April 25, 2024
A zero coupon bond, or bullet has unit cash flow at maturity t. Its price at time 0, D(t), is the discount. Its price at time s \le t is D(s, t) so D(t) = D(0,t).
Exercise. Show D(s,t) = D(t)/D(s).
Hint: Buy D(s,t) at time 0 of a zero maturing at s. At s buy D(s,t) of a zero maturing at t. It costs D(s,t)D(s) at time 0 to replicate a zero maturing at t.
The forward curve f(t) is defined by D(t) = \exp(-\int_0^t f(s)\,ds) Note D(t,u) = \exp(-\int_t^u f(s)\,ds). The par curve, or yields curve, y(t) is defined by D(t) = \exp(-t y(t)). Note the yield y(t) = (1/t)\int_0^t f(s)\,ds is the average of the forward curve so ty'(t) + y(t) = f(t). It is computationally preferable to work with the forward curve since integration smooths out variations.
An interest rate model is determined by the stochastic forward rate f_t. It represents the (unknown) repo rate available at time t. The stochastic discount is D_t = \exp(-\int_9^t f_s\,ds). The price at time s of a zero coupon bond maturing at time t is D_s(t) = E_s[D_t]/D_s, where E_s indicates the conditional expectation given information available at time s.
Bonds are fixed income instruments. A fixed income instrument has cash flows (c_j) at times (t_j), 0\le j\le n. The issue date is t_0 and maturity date is t_n.
For a standard bond with coupon c and day count basis \delta we have c_0 = 0, c_j = c\delta(t_{j-1}, t_j), 1 < j < n, and c_n = 1 + c\delta(t_{n-1}, t_n), where \delta(t,u) is the day count fraction for the interval [t,u] corresponding to the day count basis. The day count fraction is approximately equal to the time in years from t to u. For example, the Actual/360 day count basis has day count fraction equal to the number of days from t to u divided by 360.
At time t the next coupon date t\_(t) = \min\{ t_j \mid t_j > t\} and the previous coupon date \_t(t) = \max\{ t_j \mid t_j \le t\}. Note if \_t(t) \le t_j < t\_(t) then \_t(t_j) = t_j and t\_(t_j) = t_{j+1}.
The accrued interest at t is a(t) = c_j \delta(\_t(t), t). The buyer must pay this to compensate the seller since the buyer gets the full coupon at the next coupon date. The dirty price is clean price plus accrued interest. Clean price is the present value of future cash flows.
A fixed income instrument is a portfolio of zero coupon bonds. Its present value at pv date t is p(t) = \sum_{t_j > t} c_j D(t, t_j). For a stochastic interest rate model the present value at t is p_t = \sum_{t_j > t} c_j D_t(t_j).
The yield y = y(t) = y(t,p) for price p quoted at t is defined by p = \sum_{t_j > t} c_j e^{-y(t_j - t)}. This is never used in practice and yields involve a compounding frequency n. The n-th annually compounded yield y_n, or yield to maturity, if n is understood, is determined by p = \sum_{t_j > t} c_j(1 + y_n/n)^{-n(t_j - t)}. (Is this correct?).
(ytm = (c + (face - pv/2)/2)/(face + pv)/2?)
There are several types of duration. Without an adjective, duration is the derivative of price with respect to a parallel shift of the forward curve dp/df = -\sum_{t_j > t} c_j D(t, t_j) (t_j - t), the time weighted average of the discounted cash flows. Since price is a decreasing function of yield the duration is typically quoted without the minus sign.
Exercise. The duration of a zero coupon bond is its maturity.
Modified duration is the derivative of bond price with respect to yield, \begin{aligned} dp/dy_n &= \sum_{t_j > t} c_j(1 + y_n/n)^{-n(t_j - t) - 1}(-n(t_j - t)/n) \\ &= -(\sum_{t_j > t} c_j(1 + y_n/n)^{-n(t_j - t)}(t_j - t))/(1 + y_n/n) \\ \end{aligned} This is approximately equal to (dp/df)/(1 + y_n/n).
Macaulay duration is the derivative of the logarithm of the price d(\log p)/df = (dp/df)/p.
Bond issuers can make bonds cheaper by attaching a call option. The option is specified by a series of times and call prices, the call schedule. At each call date the issuer can pay the holder the call price to terminate their remaining bond cash flow obligations.
They can also make a bond more valuable by attaching a put option by specifying a put schedule allowing the bond holder to pay the issuer the put price at any put date to redeem (Are calling and putting both considered redemption?) the bond.
The yield to the first call date of a bond redeemed at the call price is the yield to call. The yield to the first put date redeemed at the put price is the yield to put. The yield to worst is the minimum of yield to maturity and yield to call.
The tax basis of a bond purchased at a premium is its preset value using the purchase date yield and cannot be deducted for tax purposes.
Let \sigma be the short term capital gains tax rate and \lambda be the long term capital gains, where \sigma > \lambda.
refund:
redeem:
Given a set of instruments I, trading times T, and information \mathcal{A}_t at t\in T, let X_t\colon\mathcal{A}_t\to\boldsymbol{R}^I be the prices and C_t\colon\mathcal{A}_t\to\boldsymbol{R}^I be the cash flows at time t\in T. A trading strategy is a finite collection of increasing stopping times \tau_j and trades \Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{R}^I. Trades accumulate to positions \Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s<t} \Gamma_s where \Gamma_s(\omega) = \Gamma_j(\omega)1(s = \tau_j(\omega)).
The value, or mark-to-market at time t of a trading strategy is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It is the amount unwinding the current position and trades just executed at prevailing market prices would produce.
The account at time t is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t. The trading strategy pays cash flows proportional to the position and trades executed at t must be paid for.