Systematic / Quant

A Tale of Two Cities · Part I — Two Prices for One Lira: The Onshore–Offshore Basis

Jun 24, 2026 · ~24 min read FX optionsmulti-curveCIPonshore/offshorevolatility smilejump-diffusionXVA

Research · FX Options · Emerging Markets

A measure-theoretic, multi-curve framework for onshore/offshore rate divergence — and its managed-crawl numerical resolution.

Abstract

In the Turkish banking system, regulatory limits on TRY transactions with foreign counterparties segment the money market into an onshore funding curve (TLREF/OIS) and an offshore funding curve (implied from FX swap points), sustaining a large, policy-driven cross-currency basis. A pervasive desk shortcut is to mix the onshore forward (for the drift) with an offshore-quoted volatility surface. Part I proves this is theoretically inconsistent and severely mispriced: because implied volatility is a quoting convention and the forward is the pricing primitive, the delta-to-strike map is intrinsically forward-dependent, so re-anchoring the smile to a mismatched forward translates it in log-strike space and corrupts the drift. Working in a multi-curve framework we separate projection from discounting, correct a dimensional base-currency discounting error, establish incomplete-market bounds for onshore-constrained hedgers, and derive the exact error decomposition. Part II tests this framework on real intraday data: delta-hedging the same USDTRY call with spot + swap rolls, it shows that rolling onshore vs offshore splits realized P&L by exactly the cumulative basis, to the cent.

Keywords: FX options · multi-curve discounting · covered interest parity · onshore/offshore segmentation · volatility smile · premium-adjusted delta · jump-diffusion · XVA

PART IThe Static Consistency Framework

01 Introduction

FX option pricing requires three primitives: a forward (the risk-neutral drift), a discount factor, and a volatility surface. In frictionless markets covered interest parity (CIP) ties the forward to domestic and foreign rates. In emerging markets with macroprudential capital controls — most notably the Turkish lira — CIP fails: regulatory limits prevent arbitrageurs from closing the gap, leaving persistently distinct onshore and offshore TRY forward curves.

Because the liquid USD/TRY option market operates offshore and is quoted in deltas by international dealers, pricing forces a choice of curves. A heavily used but fundamentally flawed heuristic combines the internal onshore TRY rate (for the drift) with the offshore broker-quoted smile. This paper formalizes the inconsistency of that mixed approach, building on the multi-curve framework and the FX-convention literature.

02 Institutional background

The Turkish regulator (BDDK) places strict dynamic quotas on the volume of TRY-providing swaps and forwards that local banks may transact with foreign entities. This friction physically segments the market into an onshore curve (TLREF/OIS, the domestic money market accessible to local banks) and an offshore curve (the synthetic TRY rate implied from deliverable offshore FX swap points and NDFs).

The wedge between them is the cross-currency basis $b$ . Depending on the regime, offshore lira can spike to extreme premiums (foreigners starved of lira, 2020) or trade at deep discounts (early-2026 curbs). Because cross-border replication is legally restricted, the CIP deviation is not an actionable risk-free arbitrage. Consequently the onshore and offshore forwards must be treated as distinct, non-fungible traded assets.

03 Notation and assumptions

Let domestic currency $d$ be TRY and foreign (base) currency $f$ be USD; $S_{t}$ is the spot in TRY per 1 USD; fix maturity $T$ . Write $P_{f} (t, T), r_{f}$ for the USD discount factor and rate; $P_{d}^{on}, r_{d}^{on}$ and $P_{d}^{off}, r_{d}^{off}$ for the onshore and offshore TRY curves, with basis $r_{d}^{off} = r_{d}^{on} + b$ . The observable forwards are

F^{on} = S_{0} e^{(r_{d}^{on} - r_{f}) T}, F^{off} = S_{0} e^{(r_{d}^{off} - r_{f}) T}, F^{off} = F^{on} e^{b T} .

