# Fixed Income Modeling Review 4

After reviewing Ito lemma and SDE, it is time to have a look at the standard market model which are forward contracts, future contracts, options on bond, interest rate caps, floors, collars and swaptions.

Forward contracts and future contracts are similar but there are also some crucial differences: forward contracts are over the counter and future contracts are traded on exchanges; future contracts are settled daily which reduces the credit risks, and after daily settlement future contracts is worth zero.

Forward price and future price are prices make the contracts worth zero at the beginning or the time after each settlement. Usually they are different but if the interest rates are deterministic, or more generally, the underlying asset price process is independent of the rate process then thery are the same.

$Fwrd(t)=\frac{S(t)}{P(t,T)}=\frac{E^B_t[D(t,T)S(T)]}{E^B_t[D(t,T)]}=E^B_t[S(T)]=Fut(t)$

One more thing is that if the underlying asset price is positively correlated with the interest rates then the futures price will be higher than the forward price and this is why the rate implied from Eurodollar futures price is higher than the corresponding forward rate.

For options on futures we have Black's model which is completely consistent with the Black-Scholes formula for stock options. Black's model is so popular that the implied volatility from the option prices has become the way most popular options are quoted, and it has also become a popular pricing and quoting tool on fixed income markets thought it assumes deterministic interest rates.

There are two basic assumptions for Black's model:
(1) The value V(T) of the option is lognormally distributed with the standard deviation of $\ln(V(T))$ equal to $\sigma(T-t)^{1/2}$.
(2) The expected value of V(T) is F(t) under T-forward measure.

And so the Black's formula becomes

$Call(t)=P(t,T)[F(t)\cdot N(d_1)-K\cdot N(d_2)]$
$Put(t)=P(t,T)[K\cdot N(-d_2)-F(t)\cdot N(-d_1)]$

Bond option is an option to buy or sell a particular bond by a certain date for a particular price. In addition to trading in the OTC market, bond options are often embedded in bonds when they are issued like callable bonds or puttable bonds. By assuming the bond price S(T) at the maturity of the option T is lognormally distributed with standard deviation of $\ln(S(T))$ equal to $\sigma(T-t)^{1/2}$ we can apply the Black's model to compute the price of European call and put options.

Interest rate cap is an OTC floating rate contract which is a series of call options with the underlying being the market rate and the strikes equal to a fixed rate. Each of the call options is called a caplet. For floors and floorlets, they are similar, only with call options changed to put options.

Every caplet can also be seen as a bond option:

$V_i(t_{i-1})=P(t_{i-1},t_i)L\tau(t_{i-1},t_i)\max[R(t_{i-1},t_i)-R_K),0]$
$=\max[L-L(1+\tau(t_{i-1},t_i)R_K)P(t_{i-1},t_i),0]$

Thus under the assumptions of Black's model we have the pricing formulas for caplets and floorlets:

$Caplet_i(t)=L\tau(t_{i-1},t_i)P(t,t_i)[F_t(t)\cdot N(d_1)-R_k\cdot N(d_2)]$ $Floorlet_i(t)=L\tau(t_{i-1},t_i)P(t,t_i)[R_K\cdot N(-d_2)-F_i(t)\cdot N(-d_1)]$

In the formulas above we use different volatilities for each caplet and there is another approach which is to use the same volatility for one maturity. Volatility in the first approach is called spot volatilities and in the latter one it is called flat volatilities. In real market we usually use flat volatility for quoting.

European swaption is another OTC contract. It gives its holder the right to enter into a swap contract at a certain time in the future called exercise time. We could take a fixed-for-float swap to examine the nature of the contract. Since at the start of the contract the value of the floating leg is equal to the notional principle of the swap, the swaption could be seen as an option on a fixed rate bond with the strike being equal to the bond priciple.

$Payoff(T_{Exe})=\max[R_K-R_{swap}(T_{Exe},T_{Stl},T_{Mat}),0]\cdot A(t,T_{Stl},T_{Mat})\cdot L$

We can easily calculate the swap rate by

$R_{Swap}(t,T_{Stl},T_{Mat})=\frac{P(t,T_{Stl})-P(t,T_{Mat})}{A(t,T_{Stl},T_{Mat})}$

then a pricing formula for the swaption is

$ReceiverSwaption(t)=L\cdot A(t,T_{Stl},T_{Mat})[R_K\cdot N(-d_2)-R_{Swap}(t)\cdot N(-d_1)]$

Sometimes we need adjustments to price options on bond and they are convexity adjustment and timing adjustment. If the bond price is given in terms of the yield, then we could not simply assume that the forward price F(t) is equal to expected bond value and this is when convexity adjustments take place. If payment date is delayed until a later time $T^>T$, then in order to compute expectation of a variable at time T with respect to \)$T^$ forward measure, we need timing adjustments.