Bond conversion factor

Answer 8

The conversion factor for a bond is set equal to the quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per annum (with semiannual compounding). The bond maturity and the times to the coupon payment dates are rounded down to the nearest 3 months for the purposes of the calculation. The practice enables the exchange to produce comprehensive tables. If, after rounding, the bond lasts for an exact number of 6-month periods, the first coupon is assumed to be paid in 6 months. If, after rounding, the bond does not last for an exact number of 6-month periods (i.e., there are an extra 3 months), the first coupon is assumed to be paid after 3 months and accrued interest is subtracted.”

As a first example of these rules, consider a 10% coupon bond with 20 years and 2 months to maturity. For the purposes of calculating the conversion factor, the bond is assumed to have exactly 20 years to maturity. The first coupon payment is assumed to be made after 6 months. Coupon payments are then assumed to be made at 6-month intervals until the end of the 20 years when the principal payment is made. Assume that the face value is $100. When the discount rate is 6% per annum with semiannual compounding (or 3% per 6 months), the value of the bond is

Summation{i = 1 to 40} 5/1.03^i + 100/1.03^40 = $146.23; [dh note: this is the same as pricing the bond with N = 40, I/Y = 3, PMT = 5.0, FV = 100 and CPT PV = $146.23]

Question 9

Answer 9

Question 10

If the volatility of the short interest rate (LIBOR) is 4.0%, what is the convexity adjustment for a five (5)-year Eurodollar futures contract?

Answer 10

convexity adjustment = 0.5*volatilty^2*T(1)*T(1+0.25).

In this case, convexity adjustment = 0.5*4%^2*5*5.25 = 2.10%

Question 11

Two differences between Eurodollar futures and FRA

Answer 11

There are two differences between a Eurodollar futures contract and an FRA. These are:

1. The difference between a Eurodollar futures contract and a similar contract where there is no daily settlement. The latter is a forward contract where a payoff equal to the difference between the forward interest rate and the realized interest rate is paid at time T1.
2. The difference between a forward contract where there is settlement at time T1 and a forward contract where there is settlement at time T2.

Question 12

Early exercise of American options

Answer 12

• It is never optimal to exercise an American call option on a non-dividend-paying stock before the expiration date.
• Unlike the American call on a non-dividend-paying stock, which is never optimal, the American put is often optimal to early exercise.

Hull: “In general, the early exercise of a put option becomes more attractive as So decreases, as r increases, and as the volatility decreases.” As (r) increases, this increases the “interest element:” the interest earned on the strike price collected (a difference from the call option!). As volatility decreases, this decreases the “insurance element” making early exercise more attractive.

Question 13

Map of “financial disasters”

Answer 13

Question 14

Impact of delivery on futures pricing

Answer 14

Whereas a forward contract normally specifies that delivery is to take place on a particular day, a futures contract often allows the party with the short position to choose to deliver at any time during a certain period. (Typically the party has to give a few days’ notice of its intention to deliver.)

The choice introduces a complication into the determination of futures prices. Should the maturity of the futures contract be assumed to be the beginning, middle, or end of the delivery period? Even though most futures contracts are closed out prior to maturity, it is important to know when delivery would have taken place in order to calculate the theoretical futures price.

• If the futures price is an increasing function of the time to maturity, it can be seen that c > y, so that the benefits from holding the asset (including convenience yield and net of storage costs) are less than the risk-free rate. It is usually optimal in such a case for the party with the short position to deliver as early as possible, because the interest earned on the cash received outweighs the benefits of holding the asset. As a rule, futures prices in these circumstances should be calculated on the basis that delivery will take place at the beginning of the delivery period.
• If futures prices are decreasing as time to maturity increases (c < y), the reverse is true. It is then usually optimal for the party with the short position to deliver as late as possible, and futures prices should, as a rule, be calculated on this assumption.

Question 15

Assume the 360-day LIBOR zero rate has been calculated as 2.40% with continuous compounding and an actual/360 day count convention. Also, the Eurodollar futures quote for a contract maturing in 360 days is 97.00. Which is nearest to the implied 450 day (15-month) LIBOR zero rate with quarterly compounding?

