Derivatives & Options Formula Cheat Sheet

Compounding & Interest Rates

Discrete Compounding

• Future Value = $P(1 + r)^T$
• Present Value = $\frac{FV}{(1 + r)^T}$
• $P$: Principal
• $r$: Interest rate per period
• $T$: Number of periods

Continuous Compounding

• Future Value = $P \times e^{rT}$
• Present Value = $FV \times e^{-rT}$
• $e$: Euler's number (≈ 2.71828)
• $r$: Continuous interest rate
• $T$: Time in years

Converting Between Rates

• Continuous rate = $\ln(1 + \text{discrete rate})$
• Discrete rate = $e^{\text{continuous rate}} - 1$

Hedging Calculations

Effective Price Received/Paid

• Long Hedge (Buyer): $\text{Effective Price} = \text{Spot Price} + (\text{Initial Futures} - \text{Final Futures})$
• Short Hedge (Seller): $\text{Effective Price} = \text{Spot Price} - (\text{Final Futures} - \text{Initial Futures})$

Hedge Ratio

• Optimal Hedge Ratio = $\rho(\sigma_S/\sigma_F)$
• $\rho$: Correlation coefficient between spot and futures prices
• $\sigma_S$: Standard deviation of spot price changes
• $\sigma_F$: Standard deviation of futures price changes

Index Futures Hedge

• Number of contracts = $\frac{\beta_P}{(\beta_F \times F \times M)}$
• $P$: Value of portfolio to be hedged
• $\beta_P$: Beta of portfolio
• $\beta_F$: Beta of futures (usually 1.0)
• $F$: Futures price
• $M$: Contract multiplier

Forward Pricing

Basic Forward Price (No dividends)

• Discrete: $F = S(1 + r_t)^T$
• Continuous: $F = Se^{rT}$
• $F$: Forward price
• $S$: Spot price
• $r_t$: Interest rate for period $T$
• $T$: Time to maturity

Forward Price with Known Income

• Discrete: $F = (S - I)(1 + r_t)^T$
• Continuous: $F = (S - I)e^{rT}$
• $I$: Present value of known income

Commodity Forward Price

• Discrete: $F = (S + U - Y)(1 + r_t)^T$
• Continuous: $F = (S + U - Y)e^{rT}$
• $U$: Present value of storage costs
• $Y$: Present value of convenience yield

Currency Forward Price

• Discrete: $F_t = S\frac{(1 + r_t)^T}{(1 + r_{f,t})^T}$
• Continuous: $F_t = Se^{(r-r_f)T}$
• $r_{f,t}$: Foreign interest rate
• $S$: Current exchange rate

Options

Put-Call Parity

• Discrete: $S + p = c + \frac{K}{(1 + r)^T}$
• Continuous: $S + p = c + Ke^{-rT}$
• $p$: Put premium
• $c$: Call premium
• $K$: Strike price

Option Payoffs at Maturity

• Long Call = $\max[S_T - K, 0]$
• Short Call = $\min[K - S_T, 0]$
• Long Put = $\max[K - S_T, 0]$
• Short Put = $\min[S_T - K, 0]$

Option Profits

• Long Call Profit = $\max[S_T - K, 0] - c$
• Short Call Profit = $c - \max[S_T - K, 0]$
• Long Put Profit = $\max[K - S_T, 0] - p$
• Short Put Profit = $p - \max[K - S_T, 0]$

Binomial Option Pricing Formulas

Single-Period Model

Basic Parameters

• $S_0$: Initial stock price
• $u$: Up factor
• $d$: Down factor
• $r$: Risk-free interest rate
• $X$: Strike price
• $T$: Time to maturity

Risk-Neutral Probability

• $q = \frac{(1 + r) - d}{u - d}$ (Discrete compounding)
• $q = \frac{e^{rT} - d}{u - d}$ (Continuous compounding)

