[5 min A Book]Options Futures and Other Derivatives by John C. Hull
9 min readApr 6, 2020
- Exchange-trade markets: individuals trade standardized contracts that have been defines by the exchange. Eg: CME group
- Over-the-counter markets: two parties agree on a trade and present it to a central counterparty (CCP) or clear the trade bilaterally. Eg: banks, funds etc.
- Forward: buy or sell an asset at a certain future time for a certain price. Traded in the OTC market aka private contrasts between 2 parties.
Gain or loss is realized at the end of the life of the contract, so some credit risk. - Future: buy or sell an asset at a certain future time for a certain price. Traded in the exchange market aka public standardized contract.
Gain or loss is realized day by day, so usually close out prior to maturity with virtually no credit risk. - Payoff for both forwards and futures:
Long position: St-K = current spot price-future delivery price. Holder must buy an asset worth St for K
Short position: K-St = future delivery price-current spot price. Holder must sell an asset worth St for K - Price for both forwards and futures:
F=S * e^(interest-yield)*time - Value of long forward contracts:
=S * e^(-yield)*time -K *e^(-interest)*time - Call option: holder’s right to buy the underlying asset by a certain date for a certain price
- Put option: holder’s right to sell the underlying asset by a certain date for a certain price
- Short hedge: short a future to neutralize the risk of an asset that will be sold in the future. Price up, asset’s gain offsets future’s loss. Price down, future’s gain offsets asset’s loss.
- Long hedge: long a future to neutralize the risk of an asset that will be purchased in the future. Price up, future’s gain offsets asset’s loss. Price down, asset’s gain offsets future’s loss.
- Basic hedging risks: (1) uncertain date of buy/sell the asset (2) uncertain close-out date of the futures (3) asset price to be hedged ≠ price used for the future contracts
- Cross hedging: want to buy/sell apples, hedge with orange futures
- Hedge ratio: # of orange futures to hedge 1 apple.
Minimum variance hedge ratio h = ρ *σS/σF
= coefficient of correlation between spot price and future price * st. dev. of ∆spot price/∆future price
= 1 if buy/sell apples and hedge with apples. - Optimal Number of Contracts N* = h * Qa/Qf
= h * size of assets/size of one futures contract - One-day hedge ratio for futures = h *( S*Qa/F*Qf) = h * Va/Vf
= h * dollar value of asset/futures - Hedging an equity portfolio that mirror the index: N* = Va/Vf
= current value of the portfolio/current value of one futures contract (futures price*the contract size) - Not mirror the index: N* = β * Va/Vf
CAPM (capital asset pricing model)
β=excess return on portfolio/excess return on index
β up = more sensitive to the movements in the index = more futures needed to hedge the portfolio - Treasury Rates: rates an investor earns on Treasury bills and Treasury bonds
LIBOR: London Interbank Offered Rate. It is an unsecured short-term
borrowing rate between banks
Fed Funds Rate: overnight rate of borrowing and lending between financial institutions
Repo Rates: secured borrowing rates
Risk-Free Rate: risk-free interest rate to valuate derivatives - Bond duration: how long to wait to receive cash payments.
An n-year Zero-coupon bond’duration=n.
Bond price = Σ cash flow at ti * e^ (-yield*ti) = PV of all cash payments
D = [Σ ti * cash flow at ti * e^ (-yield*ti) ]/bond price
= weighted average of the times when payments are made - Duration applies only to small changes in yields
∆ bond price =-bond price * duration * ∆ yield
When yield up, bond price down. When yield down, bond price up - Modified Duration: when yield is not continuous but annual compounding of m times per year
D* = D/(1+y/m)
∆ bond price =-bond price * modified duration * ∆ discrete yield - Convexity is the 2nd derivative of bond price to yield
C = [Σ ti² * cash flow at ti * e^ (-yield*ti) ]/bond price
Large C = more sensitive to ∆ yield - Swap: an OTC agreement between two companies to exchange cash
flows in the future. Forward is a simplified swap. - Interest rate swap: transform a floating rate loan into fixed rate, or a fixed rate asset into floating rate.
