CQF Comprehensive Cheatsheet - FX Module
FX MARKET FUNDAMENTALS
Currency Pair Conventions
Quote Convention: CCY1/CCY2 = "how many units of CCY2 per 1 unit of CCY1"
- Base currency (CCY1): Denominator in exchange rate
- Quote/counter currency (CCY2): Numerator
- Example: EUR/USD = 1.10 means €1 = $1.10
Major Pairs:
- EUR/USD (Euro/US Dollar) - "Euro"
- USD/JPY (US Dollar/Japanese Yen) - "Dollar Yen"
- GBP/USD (British Pound/US Dollar) - "Cable"
- AUD/USD (Australian Dollar/US Dollar) - "Aussie"
- USD/CHF (US Dollar/Swiss Franc) - "Swissy"
- USD/CAD (US Dollar/Canadian Dollar) - "Loonie"
Inverted Rates:
$$\text{USD/EUR} = \frac{1}{\text{EUR/USD}}$$Bid-Ask Spreads
Bid: Price at which dealer buys base currency (sells quote currency)
Ask/Offer: Price at which dealer sells base currency (buys quote currency)
Dealer buys EUR at 1.1000, sells EUR at 1.1003
Spread = 3 pips (1 pip = 0.0001 for most pairs)
Pip: "Percentage in point" or "Price interest point"
- Most pairs: 0.0001 (4th decimal)
- JPY pairs: 0.01 (2nd decimal)
- Pipette: 1/10 of a pip (fractional pip)
Cross Rates
Definition: Exchange rate between two currencies, neither of which is USD
Calculation via USD:
EUR/GBP = 1.1000/1.2500 = 0.8800
(€1 = £0.88)
Triangular Arbitrage: Profit from inconsistent cross rates
$$S_{AC} \neq \frac{S_{AB}}{S_{CB}} \implies \text{Arbitrage opportunity}$$SPOT FX MARKET
Spot Rate
Definition: Exchange rate for immediate delivery
Settlement: T+2 (two business days) for most pairs
Exceptions:
- USD/CAD: T+1
- Some exotic pairs: T+2 or longer
Value Date: Date on which currencies are actually exchanged
FX Dynamics under GBM
Spot rate \(S_t\) (domestic/foreign) follows:
$$dS_t = (r_d - r_f) S_t dt + \sigma S_t dW_t$$where:
- \(r_d\) = domestic interest rate
- \(r_f\) = foreign interest rate
- \(\sigma\) = FX volatility
- \(r_d - r_f\) = drift under domestic risk-neutral measure
Interpretation: Foreign currency is like a dividend-paying stock with yield \(r_f\)
Interest Rate Parity
Covered Interest Rate Parity (CIRP):
Or in continuous compounding:
$$F = S \cdot e^{(r_d - r_f)T}$$No-arbitrage condition: Forward price equals expected future spot
\(F = 1.1000 \times e^{(0.02 - 0.005) \times 1} = 1.1000 \times e^{0.015} = 1.1166\)
Euro trades at forward premium (higher rate currency depreciates)
Uncovered Interest Rate Parity (UIRP):
$$\mathbb{E}[S_{t+T}] = S_t \cdot e^{(r_d - r_f)T}$$Expected spot movement equals interest differential
Empirically violated: "Forward premium puzzle"
FX FORWARDS AND SWAPS
Forward Contracts
Definition: Agreement to exchange currencies at future date \(T\) at predetermined rate \(F\)
Forward Rate:
Forward Points (swap points):
$$\text{FP} = F - S$$Quoted in pips, added to spot rate
1Y Forward = 1.1000 + 0.0166 = 1.1166
Premium/Discount:
- \(r_d > r_f\): Foreign currency at forward premium (\(F > S\))
- \(r_d < r_f\): Foreign currency at forward discount (\(F < S\))
Forward P&L
Long forward (agree to buy base currency at \(F\)):
At maturity \(T\):
$$\text{P\&L} = (S_T - F) \times \text{Notional}$$Before maturity at time \(t < T\):
$$\text{MTM} = (F(t,T) - F) \times \text{Notional} \times e^{-r_d(T-t)}$$After 6 months: Spot = 1.1500, new 6M forward = 1.1583
MTM = (1.1583 - 1.1166) × 10M × \(e^{-0.02 \times 0.5}\)
= 0.0417 × 10M × 0.99 = $412,800 profit
FX Swaps
Definition: Simultaneous spot and forward transactions in opposite directions
