CMA Final SFM – Complete Futures & Options Derivatives Tool
CMA Final SFM - Complete Futures & Options Derivatives Tool
Interactive calculators and explanations for all key derivatives formulas
Futures Pricing Formulas
Cost of Carry Model
The Cost of Carry Model determines the theoretical price of a futures contract based on the spot price plus the cost of carrying the underlying asset until delivery.
Example: If gold spot price is $1,800, interest rate is 5%, and time to expiry is 6 months:
F = 1800 × e^(0.05 × 0.5) = 1800 × 1.0253 ≈ $1,845.54
Result:
Futures with Continuous Dividend Yield
For assets that provide a continuous dividend yield, the cost of carry is reduced by the dividend yield.
Example: Stock index with spot 3000, dividend yield 2%, interest rate 4%, time 0.5 years:
F = 3000 × e^((0.04 - 0.02) × 0.5) = 3000 × e^0.01 ≈ 3030.15
Result:
Futures with Storage Costs
For commodities that incur storage costs, these costs are added to the cost of carry.
Example: Commodity with spot $50, interest rate 5%, storage cost 3%, time 1 year:
F = 50 × e^((0.05 + 0.03) × 1) = 50 × e^0.08 ≈ 54.16
Result:
Options Pricing Models
Black-Scholes Model (Call Option)
The Black-Scholes model calculates the theoretical price of European call options using five inputs: stock price, strike price, time to expiration, risk-free rate, and volatility.
Example: Stock $100, Strike $100, Time 1 year, Rate 5%, Volatility 20%:
d1 = 0.325, d2 = 0.125, N(d1) = 0.627, N(d2) = 0.550
C = 100×0.627 - 100×e^(-0.05)×0.550 ≈ 62.70 - 52.29 = $10.41
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d2 = d1 - σ√T
Result:
Black-Scholes Model (Put Option)
The Black-Scholes formula for put options can be derived from the call option formula using put-call parity.
Example: Using same inputs as call example:
P = 100×e^(-0.05)×0.450 - 100×0.373 ≈ 42.79 - 37.30 = $5.49
Result:
Binomial Option Pricing Model
The binomial model prices options by creating a binomial tree of possible future stock prices and working backward to determine the option value at each node.
Example: Single-period model with S=100, K=100, u=1.1, d=0.9, r=5%:
p = (e^(0.05) - 0.9) / (1.1 - 0.9) = 0.756
C = e^(-0.05) × [0.756×10 + 0.244×0] ≈ 0.951 × 7.56 = $7.19
Option Value = e^(-rΔt) × [p×Value_up + (1-p)×Value_down]
Result:
Options Payoff Calculations
Call Option Payoff
The payoff for a call option buyer is the maximum of zero or the difference between the underlying price and the strike price, minus the premium paid.
Example: Buy call with K=100, premium=5. If stock price at expiry is 120:
Payoff = max(0, 120-100) - 5 = 20 - 5 = $15 profit
Short Call Payoff = Premium - max(0, S - K)
Result:
Put Option Payoff
The payoff for a put option buyer is the maximum of zero or the difference between the strike price and the underlying price, minus the premium paid.
Example: Buy put with K=100, premium=4. If stock price at expiry is 80:
Payoff = max(0, 100-80) - 4 = 20 - 4 = $16 profit
Short Put Payoff = Premium - max(0, K - S)
Result:
Covered Call Payoff
A covered call involves owning the underlying asset and selling a call option against it. This strategy generates income but limits upside potential.
Example: Buy stock at 100, sell call with K=110, premium=3. If stock at expiry is 120:
Payoff = (110-100) + 3 = 10 + 3 = $13 profit
Result:
Options Greeks
Delta (Δ)
Delta measures the sensitivity of an option's price to changes in the price of the underlying asset. Call deltas range from 0 to 1, put deltas from -1 to 0.
Example: For a call option with delta 0.6, if the stock price increases by $1, the option price will increase by approximately $0.60.
Δput = N(d1) - 1
Result:
Gamma (Γ)
Gamma measures the rate of change of delta with respect to changes in the underlying price. It's highest for at-the-money options and decreases as options move in or out of the money.
Example: If an option has a gamma of 0.05, and the stock price increases by $1, the delta will increase by 0.05.
Result:
Theta (Θ)
Theta measures the sensitivity of an option's price to the passage of time (time decay). Options lose value as expiration approaches, with theta quantifying this daily loss.
Example: If an option has a theta of -0.05, its price will decrease by approximately $0.05 per day, all else being equal.
Θput = - (S × N'(d1) × σ) / (2√T) + r × K × e^(-rT) × N(-d2)
Result:
Options Trading Strategies
Straddle Strategy
A straddle involves buying both a call and a put option with the same strike price and expiration date. This strategy profits from significant price moves in either direction.
Example: Buy call and put with K=100, call premium=5, put premium=4. Total cost=9.
If stock at expiry is 120: Payoff = (120-100) - 9 = 11 profit
If stock at expiry is 80: Payoff = (100-80) - 9 = 11 profit
Result:
Strangle Strategy
A strangle involves buying out-of-the-money call and put options with different strike prices. It's cheaper than a straddle but requires larger price moves to profit.
Example: Buy call with K=110 (premium=3) and put with K=90 (premium=2). Total cost=5.
If stock at expiry is 120: Payoff = (120-110) - 5 = 5 profit
If stock at expiry is 80: Payoff = (90-80) - 5 = 5 profit
Result:
Bull Call Spread
A bull call spread involves buying a call option at a lower strike price and selling a call option at a higher strike price. This strategy profits from moderate price increases with limited risk.
Example: Buy call with K=100 (premium=5), sell call with K=110 (premium=2). Net cost=3.
If stock at expiry is 115: Payoff = (115-100) - (115-110) - 3 = 15 - 5 - 3 = 7 profit
Result:
Arbitrage Relationships
Put-Call Parity
Put-call parity defines the relationship between the prices of European put and call options with the same strike price and expiration date. Violation of this relationship creates arbitrage opportunities.
Example: Stock=100, Call=10, Put=5, Strike=100, Rate=5%, Time=1 year.
According to parity: Call + PV(Strike) = Put + Stock
10 + 100×e^(-0.05) = 10 + 95.12 = 105.12
Put + Stock = 5 + 100 = 105 → Arbitrage opportunity exists
where PV(K) = K × e^(-rT)
Result:
Call Option Lower Bound
The price of a European call option must be at least the maximum of zero or the difference between the stock price and the present value of the strike price.
Example: Stock=100, Strike=95, Rate=5%, Time=1 year.
PV(K) = 95×e^(-0.05) ≈ 90.48
Lower bound = max(0, 100 - 90.48) = 9.52
If call price < 9.52, arbitrage opportunity exists
where PV(K) = K × e^(-rT)
Result:
Put Option Lower Bound
The price of a European put option must be at least the maximum of zero or the difference between the present value of the strike price and the stock price.
Example: Stock=100, Strike=110, Rate=5%, Time=1 year.
PV(K) = 110×e^(-0.05) ≈ 104.64
Lower bound = max(0, 104.64 - 100) = 4.64
If put price < 4.64, arbitrage opportunity exists
where PV(K) = K × e^(-rT)
Result:
Options & Futures Payoff Diagrams
Visualize how different strategies perform at various underlying prices