## On this Page

» Q&A: If a forward contract is an obligation...

» Q&A: If an instrument is priced rationally...

» Q&A: As the interest rates drop...

» Q&A: A bank has made a 10%, 5-year, $100 million fixed-rate loan...

» Q&A: A Bank issues 1,000 dual currency bonds...

» Q&A: An analyst thinks that the mean return rate for tax-exempt municipal bonds...

» Q&A: An analyst wants to calculate a 95% confidence interval for...

» Q&A: An investor enters into a 5-year interest rate swap...

» Q&A: Calculate the price of the interest rate put option using the Black model

» Q&A: compute the price of the call option using Black-Scholes-Merton model

### Q&A: If a forward contract is an obligation...

...as opposed to a right, to buy/sell, can one get out of it prior to expiration?

*Answer* : Not directly. One could, however, enter into an opposite transaction with the same or a different counterparty (with additonal credit risk). For example, if a party goes long in the original transaction, he can terminate his long position by going short in the new forward contract. aka offsetting.

Other things to know about forwards

- no money changes hands at the start.
- mostly cash settled (ie. index forwards), as opposed to physical delivery.
- common types include Equity forward (contract prices and values must take into account the fact that the underlying stock, portfolio, or index could pay dividends), bond forwards, currency forward (manage foreign exchange risk), and interest rate forwards (ie FRA).

*Category: C++ Quant > Finance > Derivatives > Forwards*

### Q&A: If an instrument is priced rationally...

...should there be any arbitrage opportunity? If yes, how much investment does it require to turn that into a profit? If not, why not?

*Answer* : there should be no arbitrage opportunity according to the no-arbitrage principle. There is no free lunch in a well-functioning market. (And it takes 0 dollars to make an arbitrage profit.)

*Long Answer*: Arbitrage is a a process through which an investor (aka arbitrageur) can buy an asset or combination of assets at one price and concurrently sell at a higher price, thereby earning a profit without investing any money or being exposed to any risk. The no-arbitrage principle

- facilitates the determination of prices: the combined actions of many investors engaging in arbitrage result in rapid price adjustments that eliminate any arbitrage opportunities, thereby bringing prices back.
- promotes market efficiency: quickly eliminate arbitrage opportunities available in the market

*Category: C++ Quant > Finance > Derivatives > Valuation*

### Q&A: As the interest rates drop...

...will you expect the yields of GNMA to pick up or drop?

*Answer* : As interest rates drop, the prepayments start to increase and the value of a GNMA tends to approach its par amount. For a noncallable Treasury, as the interest rates drop the market value increases. Thus, the Treasuries will tend to outperform the GNMAs under those conditions. Of course, refinancing rates do not necessarily move parallel with the Treasury yields.

Premium GNMAs will have the greatest prepayment risk, although the premium burnout effect should be kept in mind. The discount and par mortgage-backed securities will also show greater prepayments.

*Category: C++ Quant > Finance > Debt*

### Q&A: A bank has made a 10%, 5-year, $100 million fixed-rate loan...

...and wants to swap the fixed annual interest payment into a floating-rate annuity. If the bank could borrow at a fixed rate of 8% for 5 years, what is the notional principal of the swap?

*Answer* : 100 * 10% / 8% = $125 mil.

*Category: C++ Quant > Finance > Derivatives*

### Q&A: A Bank issues 1,000 dual currency bonds...

Principal: USD $1,000, sold and redeemed in USD

Coupon: JPY 6.5 million, semiannual.

Maturity: 5 years.

How can the bonds be synthesized?

*Answer*

- The Bank's Cash flows
- At issue: Bank receives USD 1 M.
- Every 6-mo: Bank pays JPY 6.5 M.
- At maturity: Bank pays USD 1 M.

- Cash flows can be synthesized using a corporate USD straight bond and a fixed-for-fixed currency swap. Consider the following instruments available in the market
- 7.5% corp Bonds with 5 years to maturity
- A currency swap dealer offers USD 7.5% against JPY 6.5% (swap dealer pays USD 7.5% interest payments and receives JPY 6.5%).