Typical structure:
- Spot leg: Buy EUR/sell USD at spot \(S\)
- Forward leg: Sell EUR/buy USD at forward \(F\)
Swap Points:
$$\text{Swap Points} = (F - S) \times \text{Notional}$$Purpose:
- Roll FX positions
- Funding in foreign currency
- Manage maturity mismatches
Cross-Currency Basis: Deviation from theoretical forward due to:
- Supply/demand imbalances
- Credit risk differences
- Regulatory costs
FX OPTIONS - VANILLA
Garman-Kohlhagen Model
FX Option Pricing (extension of Black-Scholes for FX):
Call Option (right to buy base currency):
Put Option (right to sell base currency):
Put-Call Parity:
$$C - P = S_0 e^{-r_f T} - K e^{-r_d T}$$Or equivalently:
$$C - P = (F - K) e^{-r_d T}$$\(r_{USD} = 2\%\), \(r_{EUR} = 0.5\%\)
\(d_1 = \frac{\ln(1.10/1.12) + (0.02-0.005+0.005)0.25}{0.10\sqrt{0.25}} = -0.2851\)
\(d_2 = -0.2851 - 0.05 = -0.3351\)
\(N(d_1) = 0.3877\), \(N(d_2) = 0.3687\)
\(C = 1.10 \times e^{-0.005 \times 0.25} \times 0.3877 - 1.12 \times e^{-0.02 \times 0.25} \times 0.3687\)
\(= 0.4261 - 0.4110 = \boxed{0.0151}\) or 151 pips
FX Option Greeks
Delta \(\Delta = \frac{\partial V}{\partial S}\):
- Call: \(\Delta = e^{-r_f T} N(d_1) \in [0, e^{-r_f T}]\)
- Put: \(\Delta = -e^{-r_f T} N(-d_1) \in [-e^{-r_f T}, 0]\)
- ATM call delta ≈ 0.5 × \(e^{-r_f T}\) (adjusted for foreign rate)
Gamma \(\Gamma = \frac{\partial^2 V}{\partial S^2}\):
$$\Gamma = \frac{e^{-r_f T} N'(d_1)}{S \sigma \sqrt{T}}$$Same for calls and puts
Vega \(\mathcal{V} = \frac{\partial V}{\partial \sigma}\):
$$\mathcal{V} = S e^{-r_f T} \sqrt{T} N'(d_1)$$Theta \(\Theta = \frac{\partial V}{\partial t}\):
Call:
$$\Theta_C = -\frac{S e^{-r_f T} N'(d_1) \sigma}{2\sqrt{T}} + r_f S e^{-r_f T} N(d_1) - r_d K e^{-r_d T} N(d_2)$$Rho (dual sensitivities):
- \(\rho_d = \frac{\partial V}{\partial r_d}\): Sensitivity to domestic rate
- \(\rho_f = \frac{\partial V}{\partial r_f}\): Sensitivity to foreign rate
Call: \(\rho_d = KT e^{-r_d T} N(d_2)\), \(\rho_f = -ST e^{-r_f T} N(d_1)\)
Option Quoting Conventions
Delta Convention: Options quoted by delta, not strike
- 25-delta call: Call with \(\Delta = 0.25 e^{-r_f T}\)
- 25-delta put: Put with \(|\Delta| = 0.25 e^{-r_f T}\)
- ATM: At-the-money (strike ≈ forward)
Strike from Delta (for call):
$$K = S e^{(r_d - r_f)T - \sigma\sqrt{T} N^{-1}(\Delta e^{r_f T}) + \sigma^2 T/2}$$ATM Strike Conventions:
- ATM DNS (Delta-Neutral Straddle): \(K\) where call delta + put delta = 0
- ATM Forward: \(K = F\)
- ATM Spot: \(K = S\)
Most liquid: ATM DNS (50-delta straddle)
FX VOLATILITY SURFACE
Volatility Smile
Implied Volatility \(\sigma_{imp}(K,T)\): Volatility that equates model price to market price
FX Smile Characteristics:
- Symmetric smile: Higher vol for OTM puts and calls
- Risk reversals: Measure of skew
- Butterflies: Measure of curvature (smile vs flat)
- ATM volatility: Vol of ATM straddle
- 25Δ Risk Reversal (RR): \(\sigma_{25Δ Call} - \sigma_{25Δ Put}\)
- 25Δ Butterfly (BF): \(\frac{\sigma_{25Δ Call} + \sigma_{25Δ Put}}{2} - \sigma_{ATM}\)
Recovering Individual Vols:
$$\sigma_{25Δ Call} = \sigma_{ATM} + \frac{RR}{2} + BF$$ $$\sigma_{25Δ Put} = \sigma_{ATM} - \frac{RR}{2} + BF$$ATM = 10%, 25Δ RR = 1.5%, 25Δ BF = 0.3%
\(\sigma_{25Δ Call} = 10\% + 0.75\% + 0.3\% = 11.05\%\)
\(\sigma_{25Δ Put} = 10\% - 0.75\% + 0.3\% = 9.55\%\)
Positive RR → call vol > put vol (bullish skew)
Vega-Weighted Smile
