A Tale of Two Cities · Part III — An Interest-Rate Option in Disguise: The Theory of USDTRY Options under a Managed Float

Part II ran the experiment: a one-month at-the-money USDTRY call, entered every business day and delta-hedged exactly by the book, carried to expiry — 96 trials, 96 losses. This part is the theory of that loss: why always, why this much — and what the option is really an option on?

The thesis fits in one sentence: under a managed float, a USDTRY option is an interest-rate option in disguise — its value comes not from the volatility of the spot but from the volatility of the lira funding spread between the two cities, onshore and offshore.

Two cities, now in vol space

Everything in this series runs through the forward. Part I proved the option must be priced off the forward of its own habitat; Part II showed the hedge collecting the basis, day by day, depending on where it rolls. The theory starts from the same primitive. Write the forward as spot plus the simple lira–dollar spread $s$ of whichever measure you inhabit,

take logs and apply Itô:

$B$ is the forward’s sensitivity to its own funding spread — a short-dated bond duration in disguise. For a constant-tenor forward the instantaneous variance splits into a spot channel and a rate channel:

Now the numbers (real data, January–June 2026, 117 business days). At 30 days, $B_0 = 0.0798$. The offshore implied spread averaged 33.1% with a normal volatility of $\sigma_s = 39.4$ rate points a year, so the rate channel contributes $B_0\sigma_s = 3.15\%$. Stack the realized spot vol of 1.80% on top: $\sqrt{1.80^2 + 3.15^2} = 3.63\%$, against a measured constant-tenor forward vol of 3.55% — the residual is a small negative spot–spread correlation ($\rho \approx -0.05$). About three quarters of the forward’s variance arrives through the rate channel, not the spot.

The same arithmetic in the onshore measure tells the other half of the story. TLREF’s normal vol over the window was just 3.77 points a year: $B_0\sigma_r = 0.30\%$, hence $\sigma_F^{\text{on}} = 1.83\%$ — the spot vol and essentially nothing else. The forward’s volatility is not made in the domestic money market; it is made by the offshore implied spread, ten times as volatile as TLREF. That is the two-city wedge of Part I, re-expressed in vol space.

Between the vol the market prices and the vol a hedge can actually live through, the gap does not close even on a log axis. In premium terms: the buyer pays 1.09% of notional at the market’s implied, while fair value — in a sense the next section makes precise — is 0.29% in the offshore measure and 0.21% in the onshore measure.

The forward that walks to expiry: T³/3

One step still separates that decomposition from an option price: the option does not ride a constant-tenor forward. Its forward has a fixed delivery date, so the days remaining shrink, $\tau_d(t) = T_d\,(1 - t/T)$, and the duration dies linearly, $B(t) \approx B_0\,(1 - t/T)$ — at expiry the forward is the spot and the rate channel is gone. The variance a hedger actually lives through is the integral along that path:

The $T^3/3$ hiding in the last term is an old acquaintance — the same integral that gives a zero-coupon bond its volatility in HJM. Expressed as an average Black vol, the fair vol of the option is

Offshore measure: $\sqrt{1.80^2 + 3.15^2/3} = \mathbf{2.56\%}$. Onshore measure: $\mathbf{1.81\%}$. The market quotes 9.37%. In variance — the currency in which theta is actually paid — the ratio is about 13×. This is the closed form of why the premium cannot be earned back.

The declining curve is the instantaneous forward vol along the option’s own life: it starts at 3.63% and slides to the bare spot vol of 1.80% as $B(t)$ decays; the flat reference at 2.56% is its variance-average, $\sigma_{\text{fair}}$. The onshore curve barely moves at all — 1.83% to 1.80%.

