Derivatives — Master Formula Sheet

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1. Forward Payoff

Long forward payoff:

${\text{Payoff}}_{\text{long}}=S_T-F_0(T)$

Short forward payoff:

${\text{Payoff}}_{\text{short}}=F_0(T)-S_T$

where $S_T$ = spot price at expiration, $F_0(T)$ = forward price agreed at inception.

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2. Option Payoff

Long call:

$c_T=\max (0,S_T-X)$

Long put:

$p_T=\max (0,X-S_T)$

Short call:

$-c_T=-\max (0,S_T-X)$

Short put:

$-p_T=-\max (0,X-S_T)$

where $X$ = strike (exercise) price.

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3. Option Profit

${\text{Profit}}_{\text{long call}}=\max (0,S_T-X)-c_0$

${\text{Profit}}_{\text{long put}}=\max (0,X-S_T)-p_0$

${\text{Profit}}_{\text{short call}}=c_0-\max (0,S_T-X)$

${\text{Profit}}_{\text{short put}}=p_0-\max (0,X-S_T)$

where $c_0$, $p_0$ = option premium paid/received at inception.

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4. Forward Pricing

No-arbitrage forward price (no income, no costs):

$F_0(T)=S_0\times (1+r{)}^T$

With carry costs and income:

$F_0(T)=[S_0-P{V}_0(I)+P{V}_0(C)]\times (1+r{)}^T$

where:

• $S_0$ = current spot price
• $r$ = risk-free rate per period
• $P{V}_0(I)$ = present value of income/benefits (dividends, coupons, convenience yield)
• $P{V}_0(C)$ = present value of carrying costs (storage, insurance)

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5. Forward Valuation

At inception ($t=0$):

$V_0=0$

During life ($0<t<T$):

$V_t(\text{long})=S_t-\frac{F_0(T)}{(1+r{)}^{T-t}}$

At expiration ($t=T$):

$V_T=S_T-F_0(T)$

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6. FRA Cash Settlement

Forward Rate Agreement — settlement at expiration of the FRA:

$\text{Settlement (to long)}=\frac{(\text{MRR}-\text{IFR})\times \text{Notional}\times \text{Period}}{1+\text{MRR}\times \text{Period}}$

where:

• $\text{MRR}$ = market reference rate at settlement (e.g., SOFR)
• $\text{IFR}$ = implied forward rate (agreed FRA rate)
• $\text{Period}$ = fraction of year (e.g., $\frac{90}{360}$)
• Denominator discounts to the settlement date

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7. Futures

Interest rate futures price:

$f=100-\text{yield}$

Basis Point Value (BPV):

$\text{BPV}=\text{Notional}\times 0.01\%\times \text{Period}$

$=\text{Notional}\times 0.0001\times \frac{\text{Days}}{360}$

Daily MTM gain/loss:

$\text{Daily P\&L}=({\text{Settlement Price}}_t-{\text{Settlement Price}}_{t-1})\times \text{Multiplier}\times \text{Contracts}$

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8. Interest Rate Swap

Net settlement per period (from fixed-rate payer’s perspective):

${\text{Settlement}}_n=({\text{MRR}}_n-F)\times \text{Notional}\times \text{Period}$

where:

• ${\text{MRR}}_n$ = floating rate set at the beginning of period $n$
• $F$ = fixed swap rate (par swap rate)

Par swap rate (from spot rates):

$F=\frac{1-\frac{1}{(1+z_N{)}^N}}{{\sum }_{i=1}^N\frac{\text{Period}}{(1+z_i{)}^i}}$

where $z_i$ = spot (zero) rate for period $i$.

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9. Option Bounds

Lower Bounds (European, no dividends)

Call:

$c_0\ge \max \left(0,S_0-\frac{X}{(1+r{)}^T}\right)$

Put:

$p_0\ge \max \left(0,\frac{X}{(1+r{)}^T}-S_0\right)$

Upper Bounds

	European	American
Call	$c_0\le S_0$	$C_0\le S_0$
Put	$p_0\le \frac{X}{(1+r{)}^T}$	$P_0\le X$

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10. Put-Call Parity

Standard (spot underlying):

$\boxed{S_0+p_0=c_0+\frac{X}{(1+r{)}^T}}$

Put-call forward parity:

$\boxed{\frac{F_0(T)}{(1+r{)}^T}+p_0=c_0+\frac{X}{(1+r{)}^T}}$

Rearranged:

$p_0-c_0=\frac{X-F_0(T)}{(1+r{)}^T}$

Synthetic Positions

Synthetic	Composition
Call	$c_0=S_0+p_0-\frac{X}{(1+r{)}^T}$
Put	$p_0=c_0-S_0+\frac{X}{(1+r{)}^T}$
Stock	$S_0=c_0-p_0+\frac{X}{(1+r{)}^T}$
Bond	$\frac{X}{(1+r{)}^T}=S_0+p_0-c_0$

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11. Binomial Option Pricing (One-Period)

Up and down prices:

$S^{+}=S_0\times u,S^{-}=S_0\times d$

Option payoffs:

$c^{+}=\max (0,S^{+}-X),c^{-}=\max (0,S^{-}-X)$

Hedge ratio (delta):

$h=\frac{c^{+}-c^{-}}{S^{+}-S^{-}}$

Risk-neutral probability:

$\pi =\frac{(1+r)-d}{u-d}$

Option price (risk-neutral pricing):

$\boxed{c_0=\frac{\pi \cdot c^{+}+(1-\pi )\cdot c^{-}}{1+r}}$

For puts, replace $c^{+},c^{-}$ with $p^{+}=\max (0,X-S^{+})$ and $p^{-}=\max (0,X-S^{-})$.

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See Also

• Forward Commitments
• Contingent Claims
• Arbitrage and Replication
• Option Valuation
• Put-Call Parity