Why vega weights? Market measures smile by vega-weighted prices, not vol difference
Strangle (25Δ call + 25Δ put):
$$\text{Strangle Price} = C_{25Δ} + P_{25Δ}$$Market BF relates to strangle price vs ATM straddle:
$$BF \approx \frac{\text{Strangle}_{\text{mid}} - 2 \times \text{ATM}_{\text{price}}}{\text{Vega}}$$Volatility Term Structure
Vol vs Maturity: \(\sigma(T)\) typically follows patterns:
- Short end (< 1M): Higher vol (event risk, liquidity)
- Medium term (1M-1Y): Intermediate vol
- Long end (>1Y): Lower vol (mean reversion)
Vol interpolation:
- Linear in variance: \(\sigma^2(T)\)
- Avoid calendar arbitrage
Sticky Delta vs Sticky Strike
Sticky Strike: Vol surface fixed in strike space
- When spot moves, delta of option changes
Sticky Delta: Vol surface fixed in delta space
- When spot moves, implied vol adjusts to maintain delta
Reality: FX markets mostly sticky delta
EXOTIC FX OPTIONS
Digital (Binary) Options
Cash-or-Nothing Call: Pays fixed amount \(Q\) if \(S_T > K\)
Asset-or-Nothing Call: Pays \(S_T\) if \(S_T > K\)
$$V = S e^{-r_f T} N(d_1)$$Delta of cash-or-nothing call:
$$\Delta = \frac{Q e^{-r_d T} N'(d_2)}{\sigma S \sqrt{T}}$$Spikes at maturity if near strike (gamma risk)
Barrier Options
Types:
- Knock-Out: Option dies if barrier hit
- Knock-In: Option activated if barrier hit
- Up: Barrier above spot
- Down: Barrier below spot
| Type | Barrier | Condition | Use Case |
|---|---|---|---|
| Up-and-Out (UOC) | \(H > S\) | KO if \(S_t \geq H\) | Cheaper call, limited upside view |
| Down-and-Out (DOC) | \(H < S\) | KO if \(S_t \leq H\) | Cheaper call, bullish view |
| Up-and-In (UIC) | \(H > S\) | Activated if \(S_t \geq H\) | Bet on break above level |
| Down-and-In (DIC) | \(H < S\) | Activated if \(S_t \leq H\) | Protection after dip |
In-Out Parity:
$$\text{Barrier In} + \text{Barrier Out} = \text{Vanilla}$$Approximate pricing (down-and-out call, \(H < S < K\)):
$$C_{DOC} \approx C_{vanilla} - \left(\frac{H}{S}\right)^{2\lambda} C(K', H^2/S)$$where \(\lambda = \frac{r_d - r_f + \sigma^2/2}{\sigma^2}\), \(K' = H^2/K\)
Vanilla call = 0.0200 (200 pips)
UOC ≈ 0.0150 (150 pips) - cheaper due to knockout
Rebate (payment if knocked out) can be added for compensation
Greeks peculiarities:
- Delta can flip sign as spot approaches barrier
- Large gamma near barrier
- Vega can be negative for out-of-money barriers
Asian Options
Payoff: Based on average exchange rate over period
Arithmetic Average:
$$\text{Payoff} = \max\left(\bar{S} - K, 0\right), \quad \bar{S} = \frac{1}{n}\sum_{i=1}^n S_{t_i}$$No closed form, use Monte Carlo or PDE
Geometric Average:
$$\bar{S}_G = \left(\prod_{i=1}^n S_{t_i}\right)^{1/n}$$Closed-form exists (log-normal distribution of geometric average)
Properties:
- Lower volatility than vanilla (averaging effect)
- Cheaper premium
- Used for hedging regular FX flows
Lookback Options
Floating Strike Call:
$$\text{Payoff} = S_T - S_{min}$$Guarantees best possible price (buying at lowest level)
Fixed Strike:
$$\text{Payoff} = \max(S_{max} - K, 0)$$Expensive (hindsight perfection) but popular for structured products
Touch Options
One-Touch: Pays if spot touches barrier before expiry
- Digital with continuous monitoring
No-Touch: Pays if spot never touches barrier
- One-Touch + No-Touch = Present Value of payment
Double-No-Touch (DNT): Pays if spot stays within corridor \([L, U]\)