- The Bank can:
- sell short USD 1,000,000 worth of corp bonds.
- Enter a fixed-for-fixed currency swap and make JPY payments.

*Bonus Points*

- Foreign exchange exposure: dual-currency bonds are purchased in terms of one currency but pay coupons or repay principal at maturity in terms of a second currency.
- By having the currency forward attached to the bond, fixed-income portfolio managers who might otherwise be restricted from trading in FX have the potential to enhance their performance if their beliefs about future market conditions prove correct.

*Category: C++ Quant > Finance > Derivatives*

### Q&A: An analyst thinks that the mean return rate for tax-exempt municipal bonds...

...is 9.1% with a standard deviation of 3%. What is the minimum percentage of return rates for tax-exempt municipal bonds with rates between 3.1% and 15.1%?

*Answer* :

- [3.7%, 14.5%] lie with +/- 2 standard deviations of the mean
- (15.1-9.1)/3 = 2
- (9.1-3.1)/3 = 2

- From Chebyshev's Inequality: 1-1/k^2 = 1-1/2^2 = 75%

*Bonus Points*

- It applies to both populations and samples, and for discrete and continuous data, regardless of the shape of the distribution.
- Empirical Rule only applies to normal distributions. It states 68% within 1s, 95% within 2s, and 99% within 3s.

- Gives a conservative estimate of the proportion of observations in an interval about the mean.
- Assume the return is normal, Empirical rule gives 95% > 75%.

*Category: C++ Quant > Finance > Quantitative Analysis > Probability*

### Q&A: An analyst wants to calculate a 95% confidence interval for...

...the projected mean return of a portfolio he manages (assuming the return is normal). His latest calculations show that this portfolio has a mean return of 8% and a variance of 36. What is the 95% confidence interval?

*Answer* : The 95% confidence interval for a normal variable X is given by: X-bar +- 1.96*s = X-bar +- 1.96*var^0.5 = 8 +- 1.96 *6 = [-3.76%, 19.76%] . That means that there is a 95% probability that the mean return lies in the range of -3.76% to 19.76%.

- Confidence interval = Point Estimate +/- Reliability Factor * Standard Error, where Reliability factor is a number based on the sampling distribution of the point estimate and the degree of confidence (1 - alpha)
- The reliability factor and the standard error, however, may vary depending on three factors:
- Distribution of population: normal or non-normal.
- Population variance: known or unknown.
- Sample size: large or small.

- For a normal variable X, the Reliability Factors are 1.645 for 90% confidence interval, 1.96 95%, and 2.58 99%.

*Bonus Points*

- We can use the sample mean to estimate the population mean, and the sample standard deviation to estimate the population standard deviation. The sample mean and sample standard deviation are point estimates.
- Confidence intervals use point estimates to make probability statements about the dispersion of the outcomes of a normal distribution. A confidence interval specifies the percentage of all observations that fall in a particular interval. ie. a 90% confidence interval means that 10% of the observations fall outside the 90% confidence interval, with 5% on each side.

*Category: C++ Quant > Finance > Quantitative Analysis > Probability*

### Q&A: An investor enters into a 5-year interest rate swap...

...The notional amount is $100 million and the reference rate is 3-month LIBOR. Payments are made quarterly. The swap rate that the investor agrees to pay is 5%. How much should the investor pay the dealer at the end of the first quarter, if for the first floating-rate payment 3-month LIBOR is 4%?

*Answer*

- The fixed-rate payment each quarter is $100mil * (0.05/4) = $1.25mil
- The first quarter floating-rate payment is $100mil * 0.04/4 = $1.0 mil
- $1.25 - $1.0 = $0.25mil: the amount the investor shuld pay the dealer.

*Category: C++ Quant > Finance > Derivatives > Swaps*

### Q&A: Calculate the price of the interest rate put option using the Black model

An interest rate put option based on a 90-day underlying rate has an exercise rate of 5.5% and expires in 150 days. The forward rate is 5.25%, and the volatility is 0.08. The continuously compounded risk-free rate is 4%. The notional principal is $10 million.