(A1) Intra-habitat no-arbitrage — the law of one price holds within the onshore habitat and separately within the offshore habitat. (A2) Segmentation — cross-habitat replication is prohibited; a hedger cannot freely borrow at $r_{d}^{on}$ to hedge a trade settling offshore. (A3) Collateralization — options are CSA-collateralized in a currency $c$ (typically USD). (A4) Habitat assignment — every option belongs to a habitat (settlement venue/counterparty) dictating the unique admissible hedging forward.

04 Theoretical framework

4.1 Multi-curve discounting and the dimensional correction

A pervasive misconception holds that under USD collateralization the TRY curve “does not enter at all,” and that the upfront TRY premium is the undiscounted TRY forward value $c_{fwd}$ times $P_{f} (0, T)$ . This is dimensionally invalid.

Proposition 1 — Valuation under third-currency collateralization

Under continuous collateralization in currency $c$ , valuation uses the forward and the collateral discount $P_{c} (0, T)$ . If $c = f$ (USD), the domestic TRY premium explicitly recovers the offshore TRY discount factor $P_{d}^{off} (0, T)$ , rigorously excluding the onshore curve.

Proof. The numéraire is the collateral bank account; let

ℚ^{f, T}

be the USD-collateral

T

-forward measure. For a call paying

(S_{T} - K)^{+}

in TRY,

V_{0}^{d} = S_{0} P_{f} (0, T) 𝔼^{ℚ^{f, T}} [\frac{(S_{T} - K)^{+}}{S_{T}}] = S_{0} P_{f} (0, T) 𝔼^{ℚ^{f, T}} [{(1 - \frac{K}{S_{T}})}^{+}] .

Under

ℚ^{f, T}

1 / S_{t}

is a martingale with mean

1 / F^{off}

(offshore makers trade

F^{off}

). The Margrabe form gives

V_{0}^{d} = S_{0} P_{f} (0, T) c_{fwd} / F^{off}

, and since

S_{0} P_{f} (0, T) / F^{off} \equiv P_{d}^{off} (0, T)

V_{0}^{d} = P_{d}^{off} (0, T) \cdot c_{fwd}

Applying

P_{f}

directly to TRY pips is a dimensional error:

P_{f}

discounts the USD equivalent

c_{fwd} / F^{off}

; converting back to spot TRY forces the offshore TRY curve exactly. ∎

4.2 Forward-dependence of the implied price

Proposition 2 — Pricing invariance

Implied volatility is defined by inverting Black–76 for a specific forward. Holding a quoted $σ$ constant while swapping the forward from $F^{off}$ to $F^{on}$ violates the law of one price.

Proof. With

V (F, σ) = F Φ (d_{1}) - K Φ (d_{2})

\partial V / \partial F |_{σ} = Φ (d_{1}) > 0

: the map is strictly monotonic. A price from

F^{on}

paired with

σ^{off}

evaluates an asset drifting at

r_{d}^{on}

but carrying the variance premium of

r_{d}^{off}

— no replicable instrument in either habitat. ∎

4.3 Delta conventions and re-anchoring the smile

For EM pairs the premium is typically paid in USD, so the quoting convention is the premium-adjusted forward delta.

Proposition 3 — Smile re-anchoring shift

Holding the quoted delta–volatility surface fixed, the implicit delta-to-strike map shifts with the anchoring forward: re-anchoring from $F^{off}$ to $F^{on}$ translates the smile in log-strike by exactly $- b T$ .

Proof. For a call,

Δ_{PA} = \frac{K}{F} Φ (d_{2})

. With moneyness

x = K / F

Δ_{PA} = x Φ (\frac{- \ln x - \frac{1}{2} σ^{2} T}{σ \sqrt{T}}) \equiv g (x; σ),

which is functionally independent of

F

. For OTM calls (

x > 1

d_{2} < 0

g^{'} (x) = Φ (d_{2}) - ϕ (d_{2}) / (σ \sqrt{T}) < 0

(Mill’s-ratio bound, small

σ \sqrt{T}

), giving a unique root

x^{*} (Δ, σ)

invariant to

F

. Hence

K^{on} / F^{on} = K^{off} / F^{off}

, so

K^{on} = K^{off} e^{- b T}

and

\ln K^{on} = \ln K^{off} - b T

. ∎

Figure 1. Proposition 3 in pictures. Identical quote data

(σ_{ATM}, {R R}_{25}, {B F}_{25})

, two forwards: re-anchoring rigidly translates the smile by

- b T

in log-strike. The same strike

K = 70

then reads

24.64 %

off the shifted curve instead of the correct

26.65 %

— and is priced off the wrong forward.