- Used for range-bound views
Payment = $100k if spot stays in range
Implied probability ≈ 60% (from pricing)
Payout (PV) = 100k × 0.60 × \(e^{-rT}\) ≈ $59.7k
Target Redemption Forwards (TRF/TARF)
Structure: Series of forwards that terminates when cumulative profit reaches target
Features:
- Participation ratio: Asymmetric (e.g., 1:2 leverage on wrong side)
- Knockout level: Target profit
- Popular in Asia (controversial due to client losses)
Risk: Unlimited loss potential with capped gains
FX VOLATILITY MODELING
Stochastic Volatility Models
Heston Model for FX:
$$dS_t = (r_d - r_f) S_t dt + \sqrt{v_t} S_t dW_t^S$$ $$dv_t = \kappa(\theta - v_t)dt + \xi\sqrt{v_t} dW_t^v$$ $$dW_t^S dW_t^v = \rho dt$$Negative \(\rho\) typical in FX (higher vol when currency depreciates)
Calibration: Fit to market smile (ATM, RR, BF)
Local Volatility Models
Dupire Formula:
Constructs volatility surface \(\sigma(S,t)\) consistent with market prices
Properties:
- Perfectly fits market vanilla options
- Deterministic volatility path (given spot path)
- Poor forward smile dynamics
SABR Model for FX
Dynamics:
$$dF_t = \alpha_t F_t^\beta dW_t^F$$ $$d\alpha_t = \nu \alpha_t dW_t^\alpha$$ $$dW_t^F dW_t^\alpha = \rho dt$$Market standard: \(\beta = 0.5\) or \(\beta = 1\) (lognormal)
Implied Vol Approximation (Hagan et al.):
For \(K \approx F\) (ATM):
$$\sigma_{BS} \approx \frac{\alpha}{F^{1-\beta}}\left[1 + \left(\frac{(1-\beta)^2}{24}\frac{\alpha^2}{F^{2-2\beta}} + \frac{\rho\beta\nu\alpha}{4F^{1-\beta}} + \frac{2-3\rho^2}{24}\nu^2\right)T\right]$$Calibration to market quotes:
- Use ATM vol to constrain \(\alpha\)
- Use RR to fit \(\rho\) (skew)
- Use BF to fit \(\nu\) (vol-of-vol, curvature)
FX RISK MANAGEMENT
Delta Hedging
Delta-Neutral Portfolio:
$$\text{Position} = V - \Delta \times S$$where \(\Delta = \frac{\partial V}{\partial S} \times \text{Notional}\)
Rebalancing frequency: Balance between transaction costs and hedge effectiveness
Delta hedge: Buy 0.50 × €10M = €5M spot
If spot moves from 1.10 → 1.11, new delta = 0.55
Additional hedge: Buy 0.05 × €10M = €500k
Vega Hedging
Vega-Neutral: Offset volatility exposure with opposite vega position
Common approach: Hedge with ATM options (highest vega)
$$\text{Hedge Ratio} = \frac{\mathcal{V}_{portfolio}}{\mathcal{V}_{hedge instrument}}$$Value-at-Risk for FX
Parametric VaR (normal returns):
$$\text{VaR}_\alpha = \text{Position} \times \sigma \sqrt{\Delta t} \times z_\alpha$$\(\text{VaR}_{99\%} = 10M \times 0.10 \times \sqrt{10/252} \times 2.33\)
= 10M × 0.0198 × 2.33 = €462k
Historical Simulation: Apply historical returns to current position
Monte Carlo: Simulate future spot paths, calculate P&L distribution
Currency Exposure Types
| Exposure Type | Description | Hedge Strategy |
|---|---|---|
| Transaction | Known future cash flow | Forward contract |
| Translation | Foreign subsidiary balance sheet | Balance sheet hedge (debt) |
| Economic | Competitive position vs FX | Natural hedge, derivatives |
FX CARRY TRADE
Carry Trade Mechanics
Strategy:
- Borrow in low-interest currency (funding currency)
- Invest in high-interest currency (target currency)
- Profit from interest differential
Carry:
$$\text{Carry} = r_{high} - r_{low}$$Total Return (unhedged):
$$R = (r_{high} - r_{low}) + \frac{\Delta S}{S}$$where \(\frac{\Delta S}{S}\) is target currency appreciation
Carry = 4.5% - 0.5% = 4.0% p.a.
