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Valuation of Forex Spot/Forward/Swap and Forex Options

上海子午线About 5 min

Valuation of Forex Spot/Forward/Swap and Forex Options

In the foreign exchange market, accurate valuation of forex spot/forward/swap transactions and forex options is critical for risk management and investment decision-making. This article, based on the logic of the Mammoth Valuation System, elaborates on how to calculate key valuation metrics: undiscounted market value (MarketValue), discounted market value (DiscMarketValue), undiscounted profit and loss (PnL), and discounted profit and loss (DiscPnL). These metrics enable traders to assess asset value and profitability from different perspectives, applicable to both forex spot/forward/swap and forex options.

1. Core Valuation Concepts

The Mammoth Valuation System defines the following four key concepts to uniformly calculate the value and profitability of forex spot/forward/swap and forex options.

  1. Undiscounted Market Value (MarketValue)

    • Definition: Measures the nominal scale of an asset’s future cash flows from the holder’s perspective, without considering the time value of money.
    • Features: Suitable for comparison with historical data, book value, or internal undiscounted data, reflecting the “nominal” value of future cash flows.

  2. Discounted Market Value (DiscMarketValue)

    • Definition: The market value of future cash flows discounted to the present using current interest rates.
    • Features: Reflects the economic value of the asset at the current time, aligning with the time value of money, commonly used for investment decisions and risk assessment.

  3. Discounted Profit and Loss (DiscPnL)

    • Definition: The profit or loss calculated at the discounted (present value) level, measuring the difference between the asset’s current theoretical value and its actual purchase price (transaction price).
    • Features: Directly reflects profitability or loss under current market conditions, suitable for real-time risk monitoring.

  4. Undiscounted Profit and Loss (PnL)

    • Definition: The discounted profit or loss restated as a nominal amount at a future time point.
    • Features: Facilitates comparison with book value or historical transaction scales, reflecting the nominal profit or loss of future cash flows.

2. Algorithmic Formulation

To uniformly calculate the valuation of forex spot/forward/swap and forex options, the Mammoth Valuation System defines the following parameters and formulas:

Parameter Definitions

  • $ P$: The discounted theoretical price calculated by a pricing function, typically based on market data or models (e.g., Black-Scholes model).
  • $ r$: The discount rate, reflecting the opportunity cost of capital or asset risk.
  • $ T$: The time to maturity (in years), i.e., the duration from the current time to the cash flow event.
  • $ DF$: The discount factor, calculated as $ DF = \exp(-r \cdot T)$, used to convert future values to present values.
  • $ TP$: The transaction price (purchase price), i.e., the actual cost paid:
    • For forex options, $ TP$ is the option premium paid at the time of the transaction (in monetary form, standardized to a currency like ccy2).
    • For forex spot/forward/swap, $ TP$ is typically 0 (no direct transaction fees, though spreads or implicit costs may apply).
  • ∅\emptyset: The transaction direction, where ∅=1\emptyset = 1 indicates a buy (long position), and ∅=−1\emptyset = -1 indicates a sell (short position).

Calculation Formulas

The following formulas are based on the transaction direction ∅\emptyset, ensuring consistent profit and loss calculations for buy and sell scenarios:

  1. Undiscounted Market Value (MarketValue)

    MarketValue=−∅⋅PDF\text{MarketValue} = -\emptyset \cdot \frac{P}{DF}

    • Explanation: The discounted theoretical price $ P$ is divided by the discount factor $ DF$ to restore it to the nominal value at a future time point. The negative sign and ∅\emptyset adjust for the holder’s perspective (negative for buys, positive for sells).

  2. Discounted Market Value (DiscMarketValue)

    DiscMarketValue=−∅⋅P\text{DiscMarketValue} = -\emptyset \cdot P

    • Explanation: Directly takes the negative of the discounted theoretical price (adjusted by transaction direction), reflecting the asset’s present value.

  3. Discounted Profit and Loss (DiscPnL)

    DiscPnL=DiscMarketValue−TP=(−∅⋅P)−TP\text{DiscPnL} = \text{DiscMarketValue} - TP = (-\emptyset \cdot P) - TP

    • Explanation: Calculates the difference between the discounted market value and the purchase price (transaction price), reflecting profit or loss at the current time. $ TP$, as a paid cost, is not discounted.

  4. Undiscounted Profit and Loss (PnL)

    PnL=MarketValue−TP=(−∅⋅PDF)−TP\text{PnL} = \text{MarketValue} - TP = \left(-\emptyset \cdot \frac{P}{DF}\right) - TP

    • Explanation: Subtracts the purchase price from the undiscounted market value, reflecting the nominal profit or loss at a future time point. $ TP$ remains a current-time value and is not reverse-discounted.

