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Numeraire: Money market vs zero coupon bond

kdmfe

Joined
6/26/18
Messages
63
Points
18
Below is the extract from Steven Shreve’s book: Stochastic Calculus for Finance II, Continuous Time Models

In the below topic I am trying to understand the difference between the instruments.

Somehow, fortunate or unfortunate, I have written all my proofs with limited understanding of the instruments used as Numeraire, hence this post.


9.2 Numeraire (Pg 376)

Associated with each Numeraire, there shall be a risk-neutral measure. For this purpose, only non-dividend paying assets shall be chosen as Numeraires.

…..( some content not typed here)

In contrast, in our model, a share of the money market account increases in value without paying a dividend.

Q1. Does the money market instrument refer to a Fixed income or equity type of instrument? Fixed income pays coupon and not dividend, as in the case of Equity.

The author lists three type of instruments as possible numeraires:

  • Domestic money-market account
  • Foreign money-market account
  • Zero coupon bond


Q2. How is a Zero coupon bond different from Fixed income related money market account?

The book also states that, the asset we take as numeraire could be one of the primary assets given by equation 9.1.1 on pg 975

Equation 9.1.1 is differential form for stock price.

Q3. How can a fixed income instrument satisfy the above differential?


Any threads, or reference to information in public domain will be very helpful.

Kindly help.

Thank you

PS: I know my posts will irritate the readers, but I am approaching this forum after trying my best to clarify these myself.
 

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Q1: Money market account is an interest-bearing deposit account, it is simply an account such that a deposit at time s produces income M(s,t)=exp(int_{s}^{t} r(u) du) at time t. In the general case one allows r(u) to evolve according to some stochastic process, and in the simple case r(u) is constant.

Q2: Normalized zero-coupon bond pays $1 at its maturity. Let P(t,T) denote the price of a ZC bond at time t that matures at time T. By a basic no-arbitrage argument, P(t,T) is related to the money market account via P(t, T) = E_t^Q [1/M(t,T)].

Q3: It is not that a fixed income instrument satisfies this SDE, it is simply that you could take S_i(t) as your numeraire instead of M(0,t). Changing your numeraire simply results in a different pricing equation than the one I gave in Q2 above.

I think you really ought to slow down and take a more methodical approach. One doesn’t learn stochastic calculus well by taking a highly nonlinear approach such as the one it seems you are taking. Sit down with Kuo and Shreve 2, focus on the first 6 chapters of Kuo and the first 5 chapters of Shreve—do not get entranced by the more advanced material.
 
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