If AUD/JPY unchanged: Return = 4.0%
If AUD appreciates 2%: Return = 4.0% + 2.0% = 6.0%
If AUD depreciates 5%: Return = 4.0% - 5.0% = -1.0%
Risk:
- Funding currency appreciation (wipes out carry)
- Crashes during risk-off events (2008, COVID)
- High kurtosis (fat tails)
Forward Premium Puzzle: Empirically, high-interest currencies tend to appreciate (violates UIRP)
Carry-to-Risk Ratio
$$\text{Sharpe Ratio} = \frac{\text{Carry}}{\sigma_{FX}}$$Higher ratio = better risk-adjusted carry
Crowded Carry Unwind
2008 Crisis: JPY carry trades unwound violently
- USD/JPY: 110 → 87 (-21%)
- AUD/JPY: 107 → 55 (-49%)
- Liquidation spiral: Losses → margin calls → more unwinding
FX MARKET MICROSTRUCTURE
Order Flow and Price Discovery
Order Flow: Net buying pressure (buy orders - sell orders)
Empirical finding: Order flow strongly predicts short-term FX movements
Price impact:
$$\Delta S \propto \text{Order Flow} \times \sqrt{\text{Volume}}$$FX Market Participants
- Interbank market: Banks trading with each other
- Prime brokers: Provide access to hedge funds
- Electronic platforms: EBS, Reuters Matching, FXall
- Corporates: Hedging commercial flows
- Central banks: Intervention, reserves management
- Retail: Via brokers, typically small size
Trading Sessions
- Asian session: Tokyo, Singapore, Hong Kong (00:00-09:00 GMT)
- European session: London, Frankfurt (08:00-17:00 GMT)
- US session: New York (13:00-22:00 GMT)
- Overlap: London-NY (13:00-17:00 GMT) - highest liquidity
Volatility patterns: Highest during overlaps, lowest during Asian afternoon
Fixing Rates
WM/Reuters Fix: 4pm London benchmark
- Based on order flow in 1-minute window
- Used for fund valuations, index rebalancing
- Can cause volatility spikes
Manipulation scandals (2013-2014): Banks colluded to move fixing rates
CENTRAL BANK INTERVENTION
Types of Intervention
Direct Intervention: Central bank trades FX in market
- Sterilized: Offset with domestic bonds (no money supply change)
- Unsterilized: Changes monetary base
Verbal Intervention: "Jawboning" - statements to influence expectations
Effectiveness
Short-term: Can move market temporarily
Long-term: Limited if fighting fundamental trends
Famous cases:
- Swiss National Bank (2011-2015): EUR/CHF floor at 1.20
- Bank of Japan: Frequent interventions in USD/JPY
- Plaza Accord (1985): G5 coordination to weaken USD
KEY FORMULAS SUMMARY
| Concept | Formula |
|---|---|
| Forward Rate (continuous) | \(F = S \cdot e^{(r_d - r_f)T}\) |
| Spot Dynamics | \(dS = (r_d - r_f)S dt + \sigma S dW\) |
| Call Option (GK) | \(C = Se^{-r_fT}N(d_1) - Ke^{-r_dT}N(d_2)\) |
| Put-Call Parity | \(C - P = Se^{-r_fT} - Ke^{-r_dT}\) |
| Call Delta | \(\Delta_C = e^{-r_fT}N(d_1)\) |
| Gamma | \(\Gamma = \frac{e^{-r_fT}N'(d_1)}{S\sigma\sqrt{T}}\) |
| Vega | \(\mathcal{V} = Se^{-r_fT}\sqrt{T}N'(d_1)\) |
| 25Δ Call Vol | \(\sigma_{ATM} + \frac{RR}{2} + BF\) |
| 25Δ Put Vol | \(\sigma_{ATM} - \frac{RR}{2} + BF\) |
| Carry Return | \(R = (r_{high} - r_{low}) + \frac{\Delta S}{S}\) |