05 Segmented-market bounds and super-replication

Proposition 4 — Super-replication in segmented habitats

If an agent can fund TRY only at $r_{d}^{on}$ , the minimal super-replication price for manufacturing an offshore option is bounded by the onshore-anchored price $V (F^{on})$ .

Proof. By (A2) the onshore hedger cannot access

r_{d}^{off}

; replicating

(S_{T} - K)^{+}

with USD cash, onshore TRY cash and spot forces

S_{T}

to drift at

r_{d}^{on} - r_{f}

under the compatible martingale measure, i.e. pricing forward

F^{on}

. Following Cvitanić–Karatzas (constrained portfolios), the super-replication cost is

c (F^{on})

. If

b < 0

then

V (F^{off}) < V (F^{on})

: selling at the “cheap” offshore price and hedging onshore yields a systematic sub-replication deficit. The offshore price is an incomplete-market equilibrium strictly outside the onshore bounds. ∎

06 The inconsistency, formalized

If a desk consumes the offshore smile $σ^{off} (K)$ but evaluates it at $F^{on}$ , the error $Δ c = c_{mix} - c_{off}$ decomposes analytically.

Proposition 5 — Error decomposition

To leading order in $b T$ , the valuation error splits into a dominant forward-drift term and a smile-slope correction:

Δ c_{fwd} \approx - b T (F^{off} Δ_{fwd} - 𝒱 \cdot 𝒮),

with forward delta $Δ_{fwd} = Φ (d_{1})$ , forward vega $𝒱 = F^{off} \sqrt{T} ϕ (d_{1})$ , and skew $𝒮 = \partial σ / \partial \ln K$ .

Proof. By Prop 3, reading the re-anchored smile at

K

equals reading the original at

K e^{b T}

, so

σ_{re-anch} (K) = σ^{off} (K e^{b T})

. A first-order expansion around

(F^{off}, σ^{off} (K))

with

F^{on} - F^{off} \approx - F^{off} b T

and

σ (K e^{b T}) - σ (K) \approx 𝒮 b T

gives the result. The drift term dominates for non-zero delta, making the inconsistency sign-independent in

b

. ∎

07 Numerical illustration

An illustrative 1Y USD/TRY call, regime “offshore lira cheap” ( $b = - 800$ bps):

S_{0} = 40

r_{f} = 4 %

r_{d}^{on} = 46 % \Rightarrow F^{on} = 60.88

r_{d}^{off} = 38 % \Rightarrow F^{off} = 56.20

surface

σ_{ATM} = 22 %

{R R}_{25} = 6 %

{B F}_{25} = 1.5 %

At $F^{off}$ the $25 Δ$ call ( $σ = 26.5 %$ ) maps to strike $69.60$ ; at $F^{on}$ the rigid map forces $75.39$ (a shift of $e^{- 0.08}$ ).

Pricing $K = 70$ : correct read $σ = 26.65 %$ off $F^{off}$ gives $c_{fwd}^{off} = 56.20 Φ (- 0.691) - 70 Φ (- 0.957) = 1.915$ TRY. The mixed approach ( $F^{on}$ drift, $σ = 24.64 %$ ) gives $c_{fwd}^{mix} = 2.852$ TRY — overpricing the undiscounted option by +48.9%, and mischaracterizing the hedge ( $Δ = 0.329$ vs the true $0.245$ ).

