Fixed Income Modeling Review 3

During the whole semester, topic 3 was probably the easiest. Because I have known the material of Ito integral and stochastic differential equation(SDE) in advance of the lecture. But it was still good as a complement to my knowledge.

Everyone should know what probability space, filtration, stochastic processes, adapted processes, martingale and Brownian motion are, cause these concepts are the very basics of mathematical finance. One should also know that Ito integrals are martingales and if the integrand is non-random then Ito integral and Stratonovich integral are the same thing, which means that the product rule for Riemann integral applies to Ito integral as well. Why do we choose Ito integral instead of Stratonovich integral given that Stratonovich integral preserves the product rule? The reason is when we construct Ito integral, we choose the left end points for the integrand so we don't look into the future, and this is essential to financial application.

One of the most important theory we need to use is Ito's lemma. It basically expands the concept of derivative we already know in calculus. We know the Taylor expansion:

\(df(x)=f'(x)dx\)

and now for Ito integral we have

\(dg(t,W(t))=g_tdt+g_xdW(t)+\frac{1}{2}g_{xx}dt\)

so essentially we are using the fact that \(dW(t)dW(t)=dt\).

In order to compare different securities and remove the price appreciation due to the time value of money effect, we introduce the concept of numeraire and prices divided by numeraire are called dicounted prices. For any numeraire there exists a corresponding martingale measure which is equivalent to the probability measure that discounted prices are martingales under such measure. Actually this is not precisely correct cause martingale measure does not always exist, so this sentence I am saying is more with respect to change of numeraire. That is if you change the numeraire then the martingale measure changes and vice versa.

The first fundamental theory of asset pricing says if there exists a martingale measure then there is no arbitrage in the market. The converse is not true and you need to add another condition called no free lunch with vanishing risk to ensure that martingale measure exists given there isn't any arbitrage opportunity. One thing new I learnt from this lecture is that if martingale measure exists for one numeraire then for another numeraire we could also find martingale measure. The most commonly used numeraire are banking account and zero coupon bond.

Change of measure is another important technique with respect to mathematical finance. You first need to know what is Radon-Nikodym derivative from real analysis. Then you should know the Girsanov's Theorem:

If W is a Brownian motion under P, then for

\(dP^*=\rho dP\), \(\rho=\exp{-\int^t_0\theta(s)dW(s)-\frac{1}{2}\int^t_0\theta^2(s)ds}\)

\(dW^(t)=dW(t)+\theta(t)dt\) is a Brownian motion under \(P^\).

Martingale measure according to banking account is called risk neutral measure which is named after the fact that the market price of risk is zero in this case. Under risk neutral measure, all the prices discounted by momey market account are martingales and in case of random interest rate model we often change the numeraire to forward measure because of simplicity of calculation(this is not the only reason!). Forward measure is the EMM with respect to zero coupon bond and one of the professors said to us that if someone knows what forward measure is then he/she could skip the course of Mathematical Finance I. Notably, forward price is a martingale under forward measure.

In fixed income modeling, one rule I didn't know is that conditional expectation of short rate is equal to the forward rate: \(E^T_t[r(T)]=f(t,T)\).

HW 3, My Solution

Fixed Income Modeling Review 2

This review will concentrate on bond mathematics and yield curves. I have come up with typical assumptions in fixed income modeling last time now we first need to examine more detailed assmptions on zero-coupon bonds, which illustrate the differences between theoretical formulas and market reality.

-- Most of the bonds pay regular coupons
-- Available bond maturities do not span the whole time axis
-- Different bonds trade at different spreads to benchmark securities because of different credit quality and liquidity level
-- Most corporate bonds are callable

After addressing the theoretical assumptions, we now turn to a very important concept in fixed-income: yield. Bond yield or yield to maturity can be defined as a constant discount rate through the whole life of a coupon bond. So we can see that the bond price is decreasing with respect to the yield. Furthermore, the bond price equals the principle if and only if yield equals the coupon rate. For short term bonds paying no coupon we define the discount yield which is a simply compounded rate. For government bonds we often have another quoting method called Bond Equivalent Yield (BEY) which is defined as a discounted rate semi-annually.

In my opinion, fixed-income market is a market of interest rate derivatives, so we definitely need to examine one derivative's interest rate risk. PVBP is one indicator to show a security's sensitivity to the change of interest rate. It is defined as

\(P=100[\sum^N_{i=1}\frac{c/2}{(1+y/2)^i}+\frac{1}{(1+y/2)^N}]\)
\(P^+=100[\sum^N_{i=1}\frac{c/2}{(1+(y+0.0001)/2)^i}+\frac{1}{(1+(y+0.0001)/2)^N}]\)
\(PVBP=P-P^+\).

PVBP can also be approximately computed by \(PVBP=-\frac{\partial P}{\partial y}0.0001\).

Duration is a similar measure to PVBP but rather in a relative way: \(D=-\frac{\partial P}{\partial y}\frac{1+y/k}{P}\).

When \(k\rightarrow\infty\), we have \(D=-\frac{\partial P}{\partial y}\frac{1}{P}\) which is also defined as modified duration. One important property of duration is that duration can be expressed as a weighted average of cash flow times.

\(D=-\frac{\partial P}{\partial y}\frac{1+y/k}{P}=\sum^N_{i=1}\frac{1}{P}\frac{CF(t_i)t_i}{(1+y/k)^{kt_i}}=w_i\cdot t_i\).

PVBP and duration measure the first order sensitivity to the change of rate while convexity measures the second order sensitivity.

\(Cx=\frac{1}{2}\frac{\partial^2P}{\partial y^2}\frac{1}{P}\)

Hence we can hedge against small parallel shifts in yield curve by making portfolio's PVBP zero and hedge against large parallel shifts by making convexity zero.

We have seen the importance of zero coupon bond curve. We can and we have to use them to calculate the present value of a cash flow. In practical world, we often use a way called bootstraping-interpolation to strip the zero-coupon bond price out of market data. Basically, it is to solve linear equations of cash flows to get the zero bond price at every node. By using this way, we could not only calculate zero bond price but also continuous forward rate, continuous spot rate and discretely compounded yields.

The last thing for this review is how to interpolate zero coupon bonds. From my point of view, the most popular way and its special cases can be called polynomial spline, which briefly speaking is to assume the zero bond price between two different time nodes could be expressed as polynomials.

HW 2, My Solution

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

第 2 页,共 2 页12