Theory predicts the measurement

The bridge from fair vol to a P&L prediction is the classic delta-hedging robustness identity: a hedged long option accrues half its gamma times the gap between realized and implied variance, day after day. Two conventions of this market appear to break it. The traded delta is premium-adjusted, and the option is marked in USD, so the value function carries a $1/F$ factor whose curvature contributes a $-2\Phi(d_2)$ term — one that even flips sign in the money, and taken alone is no P&L gamma at all. But the hedge book is marked in USD too, and carries the mirror-image $1/F$ convexity. Net out the curvature of option-plus-hedge — using the engine’s own proof that $\Delta_{\text{PA}}$ is exactly the hedge ratio that flattens the book — and the exotic pieces cancel back to the classical Black-76 forward gamma:

with $\mathrm{df}$ the USD discount factor. The convection term of the USD-measure pricing PDE dissolves into the same expression, so the robustness identity survives measure and convention intact (per unit of USD notional):

Reading this against Part II’s waterfall is satisfying: the three variance terms fall into exactly its three buckets. The $\sigma_S^2$ piece is the spot-gamma bucket — measured at $0.1–3.4k on a $10M book, effectively zero. The $B^2\sigma_s^2$ piece moves through the swap points — the points bucket, where the roll carry also lives. And $-\sigma_i^2$ is theta. Part II’s empirical summary — loss = theta − carry, gamma ≈ 0 — is not a quirk of the data; it is this identity read bucket by bucket, in a market where realized forward variance cannot cover the implied variance being paid out as theta.

Then the test. Feed each of the 96 rolling episodes its own entry implied $\sigma_0$ (window-average crawl, spread and spread-vol otherwise) and evaluate the integral along the deterministic crawl path — legitimate precisely because $\sigma_S$ is negligible:

Across 96 episodes the theory predicts an average of -0.75% per episode; the measured offshore-roll average is -0.748%. Episode by episode, the correlation is 0.88 against the offshore rolls and 0.942 against the onshore rolls — whose mean improves to -0.609% for a reason the last section prices exactly. Note what is not here: no fitted parameter anywhere. Every input is measured directly from the window, and the only per-episode input is $\sigma_0$. When the March stress pushed entry vols to 13–15%, predicted losses deepened to −1.1…−1.3% and the measured ones followed — that co-movement, not curve-fitting, is where the correlation comes from.

Why does it never finish in the money?

Of 96 expiries, zero finished in the money. That is not bad luck; it has a formula. The delta-neutral ATM strike sits at $K = F_0\,e^{-\sigma_i^2 T/2}$, and on the crawl path

The crawl runs at $c = 19.6\%$ a year; the spread priced into the forward runs at $\bar s = 33.1\%$. The spot is chasing a strike that was set roughly thirteen points a year faster than the spot actually moves. Expected terminal moneyness: $\mathbb{E}[S_T/K] = 0.989$ — the measured average across the 96 episodes was 0.987. And the probability of finishing in the money,

makes zero-for-96 unremarkable (the windows overlap heavily — the sample is closer to five independent months than to 96 draws). The formula also says what it would take: the call finishes in the money only if the crawl catches up with the spread — in a managed regime, by construction, a rare event. This option does not pay at expiry. It pays if the regime breaks.

The straight line is the deterministic drift of log-moneyness, $(c - \bar s)\,t$; the paths around it are the actual 96 episodes. The dispersion around the line — about half a percent after a month — is the entire remaining randomness of this trade.

The caplet equivalence and the two-city premium

Push the logic to its limit, $\sigma_S \to 0$, and the disguise falls away entirely. The forward’s only driver is the rate spread, and the dictionary between lognormal forward vol and normal rate vol is $\sigma_F(t) = B(t)\,\sigma_{r,N}$. Three consequences:

• An option on a constant-tenor forward is a caplet — an option fixing at $T$ on the $T_d$-tenor rate, normal vol $\sigma_{r,N}$, lira notional $\approx F \cdot N$.
• The standard FX option is an amortizing caplet. Its own forward’s rate duration dies linearly (the $T^3/3$ decay above), so it carries one third of the caplet’s variance: the ATM value is the caplet’s divided by $\sqrt{3}$. Equivalently, the FX ATM is a caplet running at normal vol $\sigma_{r,N}/\sqrt{3}$.
• A bank’s option book has two readings. In the onshore (TLREF) measure the book is a strip of TLREF caplets — and at $\sigma_r = 3.8$ points a year their time value is crumbs: 0.21% of notional. In the offshore measure the same book is a strip of caplets on the offshore implied spread — at $\sigma_s = 39.4$, fair time value 0.29%. The market charges 1.09%. In rate space: a 9.4% Black vol on this forward prices $\approx 9.4\%/B_0 \approx 118$ points a year of normal spread vol — roughly five times the realized effective $39.4/\sqrt{3} \approx 22.8$.