*Answer* :

- To price European options on futures we can use the Black model: p = e^(-r*T)*{X*[1 - N(d2)] - f(0,T)*[1 - N(d1)]}
- The time to maturity is T = 150/365 = 0.4110.
- f(0, T) = 0.0525, X = 0.055
- d1 = [ln( f(0T)/X) + (s^2/2)/T] / (s T^0.5) = [ln(0.0525/0.055) + 0.082/2 * 0.4110] / (0.08 x 0.41101/2 = -0.8815
- d2 = d1 - s T^0.5 = -0.8815 - 0.08 x 0.41101/2 = -0.9327
- N(d1) = N(-0.8815) = 1 - N(0.8815) = 0.1894.
- N(d2) = N(-0.9327) = 1 - N(0.9327) = 0.1762.
- p = e^(-0.04 x 0.4110)* [0.055 x (1 - 0.1762) - 0.0525 x (1 - 0.0.1894)] = 0.002708.

- The answer is given under assumption that the option payoff occurs at the option expiration. However, this interest rate option expires in 90 days and pays off 90 days that that. Therefore, we need to use the forward rate to discount the result back from day 240 to day 150: 0.002708 x e -0.0525 x (90/365) = 0.00265
- As the underlying rate and exercise rate are expressed as annual rates, the answer is an annual rate. However, interest rate option prices are often quoted as periodic rates. We need to convert the result to periodic rate based on a 90-day rate and using the customary 360-day year: 0.00265 x (90/365) = 0.0006625.
- the price is 10,000,000 x 0.0006625 = $6,625.

*Bonus Points*

- c = e^(-r*T)*[f(0,T)*N(d1) - X*N(d2)]
- As with the Black-Scholes-Merton formula, this model applies to European options only. We can also use the model for American options on forwards as they are never exercised early.
- volatility refers to the volatility of the continuously compounded change in the futures price.

*Category: C++ Quant > Finance > Derivatives > Options*

### Q&A: compute the price of the call option using Black-Scholes-Merton model

Assume that a stock trades at $100 and the continuously compounding risk-free interest rate is 6%. A call option on the stock has an exercise price of $100 and expires in one year. The standard deviation of the stock's returns is 0.1 per year.

*Answer*

- d1 = ( ln(S0/X) + (r+stderr^2/2)*T) / (stderr*T^1/2) = ( ln(100/100) + (0.06 + 0.12/2)*1) / (0.1*1^(1/2)) = 0.65
- d2 = d1 - (stderr*T^1/2) = 0.65 - (0.1*1^(1/2)) = 0.55
- N(d1): In Excel, Normsdist(0.65) = 0.74
- N(d2) = Normsdist(0.55) = 0.71
- C = S(0) *N(d1) - X * e^(-r*t) * N(d2) = $100 x 0.74 - $100 x e^(-0.06 x 1) x 0.71 = $7.46
- P = X*e^(-r*T)*(1-N(d2)) - S(0)*(1-N(d1)) = $100*e^(-0.06*1) * (1-0.7088) - $100*(1-0.7422) = $1.64.

*Bonus Points*

- Normsdist finds the cumulative normal values. ie. Normsdist(0.65) is the probability that a normally distributed variable with a zero mean and a standard deviation of 1.0 will have a value equal to or less than the 0.65.
- Model Assumptions
- The underlying price follows a lognormal probability distribution as it evolves through time.
- Reasonable for most assets that offer options. Additionally, the variance of the return is assumed to be constant for the life of the option.
- Interest rates remain constant and known.
- Problematic for pricing options on bonds and interest rates.

- The volatility of the underlying asset is known and constant.
- Specified in the form of the standard deviation of the log return.

- No transaction costs or taxes.
- No cash flows on the underlying.
- European exercise terms are used: not a major concern because very few calls are ever exercised before the last few days of their life.
- True because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

- The underlying price follows a lognormal probability distribution as it evolves through time.

*Category: C++ Quant > Finance > Derivatives > Valuation*