Valuation & P&L
Forward / futures position value
V = Q (F_t,T - K) e^-r (T-t)
The value of a forward or futures position is the quantity times the difference between the current forward price for delivery T and the traded (strike) price, discounted to today. For daily-margined futures the discount term is often dropped.
- V
- position value
- Q
- quantity (signed: + long, − short)
- F_t,T
- forward price at t for delivery T
- K
- traded price
- r
- discount rate
- T-t
- time to delivery (years)
Mark-to-market P&L
P&L_unrealized = V_t - V_t-1
Unrealized P&L is simply the change in mark-to-market value between two valuation points. Realized P&L is recognized at settlement.
- V_t
- value today
- V_t-1
- value at the prior mark
Discount factor
DF(t,T) = e^-r (T-t) or 1(1+r/m)^m(T-t)
Future cashflows are brought to present value with a discount factor, continuous compounding on the left, discrete (m periods per year) on the right.
- r
- zero rate
- m
- compounding frequency
Option pricing (Black-76)
Black-76 price (option on a future/forward)
c = e^-rT[F N(d₁) - K N(d₂)] p = e^-rT[K N(-d₂) - F N(-d₁)]
Black-76 is the commodity-market workhorse: it prices European options on a future or forward F. c is the call price, p the put. It is the model behind the calculator below.
- F
- forward price
- K
- strike
- r
- risk-free rate
- T
- time to expiry (years)
- N(·)
- standard normal CDF
The d-terms
d₁ = ln(F/K) + ½σ^2 Tσ√(T), d₂ = d₁ - σ√(T)
The two standardized moneyness terms. σ is the annualized volatility of the forward.
- σ
- volatility (annualized)
Put-call parity (Black-76)
c - p = e^-rT (F - K)
A model-free relationship: the call minus the put equals the discounted forward-minus-strike. Useful as a consistency check.
The Greeks (Black-76)
Delta
Δ_call = e^-rTN(d₁), Δ_put = -e^-rTN(-d₁)
Sensitivity of option value to the forward price, the slope of the value curve.
Gamma
Γ = e^-rT N^′(d₁)F σ√(T)
The rate of change of delta with the forward, the curvature of the value curve. N′ is the normal probability density.
Vega
ν = F e^-rT N^′(d₁) √(T)
Sensitivity to a change in volatility (per 1.0 change in σ; divide by 100 for per-vol-point).
Theta (call)
Θ = -F e^-rTN^′(d₁)σ2√(T) - rK e^-rTN(d₂) + rF e^-rTN(d₁)
Sensitivity to the passage of time, usually negative for long options (time decay).
Cross-gamma (multi-contract)
Γᵢj = (∂^2 V)/(∂ Fᵢ ∂ Fⱼ)
How the delta to contract i changes when a related contract j moves, essential for spread and calendar books, and surfaced in the Greeks mart.
Risk (VaR & Expected Shortfall)
Parametric VaR
VaR_c = z_c σ_P √(h) (× portfolio value)
Under a normal assumption, VaR is the confidence multiplier times the portfolio return volatility, scaled to the horizon h (in days if σ is daily).
- z_c
- normal quantile (1.645 at 95%, 2.326 at 99%)
- σ_P
- portfolio volatility
- h
- horizon
Portfolio volatility
σ_P = w^ᵀ Σ w
The portfolio volatility from position weights and the covariance matrix of risk factors, the input to parametric VaR.
- w
- position weights / sensitivities
- Σ
- covariance matrix
Historical-simulation VaR
VaR_c = - Percentile₁-c( P&Lᵢ )
Apply historical factor moves to today’s book, then read the loss at the (1−c) percentile of the simulated P&L set.
Expected Shortfall (CVaR)
ES_c = E[ L | L > VaR_c ]
The average loss given that the loss exceeds VaR, it describes the severity of the tail VaR ignores.
- L
- loss
Curves & interpolation
Linear interpolation
F(t) = F_a + (F_b - F_a) (t - t_a)/(t_b - t_a)
The simplest way to price a delivery period t between two quoted tenors a and b. Production curves use smoother schemes and seasonal shaping.
Seasonal shaping
F_month = F_period × s_month
A period (e.g. calendar) price is shaped into months by seasonal factors s that preserve the period average, essential for power and gas.
- F
- period average price
- s
- seasonal factor (averages to 1)
Demo calculator
Live, in-browser calculators using the formulas above. Nothing is sent anywhere, all math runs locally. These are illustrative; production numbers come from the governed engine.
Black-76 assumes European exercise on a forward/future with constant volatility; parametric VaR assumes normally-distributed returns. Real books use the full quant engine on governed positions.
Related
See these on your own book
The calculator is illustrative, a live walkthrough runs the real engine on your positions.
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