Answer 15

The ED futures contract, by definition, gives a 3-month forward rate. In this case, F(1.0, 1.25) = 100 - 97 = 3.0% per annum with quarterly compounding and act/360 day count. This corresponds to a continuously compounded F(1.0, 1.25) = 4*LN(1+3%/4) = 2.98881%.

The implied 450-day zero rate = (2.98881%*90 + 2.4%*360)/450 = 2.51776% with continuous, or 4*[EXP(2.51776%/4)-1] = 2.52570% with quarterly compounding

Question 16

An asset-or-nothing (binary; aka, digital) call option that pays the stock price or nothing has one year (1.0) to maturity while the risk-free rate is 4.0%. The current stock price, S(0), and the strike price (K) are both $30.00. The stock has a volatility of 36.0% and pays a dividend yield of 1.0%. N(d1) = 0.60 and N(d2) = .460. What is the price of the asset-or-nothing binary call option?

Answer 16

S(0)*exp(-qT)*N(d1) = $30*exp(-1%*1)*0.60 = $17.82

Please note: In general, dividends reduce the stock price in OPMs: S(0)*exp(-qT) N(d2) is the risk-neutral probability of the asset expiring ITM and the value of a cash-or-nothing call is given by Q*exp(-rT)*N(d2). But the asset-or-nothing call uses N(d1)!

Don’t let this confuse with BSM, where in the case of a European call option, DELTA = exp(qT)*N(d1) and N(d2) is the risk-neutral probability of expiration in-the-money (S>K).

Question 17

A zero-coupon bond with nine (9) years to maturity has a face value of $1,000. The bond’s yield (YTM) is 5.0% per annum with continuous compounding. The daily yield volatility is 70 basis points; for convenience, assume the daily yield is normally distributed. Which is nearest to the bond’s one-day linear (i.e., duration-based) 99.0% confident value at risk, VaR?

Answer 17

A zero coupon bond with nine years to maturity has Macaulay duration of 9 years. If compounding is continuous, then the modified duration is also nine: modified duration = Mac duration/(1+y/k) but k –> inf in the case of continuous compounding. But, keep in mind, we always want the modified duration here! For example, if the yield were 5.0% with semi-annual compounding, then modified duration = 9/(1+5%/2).

$1,000*exp(-5%*9)*0.0070*2.33*9 = $93.597

Question 18

A stock with a current price of $200.00 has an annual volatility of 30.0%. An at-themoney (ATM) call option has a delta, N(d1), of 0.6180 and gamma of 0.00950. If we assume 250 trading days, what is the one-day 95.0% confident delta-gamma value at risk (VaR) of a long position in the call option?

Answer 18

$3.67

The 95% VaR of the underlying stock = $200*30%*1/SQRT(250)*1.645 = $6.2423. The 95% delta-gamma VaR = $6.2423*0.618 - 0.5*0.0095*$6.2423^2 = $3.6727; i.e., the gamma adjustment decreases the VaR relative to the delta-only VaR. i.e., if the position is long a call or put, the gamma adjustment reflects a reduction in risk; but if the position is short an option, the gamma adjustment reflects an increase in risk

Question 19

Answer 19

**Covariance(A,B) = β(A)\*β(B)\*σ(M)^2** = 0.80\*1.40\*0.20^2 = 0.04480 and
correlation = 0.04480/(0.32\*0.40) = 0.350.

Question 20

Jane has a short position in 100 put option contracts (10,000 options) where, according to Black-Scholes Merton, N(d1) = 0.650 and the per option (percentage) gamma is 0.0150. The
one-day 95.0% value at risk (VaR) on a single share of the underlying non-dividend-paying stock is $4.00. Assuming i.i.d. returns, what is the 10-day 95% VaR of the short put option contract?

Answer 20

$56,272

The 10-day VaR = SQRT(10)*$4.00 = $12.65; scale per square root rule if returns are i.i.d.

The delta of the put option = 1 - N(d1) = 1 - 0.65 = -0.35;
note: the percentage delta of a put is between [-1,0]

The short put is riskier on the downside due to the gamma (curvature).

