Fixed Income Modeling Review 1

Fixed income modeling is one of the major areas of mathematical finance, which is usually more complicated than stock market modeling. Not surprisingly, "Theory and Practice of Fixed Income Modeling" is among the core courses of our program. The lecturer is Andrei Lyashenko who used to be a mathematics professor at UIC and now he is working at QRM as a quant. We were taught the theoretical part in class and our homework is done in excel as practitioners do in industry. I just finish this semester and if I let it go I would probably forget the details in 3 months. So it is better for me to have a whole review of this class.

The first thing is to justify all types of risks in fixed income market: credit risk, interest rate risk, liquidity risk, cash flow timing risk, cash flow amount risk, foreign exchange risk, inflation risk and taxation risk. Different products are exposed to different risks. For example, since the US government is unlikely to get default, the credit risk does not present for treasure bonds. However, for debt issued by corporations such as citigroup, credit risk does present. Callability, fixed or floating rate, a lot of aspects need to be taken into account as you consider the potential risk.

The very basic contract considered in fixed income modeling is zero-coupon bond and the first tricky question is does \(P(t,T)\leq 1\) for any \(t. While the first thought comes to most people is: if the market does not allow negative rates, then it is true. However, this question actually has not much to do with positive or negative rates. Since you can always hold 1 dollar instead of invest it by buying zero-coupon bond, then if the price of zero-coupon bond is higher than 1 such contract would not exist any more. So what are the assumptions when we think about fixed income modeling?

-- Zero bonds of all maturities are traded

-- No transaction cost

-- Securities are perfectly divisible

-- Short selling is permitted

-- No arbitrage

Because of no arbitrage, discount factor \(D(t,T)\) should equal the zero bond price \(P(t,T)\).

The next important thing is to clarify the kinds of rates:

-- Spot rate: is the constant rate that makes the investment of \(P(t,T)\) at current time become 1 unit of currency at maturity

-- Forward rate: is a generalization of spot rate. Only assume that the investment happens at a future time and the maturity is some time further. 3-months or 1/4-year forward rates are usually called Libor Rates. By examine the contract of FRA we can prove that \(P(t,T)\cdot P(t,T,S)=P(t,S)\). There are also instantaneous forward rates defined as \(F(t,T)=\lim_{T\rightarrow S}F(t,T,S)=-\frac{\partial\ln(P(t,T))}{\partial T}\), and it has nothing to do with day counts and compounding. Remember forward and short rate (\(r(t)=F(t,t)\)) are purely theoretical and are not observed on the market.

-- Par rate: is the bond coupon rate that makes its price equal to its principle. It can be explained as the weighted average of Libor rates.

At the same time, spot rate is an average of instantaneous forward rates. Actually, \(F(t,T)=-\frac{\partial\ln(P(t,T))}{\partial T}\) implies that \(\int^T_t F(t,s)ds=-\ln(P(t,T))\), and therefore, \(R(t,T)=\frac{1}{T-t}\int^T_t F(t,s)ds\).

Price of fixed income derivatives are often quoted by yield, and yield curve is a plot of rates against their maturities.

Last for review 1 is to introduce the fixed income derivatives:

-- Bonds

-- FRA: OTC, a rate decided ahead, lending and borrowing starting and ending in the future

-- Interest rate swap

-- Swaption: is an OTC contract giving its holder the right but not the obligation to enter into a swap contract

-- Caps and Floors: cash flows at one time is called caplet or floorlet and are defined as

\(CF(T_i)=L\tau(T_{i-1},T_i)\max[0,R(T_{i-1},T_i)-R_K]\)

or

\(CF(T_i)=L\tau(T_{i-1},T_i)\max[0,R_K-R(T_{i-1},T_i)]\).

HW 1, My Solution

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