- Market makers for swaps: usually large financial institutions; they are prepared to enter into a swap without having an offsetting swap with another counterparty.
- Swap rate: the average of (a) the fixed rate that a swap market maker is prepared to pay in exchange for receiving LIBOR (its bid rate) and (b) the fixed rate that it is prepared to receive in return for paying LIBOR (its offer rate).
- Swap value = fixed bond value -floating bond value (for floating payer)
=V-floating -V-fixed (for fixed-rate payer) - Fixed-for-fixed currency swap: transform USD bond into EUR bond
swap value = USD bond value-Spot exchange rate*EUR bond value(receive USD pay EUR)
= S0*EUR Bond-USD Bond (receive EUR pay USD) - LIBOR/Swap zero curve: used for discounting.
6-month, 12-month, and 18-month LIBOR/swap zero rates are 4%, 4.5%, and 4.8% with continuous compounding
2-year swap rate (pay semiannually) is 5%
It means that a bond with a principal of $100 and a semiannual coupon of 5% per annum sells for par. - Credit Default Swap: hedge credit risk in the same way for market risk
- ABS: a type of investment that is backed by a pool of debt, such as auto loans or home equity loans
- CDO (Collateralized Debt Obligation); a version of an ABS that may include mortgage debt as well as other types of debt.
- OIS (Overnight Indexed Swap): a fixed rate for x days is exchanged for the geometric average of the overnight rates during x days.
- OIS Zero Curve: 5-year OIS rate is 3.5% with quarterly settlements. A 5-year bond paying a quarterly coupon at a rate of 3.5% per annum would be assumed to sell for par.
- CVA (Credit value adjustment): PV of the expected loss to the bank due to a counterparty default
=Σ (prob. of early termination due to counterparty default * PV of expected loss) - DVA (Debt value adjustment): PV of the expected gain to the bank from its own default
=Σ (prob. of default by the bank * PV of expected gain) - CRA (Collateral rate adjustment): when interest rate≠ risk-free rate, PV(actual net interest-the net interest if interest rate = risk-free rate)
- constitutes an adjustment
- Portfolio value to the bank =no-default value -CVA+DVA-CRA
- Options
- Put-Call Parity: European option
call + PV(zero-coupon bond with strike price K ) =put+one share of stock
When St>K at time T, call(St-K) + K = put(0) + St
When St<K at time T, call(0) + K = put(K-St) + St
non-dividend: C+ K*e^(-r*t)=P + S0
dividend: C+D+ K*e^(-r*t)=P+ S0 - Bull spreads: bet stock price increase
long call (K1) + short call (higher K2)
most aggressive: both calls are initially out-of-money
most conservative: both calls are initially in-the-money - Bear spreads: bet stock price decrease
long put (K2) + short put (lower K1) - Box spreads: payoff always = K2-K1
When box price is low, buy box = long bull (K1, K2) + long bear (K1, K2)
When box price is high, sell box = short bull + short bear - Butterfly spreads: max profit when stock price =K2
long call/put (K1) + short call/put (K2)*2 + long call/put (K3)
When St<K1 or St>K3, payoff=0
When K1<St<K2, payoff = St-K1
When K2<St<K3, payoff = K3-St - Calendar spreads: max profit when stock price at T1 = K
short call/put (K, T1) + long call/put (K, T2)
neutral: choose K close to current stock price
bullish: higher K; bearish: lower K - Diagonal spreads: both K and T are different
short call/put (K, T) + long call/put (K’, T’) - Straddle: bet high volatility, max loss when St = K
long call (K, T) + long put (K, T) - Strips: bet large decrease in stock price, max loss when St = K
long call (K, T) + long put (K, T) *2 - Straps: bet large increase in stock price, max loss when St = K
long call (K, T) *2 + long put (K, T) - Strangles: bet high volatility, max loss when K1<St<K2
long put (K1, T) + long call (K2, T) - Binomial trees: to price an option, assuming stock price follows a random walk (formulas omitted)
- Brownian motion/Wiener process: a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per year
- Ito’s process: a generalized Wiener process in which the parameters a and b are functions of the value of the underlying variable x and time t.