Mathematically:
As dc ~= delta*dS + 0.5*gamma*dS^2, as the downside risk in a short put occurs when the stock price decreases, we have:
dc ~= -delta*-dS + 0.5*gamma*-dS^2 = +delta*dS + 0.5*gamma*dS^2; i.e., the gamma terms adds (increases the risk)

In this case, VaR (short put contract) = abs[-0.35 delta]*VaR[stock] + 0.5*gamma*VaR[stock]^2
= 0.35*$12.65 + 0.5*0.0150*$12.65^2 = $4.43 + 1.20 = $5.63.
With 100 option contracts, VaR [100 contracts] = $5.63 * 10,000 = $56,272

Question 21

Bond has a long position in a bond with a face value of $1,000.00. He assumes the daily yield volatility is 1.0% with normally distributed daily yields. The bond’s modified duration is 9.70 years and its convexity (C) is 100.0 years^2. The current price of the bond is $612.00. What is the 95% daily (quadratic) value-at-risk?

Answer 21

VaR = abs[-duration*price]*VaR(yield) - 0.5*convexity*price*VaR(yield);

i.e., the downside risk is an increase in yield, which the convexity mitigates.
As VaR(yield) = 1.0% \* 1.645 = 1.645%, VaR = 612\*[9.70 years\*1.645% - 0.5\*100\*1.645%^2] = $89.373

Question 22

Bob tries to value a three month European put option with a one-step binomial tree (one step = 0.25 years). The price of the non-dividend-paying stock is $100.00 and the put option is
at-the-money (ATM) with a strike price of $100.00. The asset volatility is 36.0% per annum with continuous compounding. The riskless rate is 4.0%. Bob decides that volatility will inform the size of the up (u) and down (d) steps according to a Cox, Ross Rubinstein (CRR) model; i.e., if the number of steps were increases the asset price would tend toward a lognormal distribution.

Answer 22

The down movement is to $100*exp[-36%*SQRT(0.25)] = $83.5270, with future put option value of max[0, 100-83.5270] = $16.4730.

u = exp[36%*SQRT(0.25)] = 1.19722 and d = 1/u = exp[-36%*SQRT(0.25)] = 0.835270; this is
per the lognormal CRR assumption.

u = exp[σ * (T/365)^0.5]

d = 1/u

Since p = [a-d]/[u-d] = [exp(rT) - d]/[u-d] = [exp(4%*0.25) - 0.835270]/[1.19722 - 0.835270]
= 0.48288. Therefore, (1-p) = 0.517112, and
The discounted option put value = $16.4730*(1-p)*exp(-4%*0.25) = $8.4336.

Question 23

Meaning of N(d1) and N(d2) in the BSM (for the call option)

Answer 23

• N(d2) is the risk-neutral probability of exercise
• N(d1) is the call option delta

Question 24

The BSM for the put option and properties of N(d1), N(d2)

Answer 24

p = X*e^-rT*N(-d2) - S₀*N(-d1)

• put delta = N(d1) - 1, and since N(d1) = 1 - N(-d1), put delta = [1 - N(-d1)] - 1 = -N(-d1).
• N(d2) is the risk-neutral probability a call option will expire ITM; therefore, 1-N(d2) is the risk-neutral probability the call option will expire OTM and the equivalent put will expire in the money. Since N(d2) = 1 - N(-d2), the probability the put will expire in-the-money = 1 - [1 - N(-d2)] = N(-d2).

Question 25

Portfolio Manager Sally has a position in 100 option contracts with the following position Greeks: theta = +25,000, vega = +330,000 and gamma = -200; i.e., positive theta, positive vega
and negative gamma. Which of the following additional trades, utilizing generally at-the-money (ATM) options, will neutralize (hedge) the portfolio with respect to theta, vega and gamma?

a) Sell short-term options + sell long-term options (all roughly at-the-money)
b) Sell short-term options + buy long-term options (~ ATM)
c) Buy short-term options + sell long-term options (~ ATM)
d) Buy short-term options + buy long-term options (~ ATM)

Answer 25

Buy short-term options + sell long-term options

For ATM options, vega and theta are increasing functions with maturity; and gamma is a decreasing function with maturity.