Discounting correction (Prop 1) The naive present value

1.915 e^{- 0.04} \approx 1.840

is dimensionally wrong. The USD upfront is

(1.915 / 56.20) e^{- 0.04} = 0.03274

USD; in spot TRY,

0.03274 \times 40 = 1.310

TRY

= 1.915 e^{- 0.38}

, recovering the offshore TRY discount curve and bypassing the onshore rate entirely.

PART IIThe Generative Dynamics — Resolving (P3)

08 The generative model: stochastic basis & devaluation risk

To explain how a heavily managed currency with near-zero realized spot volatility can simultaneously command $σ_{ATM} = 22 %$ , we specify the spot $S_{t}$ and the offshore short rate $r_{t}^{off}$ as a jump-diffusion. Under the physical measure $ℙ$ , the central bank enforces a managed crawl with suppressed diffusion, punctuated by a latent regime-break (a positive peso jump $J > 0$ ):

\frac{d S_{t}}{S_{t^{-}}} = μ_{crawl} d t + σ_{real} d W_{t}^{S, ℙ} + (e^{J} - 1) d N_{t}^{ℙ} .

The residual stochasticity of the basis enters through a mean-reverting offshore short rate:

d r_{t}^{off} = κ (θ - r_{t}^{off}) d t + σ_{r} d W_{t}^{r} .

Under the USD-collateral measure $ℚ^{f, T}$ , CIP forces the drift of $S_{t}$ to match the offshore differential, decoupling it from the physical crawl. For transparent variance accounting over $T = 1$ , map the Poisson jump to a binomial regime-switch with risk-neutral probability $p$ .

Table 1 — Input parameters and market anchors ( $T = 1$ ).
Parameter	Symbol	Value	Implication
Spot	$S_{0}$	40.00	Base USD/TRY
USD rate	$r_{f}$	4.0%	$P_{f} (0, 1) = e^{- 0.04}$
Onshore TRY	$r_{d}^{on}$	46.0%	$F^{on} = 60.88$
Offshore TRY	$r_{d}^{off}$	38.0%	$F^{off} = 56.20$ ( $b = - 800$ bps)
Implied vol	$σ_{ATM}^{off}$	22.0%	native to $F^{off}$ , $V \approx 0.0484$
Physical crawl	$μ_{crawl}$	25.0%	CB target slope
Realized diffusion	$σ_{real}$	4.0%	managed daily variance
Jump probability	$p$	15.0%	$ℚ$ -prob of peg break
Devaluation gap	$M = e^{J}$	1.628	$+ 62.8 %$ spot jump if break

Calibration check. The $ℚ^{f, T}$ martingale condition $𝔼 [S_{T}] = F^{off}$ requires

(1 - p) S_{0} e^{μ_{crawl}} + p S_{0} e^{μ_{crawl}} M = 0.85 (51.36) + 0.15 (83.61) = 43.66 + 12.54 = 56.20 .

Figure 2. The risk-neutral binomial peso calibration over

T = 1

. The call only finishes in-the-money in the

15 %

break state; the

85 %

managed-crawl state lands deep out-of-the-money. The probability-weighted terminal value recovers

F^{off} = 56.20

exactly — the option premium is almost entirely the price of the upper branch.

09 Physical vs risk-neutral drift wedge

Segmentation and peso risk drive the required risk-neutral forward drift far from the physical spot trajectory.

Table 2 — Hierarchy of depreciation slopes.
Drift metric	Annualized	Interpretation
1 · Realized crawl $μ_{crawl}$	25.0%	physical $ℙ$ -measure reality
2 · Offshore implied $r_{d}^{off} - r_{f}$	34.0%	$ℚ^{f, T}$ drift — actionable offshore
3 · Onshore implied $r_{d}^{on} - r_{f}$	42.0%	restricted super-replication bound (Prop 4)

Carry & carry-to-vol. Selling USD / buying TRY offshore earns the implied $34 %$ but bleeds the $25 %$ crawl — a $9 %$ pickup if the peg holds; against $σ_{ATM} = 22 %$ the offshore carry-to-vol is $0.41$ . An onshore-constrained bank funding at $r_{d}^{on}$ sees an artificial $17 %$ pickup ( $42 - 25$ ) and an inflated carry-to-vol of $0.77$ — a mirage that drives local entities to systematically sell USD calls (bound by Prop 4), requiring regulatory limits to curb unhedged short-gamma accumulation.