3. Calculation Features for Forex Spot/Forward/Swap and Forex Options

The product characteristics of forex spot/forward/swap and forex options differ, leading to variations in $ TP$ and profit/loss calculations. These are detailed below:

1. Forex Spot/Forward/Swap

  • Product Features:
    • Involves spot, forward, or swap transactions, typically with no direct transaction fees ($ TP = 0$), though spreads or implicit costs may exist.
    • Pricing reflects immediate market price fluctuations, with the theoretical price $ P$ calculated based on current market conditions.
  • Calculation Formulas:
    • DiscMarketValue=−∅⋅P\text{DiscMarketValue} = -\emptyset \cdot P
    • MarketValue=−∅⋅PDF\text{MarketValue} = -\emptyset \cdot \frac{P}{DF}
    • DiscPnL=(−∅⋅P)−0=−∅⋅P\text{DiscPnL} = (-\emptyset \cdot P) - 0 = -\emptyset \cdot P
    • PnL=(−∅⋅PDF)−0=−∅⋅PDF\text{PnL} = \left(-\emptyset \cdot \frac{P}{DF}\right) - 0 = -\emptyset \cdot \frac{P}{DF}
  • Example:
    • Assume $ P = 100$, $ r = 5%$, $ T = 1$, $ DF = \exp(-0.05 \cdot 1) \approx 0.9512,buy(, buy (\emptyset = 1$):
      • DiscMarketValue=−1⋅100=−100\text{DiscMarketValue} = -1 \cdot 100 = -100
      • MarketValue=−1⋅1000.9512≈−105.13\text{MarketValue} = -1 \cdot \frac{100}{0.9512} \approx -105.13
      • DiscPnL=−100−0=−100\text{DiscPnL} = -100 - 0 = -100
      • PnL=−105.13−0=−105.13\text{PnL} = -105.13 - 0 = -105.13

2. Forex Options

  • Product Features:
    • Transactions involve an option premium ($ TP$ is non-zero), priced in monetary form (standardized to a currency like ccy2).
    • The theoretical price $ P$ is typically calculated using Black-Scholes or other option pricing models, reflecting the option’s value under current market conditions.
  • Calculation Formulas:
    • DiscMarketValue=−∅⋅P\text{DiscMarketValue} = -\emptyset \cdot P
    • MarketValue=−∅⋅PDF\text{MarketValue} = -\emptyset \cdot \frac{P}{DF}
    • DiscPnL=(−∅⋅P)−TP\text{DiscPnL} = (-\emptyset \cdot P) - TP
    • PnL=(−∅⋅PDF)−TP\text{PnL} = \left(-\emptyset \cdot \frac{P}{DF}\right) - TP
  • Example:
    • Assume $ P = 100$, $ TP = 90$, $ r = 5%$, $ T = 1$, $ DF = 0.9512,buy(, buy (\emptyset = 1$):
      • DiscMarketValue=−1⋅100=−100\text{DiscMarketValue} = -1 \cdot 100 = -100
      • MarketValue=−1⋅1000.9512≈−105.13\text{MarketValue} = -1 \cdot \frac{100}{0.9512} \approx -105.13
      • DiscPnL=−100−90=−190\text{DiscPnL} = -100 - 90 = -190
      • PnL=−105.13−90≈−195.13\text{PnL} = -105.13 - 90 \approx -195.13

4. Considerations in Calculation

To ensure accurate valuation results, the following points should be noted during the calculation process:

  1. Transaction Direction (∅\emptyset):

    • Clearly specify buy (∅=1\emptyset = 1) or sell (∅=−1\emptyset = -1) direction, with the negative sign ensuring profit/loss reflects the holder’s perspective.
    • For complex transaction portfolios, calculate long and short positions separately if needed.

  2. Transaction Price ($ TP$):

    • For forex options, ensure $ TP$ is in monetary form with consistent units (e.g., ccy2).
    • For forex spot/forward/swap, $ TP$ is typically 0, but evaluate whether spreads or other implicit costs should be included.

  3. Discount Factor ($ DF$):

    • The discount rate $ r$ should be determined based on asset risk and market conditions (e.g., risk-free rate plus a risk premium).
    • The time to maturity $ T$ should be precise, in years (e.g., 6 months is $ T = 0.5$).

  4. Time Value Consistency:

    • $ TP$, as the purchase price at the current time, should not be discounted or reverse-discounted.
    • Ensure DiscPnL\text{DiscPnL} and PnL\text{PnL} calculations are based on consistent time points (current or future).

  5. Handling Anomalies:

    • If DiscPnL\text{DiscPnL} shows a loss while PnL\text{PnL} shows a profit, it may be due to a high discount rate significantly reducing the present value of future profits. Conduct sensitivity analysis to verify the reasonableness of $ r$, $ T$, and $ P$.

5. Conclusion

The valuation of forex spot/forward/swap and forex options requires a comprehensive consideration of market value and profit/loss from both discounted and undiscounted perspectives. The Mammoth Valuation System achieves unified calculations through the following formulas:

  • Undiscounted Market Value: MarketValue=−∅⋅PDF\text{MarketValue} = -\emptyset \cdot \frac{P}{DF}
  • Discounted Market Value: DiscMarketValue=−∅⋅P\text{DiscMarketValue} = -\emptyset \cdot P
  • Discounted Profit and Loss: DiscPnL=(−∅⋅P)−TP\text{DiscPnL} = (-\emptyset \cdot P) - TP
  • Undiscounted Profit and Loss: PnL=(−∅⋅PDF)−TP\text{PnL} = \left(-\emptyset \cdot \frac{P}{DF}\right) - TP

These formulas accommodate buy and sell scenarios through the transaction direction ∅\emptyset, and reflect the time value of money via the discount factor $ DF$, ensuring valuation results align with financial logic. The primary difference between forex spot/forward/swap and forex options lies in the transaction price $ TP$ (typically 0 for spot/forward/swap, and the option premium for options), but the overall calculation framework remains consistent.

In practical applications, ensure the accuracy of the pricing function, discount parameters, and transaction price, and evaluate implicit costs or conduct sensitivity analysis based on business needs. Through these methods, traders can effectively assess asset value and profitability, providing a reliable basis for risk management and investment decisions.

Pricing, Valuation and Risk Management
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