How a Discounted Money Market Position Grows to Its Maturity Value
Summary
The document clarifies the maturity value of a money market holding that is worth K multiplied by a discount factor at time t. Because the amount is invested in an account earning the risk-free rate, its value grows from time t to T by the corresponding accumulation factor. The discount and growth factors cancel, leaving K at maturity.
This timing distinction is used in a law of one price argument involving a stock and a portfolio of a discounted cash amount, a call, and a short put. At maturity, the cash grows to K, while the option payoffs combine so the portfolio has the same value as the share. The post supplies the simple factor cancellation, but does not derive the option payoff identity in detail or discuss complications such as interest-rate variation, fees, or differing borrowing and lending rates.
Key ideas
- A discounted cash amount earns interest as time advances toward maturity.
- The risk-free accumulation factor cancels the discount factor over the same period.
- The cash component therefore has value K at time T.
- The example uses this maturity value in a replication argument involving a share and options.
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# Value at maturity of long position in money market
# Value at maturity of long position in money market
This should be easy, but for some reason I am struggling with it.
Say you have a long position in the money market (you hold dollars), say you own a quantity of $Ke^{-r(T-t)}$ dollars at time $t$, where $K$ is some positive real number, $T$ is some future time $>t$ and $r>0$ is the risk-free interest rate earned by the money market account.
The question is: what is the value at time $T$ of that portfolio consisting of just some dollars?
I would think it is however much you own times $e^{-r(T-t)}$, which produces $Ke^{-2r(T-t)}$ but in my lecture slides it is actually $K$. As if the values at $T$ was computed by multiplying by $e^{r(T-t)}$. But this doesn't make sense to me since $e^{-r(T-t)}<1$.
Am I wrong or are the slides wrong?
For context, this was in a slide about the Law of one price, where portfolios $P1$ and $P2$ are created so that their value at $T$ is the same.
P1: long Ke−r(T−t) euros and one call, short one put. $V_T(P_1)=K−\max(K−S_T,0)+\max(S_T −K,0)=S_T =V_T(P_2)$
P2: long one share, $V_t(P_2) = S_t$
From this we can conclude that $V_t(P_1)=S_t\ \forall t\in[0,T]$
## Answer by Anirban Saha (score 0)
https://quant.stackexchange.com/a/60508
The value at time $t$ will be $K e^{-r(T-t)}$ times $e^{+r(T-t)}$ that is $K e^{-r(T-t)+r(T-t)}=K$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.