COMMON MISTAKES & TIPS
- Wrong currency pair convention: EUR/USD = 1.10 means €1 = $1.10, not $1 = €1.10
- Confusing base/quote: In EUR/USD, EUR is base, USD is quote
- Delta adjustment for foreign rate: ATM delta ≈ 0.5\(e^{-r_fT}\), not 0.5
- Forward points: Add to spot, don't multiply
- Forgetting foreign rate in pricing: Use \(r_d - r_f\), not just \(r_d\)
- Sign of delta: Put delta is negative
- Barrier direction: Down-and-out means barrier below spot
- RR interpretation: Positive RR = call vol > put vol (not necessarily bullish)
- Why FX smile symmetric? Both currencies are "risky" (unlike equity/zero)
- Sticky delta: FX market convention (vs equity sticky strike)
- Carry trade risk: "Picking up nickels in front of a steamroller"
- Put-call parity: Forward position = Long call + Short put
- High yield currencies: Tend to depreciate on average (UIRP), but not always
- Triangular arbitrage: Cross rates must be consistent
- CIP violations post-2008: Cross-currency basis widened significantly
- Most liquid pairs: EUR/USD > USD/JPY > GBP/USD
- Bid-ask spread: Lower for majors (1-3 pips), higher for exotics (10-50 pips)
- FX fixes: Can create manipulation opportunities (WM/Reuters scandal)
PRACTICAL EXAMPLE: COMPLETE OPTION PRICING
Market Data:
- Spot EUR/USD = 1.1000
- EUR interest rate = 0.50% p.a.
- USD interest rate = 2.00% p.a.
- Maturity = 3 months (T = 0.25)
- Volatility structure:
- ATM vol = 10.0%
- 25Δ Risk Reversal = 1.5%
- 25Δ Butterfly = 0.3%
Forward rate: \(F = 1.1000 \times e^{(0.02-0.005) \times 0.25} = 1.1041\)
ATM strike (DNS): \(K \approx F = 1.1041\)
Using Garman-Kohlhagen with \(\sigma = 10\%\):
\(d_1 = \frac{\ln(1.1000/1.1041) + (0.02-0.005+0.005) \times 0.25}{0.10 \times 0.5} = 0.0256\)
\(d_2 = 0.0256 - 0.05 = -0.0244\)
\(N(d_1) = 0.5102\), \(N(d_2) = 0.4903\)
\(C = 1.1000 \times e^{-0.005 \times 0.25} \times 0.5102 - 1.1041 \times e^{-0.02 \times 0.25} \times 0.4903\)
\(= 0.5597 - 0.5387 = \boxed{0.0210}\) or 210 pips
Task 2: Construct volatility smile
\(\sigma_{25Δ \text{ Call}} = 10.0\% + \frac{1.5\%}{2} + 0.3\% = 11.05\%\)
\(\sigma_{25Δ \text{ Put}} = 10.0\% - \frac{1.5\%}{2} + 0.3\% = 9.55\%\)
Task 3: Find 25Δ call strike
For call delta = \(0.25 \times e^{-0.005 \times 0.25} = 0.2497\):
Using iterative solver (or approximation):
\(K_{25Δ \text{ Call}} \approx 1.1350\)
Task 4: Price 25Δ call with smile vol
Using \(\sigma = 11.05\%\) (instead of flat 10%):
\(C_{25Δ} \approx \boxed{0.0092}\) or 92 pips
Task 5: Delta hedge
Notional = €10M
ATM call delta = 0.5102 × \(e^{-0.005 \times 0.25}\) = 0.5096
Hedge: Buy €5.096M spot at 1.1000
USD amount: 5.096M × 1.10 = $5.606M
Task 6: P&L after 1 week
Assume spot moves to 1.1100, vol unchanged:
New option value ≈ 0.0320 (320 pips)
Spot P&L = 5.096M × (1.1100 - 1.1000) = €0.051M
Option P&L = (0.0320 - 0.0210) × 10M = €0.110M
Total = €0.110M - €0.051M × 1.11 = €0.053M profit
(Positive due to gamma - delta hedge slightly underhedged)