Figure 3. The drift hierarchy. The physical crawl (lowest) sits below the offshore risk-neutral drift, which sits below the restricted onshore bound; the dashed line is the devaluation-tail path. The carry trade harvests the gap between the offshore drift and the crawl — until the tail fires.

10 Realized ≪ implied volatility: decomposing the gap

It is deeply counter-intuitive that an asset with $\approx 4 %$ realized volatility carries a $22 %$ implied. We decompose the total $T = 1$ log-forward variance $V_{imp} = (0.22)^{2} = 0.0484$ into three orthogonal sources:

V_{imp} \approx σ_{real}^{2} T + σ_{r}^{2} T + p (1 - p) (\ln M)^{2}

Table — Variance decomposition (target $σ_{ATM} = 22 %$ ).
Source	Contribution	Value	% of $V_{imp}$
(i) Spot diffusion	$σ_{real}^{2} = (0.04)^{2}$	0.0016	3.3%
(ii) Offshore rate vol	$σ_{r}^{2}, σ_{r} \approx 12.85 %$	0.0165	34.2%
(iii) Peso / jump	$0.15 \times 0.85 \times (\ln 1.628)^{2}$	0.0303	62.5%
Total	$\sum$	0.0484	100%

Figure 4. Spot diffusion explains essentially none (

3.3 %

) of the option’s cost. The

22 %

implied vol is almost entirely an insurance premium against a discrete

62.8 %

regime break (

62.5 %

of variance) plus the basis risk of fluctuating offshore funding (

34.2 %

). Realized-to-implied vol ratios are meaningless metrics in jump-dominated EM dirty pegs.

11 Generative skew vs re-anchoring shift

The jump $M = 1.628 > 1$ creates genuine right-tail mass in the risk-neutral density, producing a steeply bid surface for USD calls — the observed ${R R}_{25} = 6 %$ and elevated ${B F}_{25} = 1.5 %$ . This is the generative skew. It must be distinguished from the Proposition 3 re-anchoring shift: mis-anchoring the $F^{off}$ -native smile to $F^{on}$ triggers a rigid log-strike translation by $- b T = + 0.08$ .

The generative skew bends the smile (real physical tail risk). The re-anchoring shift merely misaligns the

x

-axis (a quoting-convention coordinate error) — corrupting the drift without altering the probability density. Conflating the two shatters calendar and butterfly no-arbitrage bounds.

12 Pricing contrast: realized vs implied

We price the 1Y USD call / TRY put ( $K = 70$ ). Part I (§7) established the mixed-framework error; here we add the naive realized-vol price to isolate the jump premium. By Proposition 1, TRY payoffs discount by $P_{d}^{off} = e^{- 0.38} = 0.6839$ .

Table 3 — 1Y USD call ( $K = 70$ ) pricing comparison.
Method	Forward	$σ$	$c_{fwd}$ (TRY)	Correct PV (Prop 1)
(a) Naive realized	$F^{off} = 56.20$	4.00%	≈ 0.000	0.000
(b) Mixed (§7)	$F^{on} = 60.88$	24.64%	2.852	invalid (+48.9%)
(c) Correct offshore	$F^{off} = 56.20$	26.65%	1.915	1.310

Evaluating $K = 70$ at the $4 %$ physical diffusion renders it worthless ( $d_{1} = - 5.47$ SD OTM). The $1.310$ TRY premium exists exclusively to fund the rate and jump components invisible to geometric spot modeling. The mixed approach (b) falsely inflates value by overriding the $38 %$ offshore drift with the non-replicable $46 %$ onshore drift.