To buy short-term options + sell long-term options –> negative position theta, negative position vega, and positive position gamma.

• In regard to (A), sell short-term + sell long-term –> positive theta; negative vega; negative gamma
• In regard to (B), sell short-term + buy long-term –> positive theta; positive vega; and negative gamma.
• In regard to (D), buy short-term + buy long-term –> negative theta; positive vega; and positive gamma

Note:

the above are approximately actual numbers for 100 option contracts (100 options each = 10,000 options) with the following properties: Strike = Stock = $100; volatility = 15.0%, risk-free rate = 4.0%; term = 1.0 year. Under these assumptions

• 1-year term: percentage theta ~= -5.0, vega ~= +37, gamma ~= +0.025
• 10-year term: percentage theta ~= - 2.5, vega ~= +70, gamma ~= +0.005

Question 26

Unexpected loss (UL) formula

Answer 26

UL = EAD*SQRT[PD*variance(LGD) + LGD²*variance(PD)],

such that UL is non-linear with respect to PD, volatility (PD) and LGD

Question 27

Gamma of short-life options when approaching maturity (trends)

Answer 27

Although for short-life OTM/ITM options, gamma tends to zero as maturity decreases toward zero, at-the-money options behave the opposite way: the gamma of ATM options is generally a decreasing function of maturity (i.e., gamma increases as maturity tends toward zero).

Question 28

WCDR formula

Answer 28

WCDR = N[(N^-1(PD) + sqrt(ρ)*N^-1(α))/(sqrt (1-ρ)]

α - confidence level

Question 29

UL of portfolio of loans and contribution of each loan to the portfolio UL

Answer 29

Question 30

Link between geometric and arithmetic mean returns in time series

Answer 30

The geometric mean (aka, compound annual growth rate, CAGR, in the case of annual compounding) is approximately σ²/2 less than the arithmetic mean:

µ[geometric] ≈ µ[arithmetic] - σ²/2

Question 31

GBM process, stock price and return distrubutions

Answer 31

Question 32

The spot EUR/USD exchange rate is $1.30 (i.e., USD 1.30 per 1 EUR) with a volatility of 30% per annum. The USD riskless rate is 4% per annum and the EUR riskless rate is 3% per annum. What is the delta of a one-year call option on the Euro with a strike price of EUR/USD $1.36?

Answer 32

We substitute the foreign riskfree rate for the dividend yield: d1 = [LN(S/K) + (Rf_USD - Rf_EUR+ sigma^2/2)*T] / [sigma * SQRT(T)], so that: d1 = [LN(1.30/1.36) + (4% - 3% + 30%^2/2)*1] / [30% * SQRT(1)] = 0.0329, and N(d1) = 0.5131, and again substitute the foreign riskfree rate for the dividend yield: delta of call = N(d1) * exp(-qT) = 0.5131 *exp(-3%*1) = 0.4980

Question 33

The spot price of oil is $80.00 per barrel with a volatility of 26% per annum. The riskfree rate is 5.0% per annum. What is the delta of a one-year futures contract when the one-year futures price is $90.00 per barrel?

Answer 33

Delta of forward = exp(-qT) and delta of futures = exp[(r-q)*T]. In this case, delta = exp(5%) = 1.015. The other information is unnecessary.

Question 34

8. 5. If at-the-money (ATM) options are otherwise identical, which of the following will have the LOWEST value of rho?
      a) Put with distant time to expiration
      b) Put near to expiration
      c) Call near to expiration
      d) Call with distant time to expiration

Answer 34

Rho is positive for calls and negative for puts, with interest rate having the most impact when expiration is distant. Rho (put) = -K*T*exp(-rT)*N(-d2), such that Rho(put) is a decreasing function with increasing (T).

Question 35

Expected spot vs Futures price, formula

Answer 35

F(0,T) = E[S(T)]*[exp(r - k)T]

where:

• k - the discount rate
• r - the risk-free rate

Question 36

Base/quote format for currencies?

Answer 36

X EUR/USD means X USD per 1 EURO

Question 37

Convexity of a zero-coupon bond

Answer 37

Under semi-annual compounding, convexity of a zero = T*(T+0.5)/(1+y/2)^2.