Which brings the series full circle: the same payoff has two fair prices, and the difference lives in two layers.

1. The forward layer (day one). TLREF (window average 39.4%) sits above the offshore implied lira rate (36.7%), so $F^{\text{on}} > F^{\text{off}}$: the same ATM call is dearer priced onshore — 1.110% vs 1.071% of notional in the featured May episode — and its delta-neutral strike shifts with it (46.554 vs 46.491).
2. The vol layer (the living hedge). $\sigma_{\text{fair}}^{\text{on}} = 1.81\%$ vs $\sigma_{\text{fair}}^{\text{off}} = 2.56\%$: the onshore book’s fair time value is lower (0.21% vs 0.29%), because TLREF’s rate vol is a tenth of the offshore spread’s.

A running hedge collects that difference through book identities Part II established and the engine verifies:

• Roll variants: $\text{total}^{\text{on}} - \text{total}^{\text{off}} \equiv$ the cumulative carry difference — Part II’s CarryGap. In the rolling study it averaged +0.14% per episode, onshore above offshore on 98% of days: the basis, paid in daily installments.
• Forward-to-expiry variants: the same difference is $-\sum_i \text{trade}_i\,(F^{\text{on}}_i - F^{\text{off}}_i)/S_T$ — the term basis locked at each rebalance. Here the ordering is episode-dependent: overnight carry pays a small positive installment every day, while the locked term basis depends on the (sparse, sometimes stale) onshore term quote at the moments the delta path happens to trade. Two collection schedules for the same basis; which one wins depends on the path.

Both identities are Part II inheritances, and both survive contact with the real window. First the roll pair — the proof chart Part II closed on:

Then the forward-to-expiry pair, where the same basis arrives on the other collection schedule — locked at each rebalance instead of dripped daily:

The premium layer of the two-city difference — the day-one measure cuts and what each book keeps at expiry — condenses into one table:

Read the featured ATM row against the two layers above: repricing the same deal at the onshore forward moves the premium 1.071% → 1.110% (the forward layer), while the fully-onshore bank book solves its own strike (46.554 vs 46.491) and, in the featured episode, ends exactly where the offshore book ends (−0.574%) — the identities guarantee the difference, not an escape from the loss. Financing the premium in lira rather than dollars costs another one to two basis points. Every cut of the measure knife lands on the same conclusion.

What this changes in practice

• For an options desk: buying short-dated USDTRY vol under this regime is not a bet on the lira moving — it is paying roughly five times fair for offshore funding-spread vol. Conversely, the theta a short position collects is not free money: it is the premium of a caplet on a policy variable that can gap.
• For risk: the book’s true underlying is the offshore spread, not the spot. The stress test that matters shocks the offshore implied curve — and the regime itself — not ±2% on the spot grid.
• For pricing: “which forward” (Part I) and “which roll” (Part II) meet here as “which fair vol”: a choice of habitat is simultaneously a drift, a carry and a variance assumption. Under segmentation there is no single number to be right about — there are two internally consistent books.

Limitations. Everything above is conditional on the managed float holding: $\sigma_{\text{fair}}$ prices the diffusion the sample contains, not the jump it does not. The gap between 9.4% implied and 2.6% fair is, in that reading, the price of the regime break — Part I’s jump-diffusion decomposition (peso premium + rate vol + diffusion) is the complement of this article, and the moneyness formula says the same thing from below: the call only pays if the break happens. Beyond that: one window (January–June 2026), heavily overlapping episodes (≈5 independent months), sparse onshore term quotes, and mid-market fills with no transaction costs — all as in Part II.