- The price of any derivative is a function of the stochastic variables underlying the derivative and time
- Ito’s Lemma: describe the behavior of functions of stochastic variables
- Black-Scholes Model: pricing European options
Call=S*e^(-qT)*N(d1)-K*e^(-rT )* N(d2) = stock price with dividend rate q — PV(strike price if option exercised)
Put=K*e^(-rT )* N(-d2)-S*e^(-qT)*N(-d1) = PV(strike price if option exercised )— stock price with dividend rate q
Put-Call parity: Call + K*e^(-rT ) =Put + S*e^(-qT) - d1=[ln(S/K)+(r-q+σ²/2)*T] / [volatility*sqrt(T)]
d2=[ln(S/K)+(r-q- σ²/2)*T] / [volatility*sqrt(T)] - N(d2) = the probability that a call option will be exercised in a risk-neutral world
- Forwards and Futures: change S to F in the formulas
- Delta Δ = ∂call/∂S = price sensitivity = rate of change between the option’s price and a $1 change in the stock price
- Δ(call)=N(d1)
Δ(put)=N(d1)-1 - Theta Θ = ∂call/∂T = time sensitivity =rate of change between the option price and time
- Gamma Γ = ∂²call/∂S² = rate of change between an option’s Δ and the stock price
Γ(call)=Γ(put) - Vega v =∂call/∂σ = rate of change between an option’s value and stock price’s volatility
V(call)=V(put) - Rho ρ = ∂call/∂r = rate of change between an option’s value and a 1% change in the interest rate
- VaR = value at risk
We are X percent certain that we will not lose more than V dollars in the next N days
The variable V is the VaR, X% is the confidence level, and N days is the time horizon - Stress testing & back testing
- EWMA & GARCH(1,1) model
- CVA (credit value adjustment) = sum(Prob of default * PV of expected loss to the bank during ith interval)
- DVA (debt value adjustment)= sum(Prob of default * PV of expected loss to the counterparty/gain to the bank during ith interval)
- value of outstanding transaction = no-default value-CVA+DVA
- CDS (credit default swap)= trasfer credit risk = a contract where company A buys insurance from company B against company C (the reference entity) defaulting on its obligations
- Bermudan option: early exercise may be restricted to certain dates
- Gap option
Call: European call that pays off St-K1 instead of St-K2 when St > K2
Put: European put that pays off K1-St instead of K2-St when St < K2 - Asian option: payoff depends on the arithmetic average of the price of the underlying asset during the life of the option
- interest rate derivatives
- HJM, LMM
- energy and commodity derivatives
- CAPM (capital asset pricing model): project the potential returns of an investment portfolio
- Alpha: how well or badly a stock has performed in comparison to a benchmark index; the higher the better
- Alpha = 3 or -5 or 0 means the stock or fund did 3% better or 5% worse than the index or match the benchmark
- Alpha = (end price + distribution per share — start price)/start price
- Beta: how volatile a stock’s price has been in comparison to the market as a whole
- A high beta →risk tolerant + bigger return
A low beta →risk averse + steady return - Beta = 0.5 or 1.5 or 1 means a security’s price is 50% less volatile or 50% more volatile than the market or moves exactly as the market moves
- Beta = Cov of an asset’s return with market’s return/Var of market’s return
- Covariance is used to measure the correlation in price moves of any two stocks. A positive covariance means the stocks tend to move in lockstep, while a negative covariance means they move in opposite directions.
Variance refers to how far a stock moves relative to its mean. It is frequently used to measure the volatility of a stock’s price over time.
