A Tale of Two Cities · Part II — Same Option, Two P&Ls: The Onshore/Offshore Cost of a Delta Hedge

Picture two traders. On the same day, at the same price, they buy the same USDTRY call option. Both follow the textbook recipe to the letter: sell dollars equal to the option’s delta and refresh the position daily — a “delta hedge”. A month later they open their books: their P&Ls differ. The difference is neither luck nor skill; their spot trades, expiries and options are identical. The only difference: one rolled the dollars they sold in the onshore (domestic) swap market, the other offshore.

This article is the story of that difference. Part I established the theorem; Part II runs the same theorem on real intraday data and shows, to the cent, where the difference comes from. (The vega/rho side of the basis is covered in a separate note; this part can be read independently of it.)

What did Part I say?

Briefly: regulatory limits split lira funding into two separate “cities.” In the onshore city, the reference for TL rates is TLREF and the local money market; in the offshore city, the lira available to foreigners is priced at an implied rate extracted from FX swap points. Between them — hundreds to thousands of basis points, depending on the regime — sits a spread: the basis.

Part I said three things: (1) the liquid USDTRY options market trades offshore, so the option’s pricing forward is the offshore forward; (2) mixing the onshore forward with the offshore vol surface is inconsistent and misprices; (3) because premium is paid in dollars, the market delta is “premium-adjusted”. Today we cash those three rules against data.

The three legs of a delta hedge

Anyone long an option and delta-hedging is really running three accounts at once:

1. Harvest gamma: Refreshing the hedge as the rate moves — buy low, sell high — produces small but systematic gains. The harvest grows with volatility.
2. Pay theta: The option’s time value melts every day. This is the rent on the gamma harvest.
3. Turn the roll: The leg textbooks usually dispatch in a single line. Keeping the delta hedge in spot means carrying the short-dollar/long-lira position to the next day with a swap, every day. As long as the lira rate exceeds the dollar rate this roll pays you — but how much it pays depends on which city you turn the roll in.

The first two legs are in every options book. This article is about the third: the same roll, two different prices in two cities.

The experimental setup

The design: on three separate dates (May 4, 18 and 26 — the last one deliberately on the eve of the Eid al-Adha week) we buy one-month USDTRY calls; each lives three separate lives across three strikes (at-the-money, 25-delta, 10-delta) and three hedge styles. The option is repriced every half hour off the offshore forward and the prevailing implied volatility; the hedge is refreshed hourly during Istanbul business hours; the roll accrues each morning at the previous day’s T/N quote. 27 runs in total.

An honesty note: the option itself is identical in every scenario. Same offshore pricing, same delta, same spot trades. The only thing that changes between scenarios is the market the roll is turned in. So the differences you are about to see come, by construction, from the hedge instrument alone.

Figure 1 — Same option, three paths

Three lines are three versions of the same option, in the same spot market, differing only in their roll (featured episode: the ATM call entered May 4). At month’s end the gap between onshore and offshore is 13.8 basis points (0.138% of notional) — not a rounding error, but the basis collected in monthly instalments. The forward-to-expiry variant lands between the two: the same basis, locked in one trade.

Figure 2 — The source of the difference: two prices for one lira

Over the six-month window the onshore and offshore implied TL rates diverged by 2.4 points on average — onshore on top on 90 of 117 business days, with the end-of-day gap swinging between −9 and +13 points on the noise of sparse onshore quotes. The trader turning the short-dollar position onshore collected the lira’s “expensive” rate; the one turning it offshore got the “discounted” rate for the same lira. This is the raw material of the scissors in Figure 1.

Figure 3 — The daily ledger: gamma, theta and the third leg

The textbook promises a duel between green (gamma) and red (theta); under a managed float that duel never starts. In the featured episode the gamma harvest failed to beat carry on even a single day: daily average |gamma| ≈ $0.2k, roll carry ≈$5.8k, theta ≈ −$5.2k. Even on the most volatile days (the May 21–22 vol spike, the May 26 eve-of-Eid roll) the bars owe their height to carry. The ledger’s real duel is between theta and the third leg — when spot stands still, the living side of the option is the rates leg (exactly the subject of the next part).

Figure 4 — What happens as the strike moves?

As the portfolio moves away from the money (ATM → 25Δ → 10Δ), delta and premium shrink; gamma and theta fade. Because the hedge-instrument difference is proportional to delta, it shrinks with them: the on−off gap is 13.8bp at ATM (12.9% of premium — and a quarter of the total loss), 5.5bp at 25Δ (8.1%), 1.6bp at 10Δ (4.3%). The habitat bill is largest at the money; toward the wings the difference melts away with the option itself. In this window the currency went nowhere, so the OTM deltas melted fast — a measured demonstration that the habitat difference lives and dies with delta.

Figure 5 — The proof: the difference is the roll, to the cent

Two lines, one on top of the other: the solid line is the total P&L difference between the onshore and offshore hedges; the dashed line is the cumulative difference of the roll carries alone. They coincide, because they must: in the two scenarios the option and the spot trades are identical — the only changing item is the roll. This is Part I’s “the basis is a price of its own” thesis, rendered as a ledger entry.

What this means in practice

• For an options desk: “delta-hedge cost” is not a single number; the habitat choice is itself a position. A hedger with onshore access carrying an offshore-priced option carries the basis — knowingly or not.
• For a corporate treasury: the realized cost of the same option-based hedging programme depends on which city your internal funding line is tied to. The question to ask when soliciting quotes: “where will you turn the delta?”
• For the reader: the two cities’ rate scissors is not an abstract indicator; it is a cash flow measurable within a month, traceable to the cent.

Limitations. Quotes are intermittent even at the best of times: onshore points are live only during Istanbul hours and go stale in holiday weeks (flagged step by step in the report). There is no intraday smile data; 25Δ/10Δ vols are carried from daily RR/BF quotes. Headline runs are free of transaction costs; a spread sensitivity run is reported separately. Options are struck at model mid; premium financing is not in the headline numbers. A one-month window is a sample — the basis need not be this generous every month, and regimes where it flips sign have been seen (see Part I, 2020 vs 2026).

Method appendix — conventions and engine

• Pricing: Black-76 on the offshore forward; USD-CSA value V_USD = df_USD · c_fwd / F (consistent with Part I, Prop. 1). No rate curve is built; the forward comes directly from spot + points.
• Delta: premium-adjusted forward delta (market convention); the engine also supports plain delta. ATM = delta-neutral straddle.
• Strikes: 25Δ/10Δ by numerical inversion from the inception smile (ATM + RR/BF, Malz-like 5-pillar).
• Vol: total-variance interpolation between the 1W/1M ATM pillars; flat below 1W. Smile dynamics: sticky-delta (daily RR/BF).
• Roll: every Istanbul business day at the previous end-of-day T/N quote; failing that, 1W/1M pro-rata; onshore last resort is the TLREF proxy (flagged). Weekends and holidays are inherent in the day count the quote covers (e.g. the eve-of-Eid roll carries 6 days at once).
• P&L decomposition: not a Greek approximation but a waterfall (successive full repricings: spot → points → vol → time); the items close to total MTM exactly.
• Calendar: T+1 settlement, joint TR+US holiday calendar, ACT/365.
• The engine is pure TypeScript; scenarios and invariant tests live in a separate simulation repository (the copy on this site: src/lib/hedge-sim/). Chart data is produced from that engine’s real-data run.

---

Data are derived from Bloomberg Terminal; raw series are not published. Chart series are derived quantities reduced to daily resolution. This article is not investment advice.