The Cash Account and Pricing Zero-Coupon Bonds

In this lesson, we looked at the cash account and pricing zero coupon bonds in the context of the binomial model for the short rate.

 This is our binomial lattice for the short rate:

and remember our up and down moving risk-neutral probabilities, q(u) and q(d) and pricing the non-coupon paying security:

and remember than there is no arbitrage using this formula.


We now move on to the cash-account.  The cash-account is a particular security that in each period earns interest at the short-rate.  B(t) denotes the value at time t and assume that B(0) = 1.
The cash-account is not risk-free since B(t+s) is not know at time t for any s > 1.  It is locally risk free, since B(t+1) is known at time t.

Note: B(t) = (1+ r(0,0)(1+r(1))...(1+r(t-1)), so B(t)/B(t+1) = 1/(1+r(t)).

Risk-neutral pricing for a non-coupon paying security takes the form:

rewriting this equation, we can get the following equation:

This is our risk-neutral pricing condition for a zero coupon bond.


We can do the same risk-neutral pricing for a 'coupon' paying security:

then rewrite this equation to get:

This is our risk-neutral pricing conditions for a coupon paying security.  This is simply a special case of the equation above.  This guarantees that we can price all securities with no arbitrage conditions.



We then did a simple short-rate lattice example.  consider the following lattice, where the short rate, r, which is 6% at time t=0, either grows by a factor of u=1.25 or goes down by a factor of d=0.9 in each period.

We then went on to price a zero coupon bond that matures at time t=4.

Similarly to the original way we priced binomial models, we start at time t=4 and work backwards using the formula 1 period at a time.  We use the following formula:

So for node N(3,3) the formula would be:
Z(3,3) = 1/(1+.1172) * [0.5*(100) + 0.5*(100)] = 89.51.  We can continue using the formula from period t=4 to period t=0 to find the price of the zcb.

Having calculated the zcb at time t=0, we can find the compound interest rate: 77.22(1+s(4))^4 = 100 leads us to s(4) = 6.68% compounded interest rate for borrowing or lending for four periods.

Therefore we can compute all the coupon bond prices for each time period and by using the same formula as above, we can also compute the compound interest rates for each length of periods.  The different rates for the different time periods is known as the term-structure of interest rates.   This is known as the relationships between interest rates and different time periods until maturity.  There will be different term structures at every single node N(i,j).  Therefore, by modeling the short rate, we define a model for the term-structure of the zcb.



There is also an excel spreadsheet for calculating term-structure of interest rates, which can be found here:
https://d396qusza40orc.cloudfront.net/fe1/class_resources/Term_Structure_Lattices.xlsx