Nonlinear Exposure and FX Options
When is linear hedging (forwards) insufficient?
Sources of nonlinear exposure: operating decisions create option-like payoffs
FX option strategies: protective put, collar, risk reversal
FX option pricing: Garman-Kohlhagen framework
Risk information in option prices: implied volatility, the smile, and the volatility risk premium
In Lecture 7, we measured exposure with the regression:
\[V_T = a + b \cdot S_T + u_T\]
\(b\) = exposure (FC units). Hedge: sell \(b\) forward.
This works when the relationship between firm value and the exchange rate is linear.
But what if the relationship is nonlinear?
Exposure may change depending on where \(S\) is
The slope \(b\) is not constant — it varies across states
Why might firm value respond nonlinearly to FX?
Pricing power thresholds: Firm absorbs small FX moves but adjusts prices for large ones
Competitive thresholds: Below a certain rate, you lose the market entirely
Contractual features: Quantity adjustments, renegotiation triggers, price caps
Pass-through asymmetries: Firms pass through depreciations faster than appreciations
In all these cases, the firm’s response to FX changes is different in different states — creating kinks, curves, and option-like payoffs.
A forward locks in both upside and downside equally.
If exposure has a kink:
A forward over-hedges on one side of the kink
And under-hedges on the other
You need an instrument with an asymmetric payoff — one that pays off in some states but not others.
That instrument is an option.
Egress is a US firm that can sell 1 unit of its product:
At home for USD 1, or
In Canada for CAD 1 (worth \(S\) in USD terms)
The firm exports only when it’s profitable: when \(S > 1\) (USD/CAD).
Revenue in USD:
\[V(S) = \begin{cases} 1 & \text{if } S \leq 1 \text{ (sell at home)} \\ S & \text{if } S > 1 \text{ (export)} \end{cases}\]
This can be written as: \(V(S) = 1 + \max(S - 1, 0)\)
\[V(S) = 1 + \max(S - 1, 0) = 1 + \text{Call}(K = 1)\]
The firm’s operating decision creates a natural long call position:
Below the kink: exposure = 0 (sells at home, no FX sensitivity)
Above the kink: exposure = 1 (exports, fully exposed to USD/CAD)
A forward cannot hedge this.
A forward has constant slope (delta = 1). This exposure has changing slope (delta = 0, then delta = 1).
To hedge: sell (write) a call with strike \(K = 1\).
Hedged value: \(V(S) - \text{Call}(K=1) = 1\) — perfectly flat.
When exposure is kinked or nonlinear, identify the option-like component:
The kink in operating exposure corresponds to a real business decision (export or not, enter market or not, adjust prices or not)
The strike price of the embedded option is the threshold at which the decision changes
Hedge with the matching option (call or put, appropriate strike)
The operating flexibility of the firm IS an option — and option pricing theory tells us how to value and hedge it.
Not all exposures have a clean kink. Some are smoothly curved.
Example: Both export quantity AND price increase with \(S\):
\[V(S) = S \times S = S^2\]
This is convex exposure — the slope increases as \(S\) rises.
A single forward (linear) cannot capture the curvature.
Any smooth curve can be approximated by a portfolio of calls at different strikes
Each call adds a kink, changing the slope — more calls means better fit
In practice:
Most firms don’t need perfect replication — 1-2 options can capture the main nonlinearity
Key question: where is the kink in your exposure? That determines the strike price.
If exposure is approximately linear over the relevant range \(\Rightarrow\) use a forward (simpler, cheaper)
If there’s a threshold, cliff, or significant curvature \(\Rightarrow\) use options
Call = right to buy FC at strike \(X\). Hedges FC outflows (caps the cost).
Put = right to sell FC at strike \(X\). Hedges FC inflows (sets a floor).
Cost: premium paid upfront (unlike forwards, which cost nothing to enter).
Firm will receive FC \(\Rightarrow\) buys a put to set a floor on HC value.
