title: Unified Mathematical Finance author: Keith A. Lewis ...

Mathematical Finance provides a rigourous model for quanifying the problems of how to move cash flows through time, how much it costs, and how risky that is. The classical Black-Scholes/Merton approach assumes optimal hedging strategies executed in "continuous" time. A model more faithful to trading reality should allow for non-optimal strategies and the fact that continuous time hedging is not possible.

A date, or datetime, is an absolute time. Universal Coordinated Time, or UTC, can be used for this. The permuted acronym has an interesting history, as do all matters involved with measuring dates and times.

The difference of two dates is a time period. An epoch date can be used to convert between dates and times. The epoch corresponds to time 0 and the date is a time period offset from that. It is common to work with time in years, although years is not a proper time period since it may involve leap days. Likewise, using seconds has a similar problem because of leap seconds.

Instruments are contracts that provide ownership of a financial asset to the holder and a financial liability of the instrument issuer.

The most fundamental instrument is cash in a particular currency. Stocks provide equity in the issuing company. Bonds are a contractual right to coupons and final principal repayment by the issuer. A commodity is not an instrument, but a forward rate agreement to exchange cash for the commodity at the termination date of the contract is. These instruments are backed by tangible assets. Convertible bonds, asset backed securities, and structured products are hybrid intruments. Classical derivitive instruments are options, forwards, and futures.

A derivative contract specifies the cash flows between the buyer and the seller as a function of other instruments. If a derivative is not traded in the market then there is a mathematical theory originally developed by Black, Scholes, and Merton to approximate its value. A sell-side trader can use that, plus vigorish, to quote prices.

There is a credit component to every intrument. Either party might default on their contractual obligations and only provide partial repayment. This can be modeled using default time and recovery.

The atoms of finance are holdings, an amount of some instrument and who owns it. Holdings interact via transactions, a swap of holdings between a buyer and a seller on some date.

A transaction is a buyer and a seller exchanging holding at a transacton time. The transaction $\chi = (t;a, i, o; a', i', o')$ indicates the buyer swapped the holding $(a, i, o)$ for the holding $(a', i', o')$ with the seller at time $t$. At settlement $u > t$ the buyer holds $(a',i',o)$ and the seller holds $(a,i,o')$. The price of the transaction is $X = a/a'$ so $\chi = (t;Xa', i, o; a', i', o')$. The price depends on the time executed, the buyer instrument, and the amount of the seller instrument. Prices are determined by the seller. The buyer decides when to exchange holdings based on the seller’s price, among other considerations.

Owning an instrument entails cash flows. Stocks pay dividends or incur borrow costs when shorted. Bonds pay coupons. Currencies do not have cash flows. Commodities may have storage costs. Futures cash flows are the daily margin adjustents. The price of a futures is always zero. Cash flows are added to holdings in proportion to the amount of the instrument.

Cash flows are transactions between the buyer and the issuer that are triggered by contractual obligations instead of trading events. If issuer $o'$ pays cash flow $C$ in units of instrument $i$ at time $t$ then all holders of $(a, i', o)$ are subject to a transaction $(t;0, i, o; Ca, i, o')$. After payment the buyer holds ${(a,i',o),(Ca,i,o)}$ and the issuer holds $(0,i,o')$.

Given a portfolio of holdings ${(a_j, i_j, o_j)}$ define the net amount of asset $i$ held by entity $o$ to be $$ N(i, o) = \sum {a_j\mid i_j = i, o_j = o}. $$ Transactions and cash flows change a set of holding over time. If transaction $\chi = (t;a, i, o; a', i', o')$ settles at $u > t$ then the set of holdings at $u$ now includes ${(-a, i, o), (a', i', o), (-a', i', o'), (a, i, o')}$. The holdings with negative signs are often netted with existing buyer and seller holdings. This assumes the holdings $(a_0, i, o)$ and $(a_1, i, o)$ can be combined to the holding $(a_0 + a_1, i, o)$. This is close to being true when $a_0 > 0$ and $a_1 > -a_0$, but that, and every, assumption should be made explicit.

To mark-to-market a portfolio fix an instrument $i_0$ and posit a set of "prices" $X(i_0,i)$ for converting each instrument $i$ to $i_0$. We assume $X(i_0,i_0) = 1$. The mark-to-market value in terms of $i_0$ for entity $o$ is $$ M(i_0, o) = \sum_{i} X(i_0, i) N(i,o). $$ This is the net value after converting each holding $(a,i,o)$ to $X(i_0,i)a,i_0,o)$ at price $X(i_0,i)$, assuming that is possible. We use "price" since there may be no market price available.

The profit-and-loss (P&L) per instrument and holder over the period $[t, u]$ is $N_u(i,o) - N_t(i,o)$. This can also be marked-to-market using prices specified at the beginning and end of the period to get the P&L $M_u(i_0,o) - M_t(i_0,o)$ in terms of $i_0$.