Transaction costs and no arbitrage price intervals

Definition At each date $t$ the stock can be bought at the ask price $S^{a}_t$ and sold at the bid price $S^{b}_t$, where $0 < S^{b}_t \le S^{a}_t$. The bid-ask spread is $S^{a}_t - S^{b}_t$, and the mid-price is $(S^{a}_t + S^{b}_t)/2$. On this page the cash account $B_t$ has no friction. It starts at $B_0 = 1$ and borrows and lends at the same growth factor $R > 0$ per period, so $B_1 = R$. A position of $\theta > 0$ shares bought and sold again at once loses $\theta(S^{a}_t - S^{b}_t)$. With a spread the cash flow of a trade depends on its direction, so a portfolio no longer has one value that is linear in its holdings, and the self-financing condition must use the cash actually paid or received.

Definition In a model with proportional transaction costs there is a mid-price process $S_t > 0$ and a constant $\lambda$ with $0 \le \lambda < 1$, and

The value $\lambda = 0$ is the frictionless market. Write $x^{+} = \max(x, 0)$ and $x^{-} = \max(-x, 0)$. Changing a stock holding by $\Delta\theta$ shares at date $t$ costs the cash

and liquidating a holding $\theta$ at date $t$, by selling a long holding at the bid or buying back a short holding at the ask, brings in

The trading cost is convex in $\Delta\theta$, the liquidation value is concave in $\theta$, and both are multiplied by $k$ when the position is multiplied by a number $k > 0$. In the one-period model of this page a portfolio $(\Delta, b)$ buys $\Delta$ shares at date 0, a short sale when $\Delta < 0$, and holds $b$ in the cash account, a debt when $b < 0$. Its initial cost is

and at date 1 it brings in $L_{1,j}(\Delta) + bR$ in terminal state $j$, where $L_{1,j}$ is the liquidation value at the prices of that state.

Definition Let a claim pay the cash amount $X_j$ in terminal state $j$. A covering hedge of $X$ is a portfolio $(\Delta, b)$ whose liquidation cash at maturity covers the claim, so that

in every terminal state $j$. A seller of $X$ needs a covering hedge of $X$. A buyer of $X$ who wants to lock in the purchase needs a covering hedge of the negative claim $-X$, which pays $-X_j$, because the claim together with that hedge leaves a nonnegative amount in every state. A covering hedge that holds with equality in every state replicates the claim after costs. Long holdings are valued at the terminal bid and short holdings at the terminal ask, so a hedge computed from mid-prices is in general not a covering hedge.

Definition Superreplication means covering a claim at the lowest possible initial cost. The superreplication price of $X$ is

where the inequality must hold in every terminal state. For a fixed holding $\Delta$ the cheapest cash position is $b = \max_j (X_j - L_{1,j}(\Delta))/R$, so $\overline{V}(X)$ is the infimum over $\Delta$ of

In this one-period model $f$ is convex and piecewise linear, so a point where its slope changes from negative to positive is a minimum. Matching the claim in every state gives a covering hedge, and this slope check shows whether it is the cheapest one. A consistent price system that values the claim at the hedge’s cost, as in the last card, is another certificate of optimality. With $\lambda = 0$ and a claim that can be replicated, the superreplication price in a market without arbitrage is the replication price.

Definition The seller price bound of a claim $X$ is $\overline{V}(X)$, and the buyer price bound is

the largest price a buyer can pay for $X$ and still cover the position. Three facts follow from the definitions. If the claim can be sold at a price $p > \overline{V}(X)$, selling it and buying a covering hedge that costs less than $p$ leaves positive cash today and a nonnegative amount in every state at maturity, an arbitrage. If the claim can be bought at $p < \underline{V}(X)$, buying it and setting up a covering hedge of $-X$ is an arbitrage in the same way. If the stock and cash market has no arbitrage, then $\underline{V}(X) \le \overline{V}(X)$. Otherwise hedges of $X$ and of $-X$ could be found whose total cost is negative, and combining positions never raises the trading cost and never lowers the liquidation cash, so the combined position would cost less than zero and bring in at least $X - X = 0$ in every state, which is an arbitrage. The bounds come from covering arguments only, and prices between them cannot be excluded by those arguments. When $\lambda = 0$, the market has no arbitrage and $X$ can be replicated, both bounds equal the replication price.

Definition A shadow price is a process $\tilde S_t$ with

at every date and in every state, together with a probability measure $\mathbb{Q}$ that gives every state positive probability and makes $\tilde S_t/B_t$ a $\mathbb{Q}$-martingale. The pair is called a consistent price system. In one period with growth factor $R$ the martingale condition reads $\tilde S_0 = \mathbb{E}^{\mathbb{Q}}[\tilde S_1]/R$. If a consistent price system exists, the bid-ask market has no arbitrage. To see this, compare each trade with a frictionless market at the shadow price. Purchases cost at least the shadow price and sales bring in at most the shadow price, so $c(\Delta, b) \ge \Delta \tilde S_0 + b$ and $L_1(\Delta) \le \Delta \tilde S_1$ in every state, hence

A portfolio with $c(\Delta, b) \le 0$ whose liquidation cash is nonnegative in every state therefore has liquidation cash zero in every state, because $\mathbb{Q}$ gives every state positive probability. The same chain shows that every covering hedge of $X$ costs at least $\mathbb{E}^{\mathbb{Q}}[X]/R$. So every claim value $\mathbb{E}^{\mathbb{Q}}[X]/R$ from a consistent price system lies between $\underline{V}(X)$ and $\overline{V}(X)$, and trading the claim at such a value adds no arbitrage. The converse, that no arbitrage produces a consistent price system, is not proved here, and the stronger notion that asks for shadow prices strictly inside the spread belongs to the dynamic theory of transaction costs, which is outside this lesson.