If \(S_T < X\): exercise put, receive \(X\) per FC unit (protected)
If \(S_T > X\): let put expire, convert at market rate (upside preserved)
Outcome: downside protected, upside preserved.
Cost: option premium (paid upfront). This is the price of asymmetric protection.
The protective put is the simplest and most common option strategy for corporate hedging.
Buy a put (floor) AND sell a call (cap) on the same FC amount.
The call premium received offsets (part of) the put premium paid
Zero-cost collar: choose strikes so premiums exactly offset
Outcome: bounded range — protected below the put strike, capped above the call strike.
Trade-off: give up upside to reduce cost.
Buy OTM put (deep downside protection) + sell OTM call (give up far upside).
Near-zero premium: the call premium funds the put
Provides cheap tail protection against extreme adverse moves
Common in practice for event risk (elections, central bank decisions, geopolitical shocks)
Key insight: The risk reversal lets the firm insure against disaster scenarios while paying very little upfront — at the cost of giving up gains in the best-case scenarios.
Use forwards when: exposure is certain (known amount, known date), FX view is neutral, cost minimization is the priority.
Use options when:
Layered hedging: forwards for near-term certain flows + options for tail risk or uncertain flows further out.
Setup: Firm submits a bid in FC for a contract.
Forward hedge is dangerous: if bid rejected, firm has a naked forward position (must deliver FC it doesn’t have).
Put option solves this:
If bid accepted \(\Rightarrow\) exercise put to lock in exchange rate
If bid rejected \(\Rightarrow\) let put expire (lose premium only)
This is why option markets exist for corporates — they hedge contingent exposures where the amount itself is uncertain.
The Black-Scholes model adapted for FX (Garman-Kohlhagen, 1983):
\[c = e^{-rT}\left[F \cdot N(d_1) - X \cdot N(d_2)\right]\]
\[p = e^{-rT}\left[X \cdot N(-d_2) - F \cdot N(-d_1)\right]\]
where \(d_1 = \frac{\ln(F/X) + \frac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}\), \(d_2 = d_1 - \sigma\sqrt{T}\)
Uses the forward rate \(F\) as the underlying (accounts for interest rate differential via CIP)
Key input: volatility \(\sigma\) — the only unobservable
Five inputs determine the option price:
| Input | Effect on call price | Effect on put price |
|---|---|---|
| Forward rate \(F\) \(\uparrow\) | \(\uparrow\) | \(\downarrow\) |
| Strike \(X\) \(\uparrow\) | \(\downarrow\) | \(\uparrow\) |
| Volatility \(\sigma\) \(\uparrow\) | \(\uparrow\) | \(\uparrow\) |
| Time to expiry \(T\) \(\uparrow\) | \(\uparrow\) | \(\uparrow\) |
| HC interest rate \(r\) \(\uparrow\) | \(\downarrow\) | \(\downarrow\) |
Volatility is the critical input: higher vol \(\Rightarrow\) more expensive options (more potential payoff).
The forward rate (not spot) matters — this connects back to CIP (Lecture 4).
A call and put with the same strike and expiry are linked by arbitrage:
\[c - p = \frac{F - X}{1 + r}\]
If you know the call price, you get the put price for free (and vice versa)
A forward is equivalent to: long call + short put at strike \(F\)
At-the-money forward (\(X = F\)): \(c = p\) (call and put have equal value)
Put-call parity is a no-arbitrage condition — it holds by the same logic as CIP.
An option can be replicated by a dynamic portfolio of a forward + a bond
The hedge ratio (delta, \(\Delta\)) tells you how many forwards you need:
Key difference from forwards:
A forward has constant delta = 1
An option has changing delta — it varies with \(S\)
This connects to the nonlinear exposure discussion: the changing delta of an option is what allows it to match nonlinear exposure that a constant-delta forward cannot.
Setup: \(S_0 = 1000\), \(r = 5\%\), \(r^* = 4\%\). Call with strike \(X = 1050\).
