Future vs Synthetic Forward

What’s the best strategy to minimize market impact?

Let’s say we want to go long Q units of asset S immediately, on a forward basis (at some maturity). We have the choice between buying futures with a market order (since we want to execute immediately, we’ll rule out working limit orders ourselves) and buying a synthetic forward (long call / short put) from an option market-maker (OMM for short). Which of the two is cheaper? Or, to put it more precisely, what is the fair value of the spread we should pay on such an option trade? What we will show is that it’s always cheaper to buy a synthetic forward (provided it’s priced at fair value).

For simplicity, let’s assume there’s enough liquidity on the best bid/offer for our market order, and that the half-spread from mid price s_0 of asset S is δ_0 > 0. We’ll also assume the OMM’s inventory in futures (which they use as a delta hedge for their clients’ option orders), denoted by q_t, follows an Ornstein-Uhlenbeck process

\((1)\ \ \ \ \ \ \begin{equation*} \begin{split} dq_t &= -\theta q_t dt + \sigma dW_t\\ \\ q_t &\sim {\cal N}\left(0,\frac{\sigma^2}{2\theta}\right)\ \ \ \ \hbox{asymptotically}, \end{split} \end{equation*}\)

(where σ > 0 is the volatility, and θ > 0 is the speed of mean-reversion, related to the risk aversion of the OMM) which encodes the common practice of having an internal “delta-one” desk that internally posts to option traders a best bid/offer inside the market’s, and then works out the resulting inventory out to the market (thus avoiding crossing a spread each time a client order in options needs to be hedged). We make the further assumption that this inventory is worked out in the market using limit orders, unless its absolute value reaches

\((2)\ \ \ \ \ \ \bar{Q} = \frac{\sigma}{\sqrt{2\theta}},\)

after accepting (the hedge of) a client order (and in that situation only), in which case the inventory in excess of that threshold will be offloaded on the market with a market order (thus incurring a unit cost of δ_0, see above), for risk-management purposes. For simplicity, if that happens, we’ll assume the OMM charges δ_0 for the whole order (not just on Q - Q bar). Finally, for order book modeling, we’ll follow our previous post. In particular, the arrival of orders will follow the power law

\((3)\ \ \ \ \ A\frac{1}{(\delta + \delta_0)^k} \ \ \ (A>0, \ k>1).\)

where δ > 0 is the distance to the mid.

WIth this in mind, there are two cases. With probability p

\((4)\ \ \ \ \ \ \begin{equation*} \begin{split} p &= P\left(q - Q > \bar{Q}\right) + P\left(q - Q < -\bar{Q}\right)\\ \\ &= P\left(q > Q + \bar{Q}\right) + P\left( q < Q - \bar{Q} \right)\\ \\ &= N\left(\frac{\sqrt{2\theta}Q}{\sigma} - 1\right) + N\left(-\frac{\sqrt{2\theta}Q}{\sigma} - 1\right) \end{split} \end{equation*}\)

(with N the normal cdf) the unit cost of the hedge will be δ_0 (see above), and with probability (1-p), the unit cost of the hedge will be the same as getting out of a position -Q with limit buy order (examined in our previous post). Putting everything together, we find that the fair unit cost (hence the fair half-spread that the OMM should charge for a synthetic forward) is given by

\((5)\ \ \ \ \ \ \begin{equation*} \begin{split} &\boxed{p\delta_0 + (1-p)\left( \delta_0 - \phi(Q,\tau)\right) < \delta_0}\\ \\ &\hbox{where}\\ \\ &\phi(q,\tau) = \left(Aq_0\right)^{1/k}\tau^{1/k}q^{-1/k}\ \hbox{if}\ \ \frac{q}{\tau} < A q_0\delta_0^{-k}, A q_0\delta_0^{1-k}\tau/q\ \hbox{otherwise,} \end{split} \end{equation*} \)

where τ > 0 is the time remaining till the end of the trading session and q_0 the limit order size. As announced in the beginning, we find that the fair cost (half-spread) of buying a synthetic forward is therefore always less than the prevailing half-spread for the underlying future. However, as our order gets larger, φ(Q,τ) —> 0, and the unit costs will be δ_0 in the limit, meaning the improvement the OMM can provide by selling a synthetic forward over buying the future outright will become negligible for very large orders (as can be expected).

Quantitatively Yours,

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