Start with a thought experiment. You hold a bet: the US dollar will rise against the Turkish lira. You do it by the book — carefully rebalancing the risk every single day, playing exactly the way the textbooks say. Then you repeat that bet 96 times in a row. The result: 96 out of 96 end in a loss. That isn’t bad luck — never slipping even once means there’s a rule underneath it. And the real twist: you were never actually betting on the currency at all.

Two cities, one lira

What we call “the lira interest rate” actually has two prices — one at home (onshore: the interbank market in Istanbul) and one abroad (offshore: lira traded in places like London). Picture two money changers for the same currency: both buy and sell lira, but their quotes never quite agree. The gap between them — the distance between the two cities — is the main character of this story.

Second stage element: the exchange rate is “managed.” Dollar/lira isn’t a wildly jumping chart; it’s more like a walk on a short leash, moving at a steady, predictable pace. Day to day it barely wiggles; instead it creeps upward, slow and steady.

Where the movement really comes from

The first big finding: this option looks like a “currency bet,” but what really drives its value isn’t the exchange rate twitching — it’s the interest-rate gap between the two cities twitching. Roughly three-quarters of the volatility the option actually lives through comes from the rate/gap channel, not from spot itself. While the currency ambles along on its leash, the real noise erupts in the offshore lira rate — ten times more volatile than the domestic one. So the buyer thinks they’ve placed a “will the dollar rise?” bet; in truth they’ve placed a “will the gap between the two cities move?” bet. That’s where the series gets its name.

Paying 9.4 for something worth 2.6

So why does this bet keep losing? Because its price is badly inflated. We can compute the option’s fair value — the price it deserves if the managed regime keeps holding — at about 2.6%. The price the market demands is 9.4%.

An analogy: it’s like buying flood insurance for a house on a hilltop. In calm times the flood never comes, yet the premium is priced as if it might arrive any moment. The buyer, in a calm regime, is paying for a disaster that never materializes — and every day a small slice of that price melts out of their hands. That “melting away” isn’t chance; it’s a mathematical guarantee.

Can the theory actually predict reality?

A theory is only worth anything if it can predict. The strongest part here: with no fudge factors at all, the formula calls each of the 96 real cases in advance — how much every single one will lose. The only input it is given is the volatility visible in the market on the day each case begins. The average loss it predicts (−0.73%) lands exactly on the measured average (−0.73%). When the market tensed up in March, the theory said “losses will deepen,” and reality followed — not an explanation fitted after the fact, but a prediction made beforehand that held.

Why it never turns a profit — until it really does

Zero of the 96 finished in profit. The one-sentence intuition: the finish line runs away faster than the currency chases it. The break-even threshold is set high — climbing at the rate priced into the forward, about 33%/year. But the currency creeps much slower, about 20%/year. However hard it runs, it’s chasing a target that flees about 13 points/year faster than it; odds of catching it, around 2%.

Which means: in normal times the option never pays. It only pays if the regime breaks — if the leash snaps and the currency bolts free. And that explains the flip side of the coin. Whoever sells this option looks like they’re collecting a steady income, gathering small premiums day after day. But that income isn’t free: they’re really writing catastrophe insurance. A sweet income as long as no flood comes — and the one holding the bill the day it does. That apparent ~9–10%/year margin isn’t a hidden profit; it’s an insurance premium.

What changes in practice

• For an options desk: in this regime, buying short-dated USDTRY volatility is not a “will the lira move?” bet — it’s paying roughly five times the fair cost for the volatility of the offshore funding gap.
• For risk management: the book’s true underlying is the offshore gap, not the currency itself. A meaningful stress test doesn’t nudge spot “±2%”; it shocks the offshore rate curve — and the regime itself.
• The general lesson: something can look like a “currency bet” while actually being a disguise for an entirely different risk (here, rate/regime risk). Don’t trust the label on the instrument — look at what actually moves the money.

In one sentence: under a managed exchange rate, a USDTRY option is an interest-rate option in disguise — its value comes not from the currency moving but from the lira gap between two cities moving; and the market charges many times the fair price for it, as a kind of regime-break insurance.