Replicating portfolio: \(\Delta\) forwards + \(B\) in bonds.
\[\Delta(S_u - F) + B(1+r) = c_u \quad \Rightarrow \quad \Delta(1100 - 1010) + 1.05B = 50\] \[\Delta(S_d - F) + B(1+r) = c_d \quad \Rightarrow \quad \Delta(950 - 1010) + 1.05B = 0\]
Step 1 — Delta (exposure):
\[\Delta = \frac{c_u - c_d}{S_u - S_d} = \frac{50 - 0}{1100 - 950} = \frac{1}{3}\]
The call’s exposure is \(\frac{1}{3}\) of a forward — it moves \(\frac{1}{3}\) as much as the spot rate.
Step 2 — Bond position: From the down-state equation: \(B = \frac{60/3}{1.05} = \frac{20}{1.05} \approx 19.05\)
Step 3 — Call price: Cost of replicating portfolio (forward costs zero to enter):
\[c_0 = B = 19.05\]
Key insight: We priced the option without knowing the true probability of up/down. Only the no-arbitrage condition and the ability to replicate matter.
Given an observed option price, invert the GK formula to extract the market’s expectation of future volatility.
This is implied volatility (\(\sigma_{\text{imp}}\)):
\[c_{\text{market}} = \text{GK}(F, X, T, r, \sigma_{\text{imp}})\]
Unlike historical (realized) volatility, implied vol is forward-looking
It reflects the market’s best assessment of future FX uncertainty
It is the single most important quantity in option markets
The GK model assumes constant \(\sigma\) across all strikes. In reality, \(\sigma\) varies:
Smile: OTM puts and OTM calls are both more expensive than ATM \(\Rightarrow\) U-shape
Skew: OTM puts are more expensive than OTM calls \(\Rightarrow\) the market prices crash risk more heavily
On average, implied volatility exceeds realized volatility:
\[E[\sigma_{\text{imp}}] > E[\sigma_{\text{realized}}]\]
Option sellers earn a premium for bearing volatility risk
The wedge is the volatility risk premium (VRP)
VRP = compensation for the risk that volatility spikes unexpectedly
Why it matters: When a firm buys a put to hedge, it pays the VRP on top of “fair” value. Hedging with options is more expensive than a pure probability calculation would suggest — because the seller demands compensation for risk.
Risk reversal = IV of 25\(\Delta\) call \(-\) IV of 25\(\Delta\) put
Positive RR: calls more expensive \(\Rightarrow\) market expects FC appreciation
Negative RR: puts more expensive \(\Rightarrow\) market expects FC depreciation
Corporate treasurers and traders use risk reversals as a real-time sentiment gauge:
A sharp move in the RR signals changing market expectations
Useful for timing hedge decisions (though not for speculation!)
Risk reversals are quoted directly in the interbank FX options market — they are one of the primary instruments traded.
The volatility risk premium is observable evidence that FX risk is priced.
UIP failure (Lecture 5): high-rate currencies don’t depreciate as predicted \(\Rightarrow\) carry trade earns risk premia
Vol risk premium (this lecture): implied vol > realized vol \(\Rightarrow\) option sellers earn risk premia
Both are compensation for bearing risk — the unifying theme of the course.
Every time the firm hedges (with forwards or options), it is paying or receiving these risk premia — whether it realizes it or not.
Preview: Full treatment of risk premia, the ICAPM, and carry trade returns in a later lecture.
Linear exposure \(\Rightarrow\) hedge with forwards. Nonlinear exposure \(\Rightarrow\) need options.
Operating decisions (export thresholds, pricing power) create option-like exposure
Key strategies: protective put (floor), collar (bounded range), options for contingent exposures (tenders)
Garman-Kohlhagen prices FX options using the forward rate and volatility
Option prices contain risk information: implied vol, skew, vol risk premium
The vol risk premium connects to UIP failure — both are evidence that FX risk is priced
