# Valuation

## Valuation of linear deals

Fundamentally all commodities deals are swaps, whereby one receives one leg and pays the other one. Tying it back to the generic trade template, the mark to market of a trade results from the steps detailed below. Upfront payments such as premiums or fees come on top of MTM to obtain the overall profit & loss.

Swap valuation
l
Swap valuation
Level Component Description Notation
Trade net present value Discounted PnL. The difference of the NPV of both legs. $$\rm NPV = ( NPV_{\space l_0} - NPV_{\space l_1} ) \times LS$$
Trade value Non discounted trade value. $$\rm V = ( ( V_{\space l_0} \times Q_{\space l_0}) - ( V_{\space l_1} \times Q_{\space l_1} ) ) \times LS$$
Trade intrinsic value The difference between the market rate of both legs.
Moneyness, unitary 'margin', non-probability weighted 'premium'.
The sign depends on trade direction: long minus short leg.
$$\rm IV = ( F_{\space l_0} - F_{\space l_1} ) \times LS$$
Legl
Leg net present value The sum of the discounted estimated future cash flows of all periods of the leg. $$\rm NPV_{\space l} = \frac {\space \sum_{ p=1 }^P \space ( NPV_{\space l,p} \times Q_{\space l,p})} { \sum_{ p=1 }^P Q_{\space l,p} }$$
Leg rate
• For single-period trades leg rate means period rate.
• For multi-period rates, it equals the quantity weighted average of the period rates.
$$\rm R_{\space l} = \frac {\space \sum_{ p=1 }^P \space ( R_{\space l,p} \times Q_{\space l,p})} { \sum_{ p=1 }^P Q_{\space l,p} }$$
Leg market rate $$\rm F_{\space l} = \frac {\space \sum_{ p=1 }^P \space ( F_{\space l,p} \times Q_{\space l,p})} { \sum_{ p=1 }^P Q_{\space l,p} }$$
Periodp
Period net present value Value corrected by discount factor between flow settlement date and evaluation date. $$\rm NPV_{\space l,p} = V_{\space l,p} \times D_{\space l,p}$$
Period value Rate times quantity. $$\rm V_{\space l,p} = R_{\space l,p} \times Q_{\space l,p}$$
Period rate Net rate, multiplied by rate factor then incremented by rate margin.
Values of those are most often respectively 1 and 0, hence equal to net rate.
$$\rm R_{\space l,p} = (F_{\space l,p} \times RF_{\space l,p}) + RM_{\space l,p}$$
Period market rate, Price
• For a floating leg, the market rate for the observed period.
• For a fix leg, the trade price (strike).
$$\rm F_{\space l,p}$$

Model Approach
Black 66
Bachelier

## Profit & Loss Taxonomy

The valuation approach in trading is mark to market: exposures are valued at quotes released in the market for the observed (delivered) underlying. Depending on regulatory requirements, accounting standards or risk reporting needs, several categorizations of P&L may coexist.

### Capital gain, Fees

This classification of profit & loss pertain capital gain, on top of which one should add or subtract the various transaction costs such as brokerage, duty and other fees.

### Market value, Future cash, Past cash

 Event Calendar MT-1 Current month MT+1 MT+2 30 31 01 02 03 04 … 28 29 30 01 02 03 04 05 … 31 01 02 03 04 05 … 30 Settlement Flow PC FC MV

Depending on when the evaluation date is relative to flow dates (price fixing, settlement dates) distinction is made between:

• Prior to delivery and/or price fixing, the profit & loss figures are called market value. The market rate is still an assessment and the amount of the future flow uncertain.
• Once the deal is delivered and the price is fixed, prior to the settlement due date, the profit & loss figures are called future cash. The delivered quantity is measured, closing prices have been published, the flow amount is known. There is no longer market risk, but still settlement risk.
• Having reached the payment date and released or received the payment, the profit & loss figures are called past cash. The cash flow is realized.

## Greeks

The sensitivity of the value of a option trade or portfolio to a small change in a given underlying parameter are the so-called option greeks. They are important tools in risk management.

Option Greeks
Attribute Delta Gamma Vega Theta Rho
Greek Delta Gamma Vega Theta Rho
Notation, given:
$$V = \enspace \textsf{Value of the security}$$
$$S = \enspace \textsf{Price of the underlying}$$
$$\nu = \enspace \textsf{Volatility of the underlying}$$
$$r = \enspace \textsf{Risk free rate}$$
$$t = \enspace \textsf{Tenor, Time to maturity}$$
$$\Delta = \frac{\delta \space V} {\delta \space S}$$ $$\Gamma = \frac{\delta \space \Delta} {\delta \space S} = \frac{\delta \space V^2}{\delta \space S^2}$$ $$\Upsilon = \frac{\delta \space V} {\delta \space \nu}$$ $$\Theta = - \space \frac{\delta \space V} {\delta \space t}$$ $$\Rho = \frac{\delta \space V} {\delta \space r}$$
Measure Trade value (NPV) change in response to 1-point movement of the underlying asset rate (speed). Delta change in response to 1-point movement of the underlying asset rate (acceleration). Trade value (NPV) change in response to 1-point movement of the underlying implied volatility. Trade value (NPV) change in response to 1-point movement of the underlying implied volatility. Trade value (NPV) change in response to 100-basis point (1% per annum) movement of the risk free interest rate.
Mathematical First derivative of the option value with respect to the change in the underlying price. Second derivative of the option value with respect to the change in the underlying price. First derivative of the option value with respect to the change in the underlying volatility.
Unit UOQ CUR
Position Direction →
Right ↓
Call + + + + + + +
Put + + + + +
Behaviour Moneyness
(S - K)
Close to 0 when deep OTM, around 0.5 when ATM, approaching 1 when deep ITM. Highest around ATM, decreases when deep OTM or ITM. Vega has no effect on the intrinsic value, but only affects time value of the option price. Higher for ATM options (have the most time value built in their premium).
Volatility Low volatility (little time value) increases the gamma, the more so for ATM options.
Time to expiry (days) As expiration approaches, change in underlying will cause more dramatic changes in delta, due to increased probability to be ITM or OTM. Gamma is more significant for near-term (close to expiry) options than for longer-term ones. Vega is higher for long-term options (time value contributes more to premium) compared to close to expiry. Time value melts away (theta increases) at an accelerated rate as expiration approaches. Obviously rho will be more significant for longer-term options (greater cost of carry).
PnL Predict (Contribution) $$Delta \space_{Yesterday} \times (Rate \space_{Today} - Rate \space_{Yesterday})$$ $$Gamma \space_{Yesterday} \times \frac {\space (Rate \space_{Today} \space - \space Rate \space_{Yesterday})^2}{2}$$ $$Vega \space_{Yesterday} \times (Volatility \space_{Today} - Volatility \space_{Yesterday})$$ $$Theta \space_{Yesterday} \times Time \space decay \space_{Days}$$