## What price Brexit?

On next Thursday it looks like I may well have to abandon at least one somewhat-cherished belief: (1) democracy basically works; (2) leaving the EU will be terrible, economically, socially and culturally, for the UK; (3) my own beliefs on this matter have no gaping inconsistencies. Still, at least the campaign will be over. I thought nothing could be more painful than the pro-Brexit leaflet I was sent last week, including a truly-beyond-parody map of Europe, suitably enlarged to include Iraq and Syria, coloured in a threatening shade of blood, until I saw Farage’s latest poster (no link).

At the time of writing (Saturday 18 June), the FT poll-of-polls has Remain on 43% and leave on 48%, with the remainder undecided. The best odds available from conventional bookmakers are 1:2 for remain and 7:4 for leave, corresponding to 66.7% and 36.4% (with a 3% profit margin). The Betfair prices are 1.51 for remain and 2.94 for leave, corresponding to 66.2% and 34.0%.

Here is a highly inexpert attempt to work out the implied odds from option prices. The GBP/USD exchange rate closed on Friday at 1.4358, having bounced back slightly from a low of 1.4114 on Tuesday. According to the minutes of the MPC meeting on Thursday, if the UK votes to leave the EU then ‘Sterling is also likely to depreciate further, perhaps sharply’. Maybe there is a code, known to experts, that translates phrases such as ‘depreciate further, perhaps sharply’, ‘drop significantly’, ’cause the end of civilisation as we know it’ into approximate percentages. For this post, I’ll assume that a leave vote means the rate is $1.44(1-e)$, and a remain vote means the rate is at least $1.44$, at any time when this matters. Let $p$ be the probability of a vote to leave. From now on, all prices are in dollars.

The most obvious way to hedge Brexit is to buy a put option with strike price 1.44, expiring after the referendum. According to these prices, a put option with strike 1.44 expiring in July costs 0.0536. So for about 0.05, I (or rather, an investment bank) can purchase the right to sell 1 GBP for 1.44 at some date (I’m not sure when) in July. This option is worthless after a vote to remain. By assumption, it pays off

$1.44 - 1.44(1-e) = 1.44e$

after a vote to leave. No-arbitrage implies that $1.44 ep = 0.0536$, giving $p = 0.0372/e$. Taking $e = 0.1$, corresponding to a post-Brexit rate of 1.296, gives $p = 0.372$.

A slightly more sophisticated hedge also sells a put option at a lower price. By assumption, the rate does not go below $1.44(1-e)$, so selling a put option at this price gives some free extra cash. (Of course this contradicts no-arbitrage, but never mind.) Assuming that $e \le 0.1$, the pay-off is maximised by selling a put option with strike 1.295, netting 0.0069. No-arbitrage now implies that

$1.44 ep = 0.0536-0.0069 = 0.0467,$

giving $p = 0.0324/e$. The new hedging strategy makes Brexit more profitable, so the implied odds go down: if $e =0.1$ then $p=0.324$.

Going the other way, if any of this is to be believed, the betting markets predict a 10% drop in the pound on Brexit, and the opinion polls predict a 6% drop ($0.061425 = 0.0324 \times (48 + 43)/48$).

Post Brexit update, 9th July. At least one somewhat-cherished belief has duly been abandoned. The pound/dollar rate is now 1.2953, almost exactly 10% down.