Fixed Income (General) Flashcards
Explain the principle of risk-neutral valuation
The principle of risk-neutral valuation is one of the main reasons behind the BSM models popularity and wide-spread use. Prior economists had developed models for pricing options before the BSM model was introduced in 1973, but these prior models all had the same problem: the interest rate at which a known payoff at time T had to be discounted with was unknown, because investors have different risk preferences. This problem was solved with the introduction of the principle of risk-neutral valuation. The principle involves that the risk-free interest rate in the economy is used for discounting cash flows. This means that the price of a derivative when expressed relative to the price of the underlying is independent of risk preferences. In this risk-neutral world all traded derivates have an expected payoff equal to the risk-free interest rate.
Provide the definition of delta and describe how the delta changes with the stock price, i.e. sketch the delta curve
See answer below
Provide the definition of gamma and describe how the gamma changes with the stock price, i.e. sketch the gamma curve
See answer below
Provide the definition of theta and describe how the theta changes with the stock price
In general theta (Θ) is defined as the rate of change in the call price with respect to the passage of time. Theta will generally be negative for an option since, all else being equal, the option value is positively dependent of the time to maturity so as time passes and the maturity date draws closer the value of the option will decrease. When the stock price is low the value of theta is small because even as time passes there’s only a small likelihood that the option will end up ITM. When the option is ATM, at the stock price is close to the strike price, theta will be negative as the passage of time reduces the value of choice, which is particularly valuable when the option is ATM. Also, when the stock price is high and the option is ITM the passage of time will again reduce the value of the option but less so compared to an option that is ATM, because the likelihood of the ITM option being exercised is large.
Explain what vega is. Can a portfolio both be made vega and gamma neutral?
In practice, volatilities change over time. This means that the value of a derivative is liable to change because of movements in volatility as well as because of changes in the asset price and the passage of time. The vega of a portfolio of derivatives, V, is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. If vega is highly positive or highly negative, the portfolio’s value is very sensitive to small changes in volatility. If it is close to zero, volatility changes have relatively little impact on the value of the portfolio. A position in the underlying asset has zero vega.
Unfortunately, a portfolio that is gamma neutral will not in general be vega neutral, and vice versa. If a hedger requires a portfolio to be both gamma and vega neutral, at least two traded derivatives dependent on the underlying asset must usually be used.
Explain how you would go about calculating the zero rates for below bonds
The first two bonds are zero coupon bonds so their zero rates are easily calculated For an example, the first bond turns an investment of $98 into $100 over a 6 month period so you know that 100 = 98e^r*0,5. From this expression you just have to back out r. The same procedure is used for the 12 month bond which also does not pay any coupons.
The third and fourth bonds does pay coupons so to value these the bootstrap method is used. For an example, for the third bond you know it pays a semi-annual coupon of $6,2 per year so the semi-annual coupon is $3,1, its time to maturity is 18 months, its principal is $100, and its price is $101. Using the calculated zero rates for the zero coupon bonds the following expression can be constructed: 3,1e^-r(1)0,5 + 3,1e^-r(2)1 + 103,1e^r*1,5 = 101. From this expression you simply have to back out the 18 month zero rate that is, r. The same procedure is used for calculating the zero rate for the 24 month bond.
What is forward rates and how are they calculated?
Forward interest rates are the future rates of interest implied by current zero rates for periods of time in the future.
If we for an example have a forward interest rate for year 2 of 5% per annum. This is the rate of interest that is implied by the zero rates for the period of time between the end of the first year and the end of the second year.
It can be calculated from the 1-year zero interest rate of 3% per annum and the 2-year zero interest rate of 4% per annum. It is the rate of interest for year 2 that, when combined with 3% per annum for year 1, gives 4% overall for the 2 years.
The formula for calculating the forward rate is seen below
How is the price of a 2-year bond providing a semi-annual coupon of 7% per annum calculated?
In words you basically just have to discount the four coupon payments with he calculated zero rates back to time 0. An example is seen below
What is the bond yield and how is it calculated?
A bond’s yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price. If y is the yield on the bond, expressed with continuous compounding, it must be true that the below expression holds.
The bond yield however has to be calculated in Excel so it’s probably not relevant for the exam