Option Values at Maturity

• Up state stock price: $S_u = S_0 \times u$
• Down state stock price: $S_d = S_0 \times d$
• Call payoff up: $c_u = \max(S_u - X, 0)$
• Call payoff down: $c_d = \max(S_d - X, 0)$
• Put payoff up: $p_u = \max(X - S_u, 0)$
• Put payoff down: $p_d = \max(X - S_d, 0)$

Single-Period Option Pricing

Discrete Compounding

• Call: $c = \frac{1}{1 + r}[qc_u + (1-q)c_d]$
• Put: $p = \frac{1}{1 + r}[qp_u + (1-q)p_d]$

Continuous Compounding

• Call: $c = e^{-rT}[qc_u + (1-q)c_d]$
• Put: $p = e^{-rT}[qp_u + (1-q)p_d]$

Two-Period Model

Stock Price Movements

• $S_{uu} = S_0 \times u^2$ (Up-Up)
• $S_{ud} = S_{du} = S_0 \times u \times d$ (Up-Down or Down-Up)
• $S_{dd} = S_0 \times d^2$ (Down-Down)

Option Values at Maturity

• Call payoffs:
  • $c_{uu} = \max(S_{uu} - X, 0)$
  • $c_{ud} = \max(S_{ud} - X, 0)$
  • $c_{dd} = \max(S_{dd} - X, 0)$
• Put payoffs:
  • $p_{uu} = \max(X - S_{uu}, 0)$
  • $p_{ud} = \max(X - S_{ud}, 0)$
  • $p_{dd} = \max(X - S_{dd}, 0)$

Two-Period Option Pricing

Discrete Compounding

Working backwards:

1. Middle node values (at time T₁):

   • Call up: $c_u = \frac{1}{1 + r}[qc_{uu} + (1-q)c_{ud}]$
   • Call down: $c_d = \frac{1}{1 + r}[qc_{ud} + (1-q)c_{dd}]$
   • Put up: $p_u = \frac{1}{1 + r}[qp_{uu} + (1-q)p_{ud}]$
   • Put down: $p_d = \frac{1}{1 + r}[qp_{ud} + (1-q)p_{dd}]$

2. Initial value (at time 0):

   • Call: $c = \frac{1}{1 + r}[qc_u + (1-q)c_d]$
   • Put: $p = \frac{1}{1 + r}[qp_u + (1-q)p_d]$

Continuous Compounding

Working backwards:

1. Middle node values (at time T₁):

   • Call up: $c_u = e^{-r\Delta t}[qc_{uu} + (1-q)c_{ud}]$
   • Call down: $c_d = e^{-r\Delta t}[qc_{ud} + (1-q)c_{dd}]$
   • Put up: $p_u = e^{-r\Delta t}[qp_{uu} + (1-q)p_{ud}]$
   • Put down: $p_d = e^{-r\Delta t}[qp_{ud} + (1-q)p_{dd}]$

2. Initial value (at time 0):

   • Call: $c = e^{-r\Delta t}[qc_u + (1-q)c_d]$
   • Put: $p = e^{-r\Delta t}[qp_u + (1-q)p_d]$

Where $\Delta t = T/n$ is the time step (T/2 for two-period model)

Direct Formula to get from end to beginning value:

1. Discrete Compounding: $$c = \frac{1}{1+r^2}(q^2c_{uu} + 2q(1-q)c_{ud} + (1-q)^2c_{dd})$$ $$p = \frac{1}{1+r^2}(q^2p_{uu} + 2q(1-q)p_{ud} + (1-q)^2p_{dd})$$
2. Continuous Compounding: $$c = e^{-2r\Delta t}(q^2c_{uu} + 2q(1-q)c_{ud} + (1-q)^2c_{dd})$$ $$p = e^{-2r\Delta t}(q^2p_{uu} + 2q(1-q)p_{ud} + (1-q)^2p_{dd})$$

Delta Hedging Parameters

• $\Delta_{call} = \frac{c_u - c_d}{S_0(u-d)}$
• $\Delta_{put} = \frac{p_u - p_d}{S_0(u-d)}$