13 Dynamic hedging & rho/basis risk — the steamroller

Selling the $K = 70$ call for $1.310$ TRY and spot-delta-hedging does not face standard diffusion risk. Because $F^{off}$ is stochastic, even a spot-delta-neutral book retains heavy offshore-curve exposure (rho / basis). The initial spot delta is $Δ_{S} = Φ (d_{1}) e^{- r_{f} T} = 0.2448 \times e^{- 0.04} = 0.235$ USD (buy $0.235$ USD at $40$ ).

Table 4 — Representative instantaneous P&L, short call, spot-delta-neutral.
State	Event	Option liab.	Spot hedge	Net P&L
(i) Peg holds	crawl 25%, diffuse 4%	+0.003	−0.002	+0.001 · pennies
(ii) Swap shock	$r_{d}^{off} + 500$ bps → 43%	−0.453	0.000	−0.45 · basis loss
(iii) Devaluation	spot jumps $+ 62.8 %$	−14.58	+5.91	−8.67 · steamroller

The spot-delta-neutral book is an illusion. The trader harvests pennies of carry in (i), but is naked to the stochastic basis in (ii) — hedging which requires executing offshore FX swaps, precisely where BDDK limits restrict access — and the linear delta provides minimal protection against the catastrophic break in (iii). (State (ii): $F = 40 e^{0.39} = 59.08$ , liability PV $1.76$ ; state (iii): post-jump $F = 91.49$ , liability PV $15.89$ .)

14 Synthesis: the cross-currency basis as an XVA reserve

This exercise bridges the static consistency logic of Propositions 1–5 into the valuation-adjustment (XVA) paradigm, resolving open problem (P3). The $1.310$ TRY premium — wildly inflated against the $0.000$ realized expectation — is the rigorous counterpart of a Funding Valuation Adjustment (FVA) and jump-risk reserve. Because the basis $b_{t}$ is highly stochastic and legally constrained by capital controls, the premium is the market-clearing price required to warehouse unhedgeable gap and basis risk.

A practitioner who mis-anchors the smile to the onshore forward not only triggers a coordinate error — they mathematically bypass this economic reality, underpricing the requisite XVA reserve and adopting a structurally toxic short-tail profile.

15 Correct procedure

Identify the habitat. Read the forward $F$ from the habitat (onshore deliverable vs offshore NDF) where the physical delta-hedge executes.
Match the volatility. Anchor the surface strictly to the forward native to the quoting habitat. Never cross-pollinate $F^{on}$ with $σ^{off}$ .
Discount properly. Under a USD CSA, discount the USD-equivalent premium $c_{fwd} / F$ by $P_{f}$ ; equivalently apply $P_{d}^{off} = e^{- r_{d}^{off} T}$ to the TRY value. Do not apply $e^{- r_{f} T}$ to the TRY forward amount.
Translate via preserved price. To internalize an offshore price on an onshore-calibrated system, hold the base-currency price invariant and root-solve the implied $σ$ against $F^{on}$ (here $20.24 %$ , not $26.65 %$ ).
Reserve the basis. Treat the segmentation wedge as a stochastic, jump-driven FVA/gap reserve — stress it; it is policy-driven, not static.

16 Remaining open problems

(P4) NDF vs deliverable smiles. Offshore TRY curves from NDFs (settling in USD against the CBRT fixing) and deliverable FX swaps can bifurcate during capital-control events; the exact convexity adjustment mapping an NDF-implied smile to a deliverable smile remains open. (P5) Static arbitrage under re-anchoring. Exact log-strike translation preserves the density: by linear homogeneity $c_{new} (K) = e^{- b T} c_{old} (K e^{b T})$ , and differentiating twice ${c_{new}^{'}}^{'} (K) = e^{b T} {c_{old}^{'}}^{'} (K e^{b T}) \geq 0$ , preserving butterfly convexity. But shifting only the discrete $Δ$ pillars and re-interpolating via SABR / vanna-volga can violently violate calendar and butterfly bounds.

17 References

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