No-Arbitrage Conditions

• $d < 1 + r < u$ (Discrete compounding)
• $d < e^{rT} < u$ (Continuous compounding)

Option Price Bounds

European Call Bounds

• $\max(0, S - Ke^{-rT}) \leq c \leq S$

European Put Bounds

• $\max(0, Ke^{-rT} - S) \leq p \leq Ke^{-rT}$

Option Value Components

• Total Value = Intrinsic Value + Time Value
• Intrinsic Value (Call) = $\max(S - K, 0)$
• Intrinsic Value (Put) = $\max(K - S, 0)$
• Time Value = Option Price - Intrinsic Value

Margin Calculations

Margin Call Thresholds

For Short Position:

• Margin Call Trigger Price = Initial Price + ((Initial Margin - Maintenance Margin)/(Units × Price per Unit))
• Where:
  • Initial Price = Original futures price
  • Initial Margin = Initial deposit required
  • Maintenance Margin = Minimum required balance
  • Units = Number of units in contract

For Long Position:

• Margin Call Trigger Price = Initial Price - ((Initial Margin - Maintenance Margin)/(Units × Price per Unit))

Example Calculation

For Q3:

• Initial Price = $0.70
• Initial Margin = $4,000
• Maintenance Margin = $3,000
• Units = 50,000
• Price Change Threshold = ($4,000 - $3,000)/(50,000) = $0.02
• Margin Call Price (Short) = $0.70 + $0.02 = $0.72

Therefore, if futures price rises above $0.72, there will be a margin call.

General Rules

• Short Position: Margin call when price rises above threshold
• Long Position: Margin call when price falls below threshold
• Loss Threshold = Initial Margin - Maintenance Margin
• Price Move for Margin Call = Loss Threshold ÷ (Units × Price per Unit)

Daily Price Movement

Standard Deviation of Daily Price Changes

• $\sigma_{daily} = \sigma_{annual} \times \sqrt{\frac{1}{365}}$
• $\sigma_{daily}$: Daily standard deviation
• $\sigma_{annual}$: Annual volatility

Black-Scholes Option Pricing Formulas

Core Equations

European Call Option

$c = SN(d_1) - Xe^{-rT}N(d_2)$

European Put Option

$p = Xe^{-rT}N(-d_2) - SN(-d_1)$

Where: $d_1 = \frac{\ln(S/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T}$

Parameters

• $S$ = Current stock price
• $X$ = Strike price
• $r$ = Risk-free interest rate (continuous compounding)
• $T$ = Time to maturity (in years)
• $\sigma$ = Stock price volatility (annual)
• $N(x)$ = Cumulative standard normal distribution function

Geometric Brownian Motion (GBM)

Stock Price Process

$\frac{dS}{S} = \mu dt + \sigma dZ$

Distribution Properties

• Returns distribution: $\frac{dS}{S} \sim N(\mu dt, \sigma^2dt)$
• Log price distribution: $\ln(S_T) \sim N[\ln(S_0) + (\mu - \frac{\sigma^2}{2})T, \sigma^2T]$

Parameter Effects on Option Prices

Parameter Increase	Call Price	Put Price
Stock Price ($S$)	↑	↓
Strike Price ($X$)	↓	↑
Volatility ($\sigma$)	↑	↑
Time to Mat. ($T$)	↑	↑
Interest Rate ($r$)	↑	↓

Important Properties

American vs European Options

For non-dividend paying stocks:

• American calls = European calls
• American puts > European puts

Implied Volatility

• Implied volatility is the value of $\sigma$ that makes the Black-Scholes price equal to the market price
• Used as a measure of market expectations of future volatility
• Often used to quote option prices in markets

Model Assumptions

1. No arbitrage opportunities
2. Geometric Brownian motion for stock prices
3. Constant volatility
4. No dividends
5. European exercise
6. Continuous trading