Forwards & Futures

Forward Contracts

Definition: A commitment to purchase at a future date a given amount of a

commodity or an asset at a price agreed on today.
● The price fixed now for future exchange is the forward price
● The buyer obtains a ‘long position’ in the asset/commodity

Features of forward contracts:

● Traded over the counter (not an exchange)
● Custom tailored
● No money changed hands until maturity
● Non-trivial counterparty risk

Example: Buying 10,000 bushels of soybeans at $3.50/bushel for delivery in 3

months

Future Contracts
Definition: An exchange-traded, standardised, forward-like contract that is marked to
the
market daily.

Features of futures contracts:

● Standardised contracts:
● Traded in exchanges
● Guaranteed by the clearing house - little counterparty risk.
● gains/losses settled daily - marked to market
● Margin account required as collateral to cover losses

Example: Buying 10 December live-cattle contracts (40,000 lbs each) at $0.7455/lb.

Pricing Forwards and Futures

Arbitrage ensures that the cost of buying and storing the underlying asset must
equal the price of entering into a forward or futures contract. If this relationship
doesn’t hold, there would be an arbitrage opportunity:

 For example, if the forward price is too high relative to the storage cost and
benefits, investors could buy the asset in the spot market, store it, and sell a
forward contract to lock in a risk-free profit.
 Conversely, if the forward price is too low, investors could sell the asset in the
spot market, enter into a forward contract to buy it at the lower price, and
make a profit.
Commodity Futures

Commodity Futures Market Conditions

Financial Futures

● Since the underlying asset is a portfolio in the case of index futures, trading in the
futures market is easier than trading in cash market.
● Thus, futures prices may react quicker to macro-economic news than the index
itself.
● Index futures are very useful to market makers, investment bankers, stock portfolio
managers:
○ Hedging market risk in block purchases and underwriting
○ Creating synthetic index fund
○ Portfolio insurance

EXAMPLE
Hedging with Forwards
Hedging with forward contracts is simple, because one can tailor the contract to
match the maturity and size of position to be hedged

Example.
Suppose that you, the manager of an oil exploration firm, have just struck oil. You
expect that in 5 month’s time you will have 1 million barrels of oil. You are unsure of
the future price of oil and would like to hedge your position.

 Regardless of the actual spot price S5 in 5 months, the firm is guaranteed to

sell the oil at the agreed forward price F(5).

Hedging with Futures

One problem with using forwards to hedge is that they are illiquid. Thus, if after 1
month you discover that there is no oil, then you no longer need the forward contract.
In fact, holding just the forward contract you are now exposed to the risk of oil-price
changes. In this case, you would want to unwind your position by buying back the
contract. Given the illiquidity of forward contracts, this may be difficult and expensive

To avoid problems with illiquid forward markets, one may prefer to use futures
contracts.

Example. In the above example, you can sell 1 million barrels worth of futures.
Suppose that the size of each futures contract is 1,000 barrels. The number of
contracts you want to short is

Futures Hedging Mismatches

Futures are standardised, so may not perfectly match your hedging need. The
following mismatches may arise when hedging with futures:
● Maturity
● Contract size
● Asset

Thus, a perfect hedge is available only when

1. the maturity of futures matches that of the cash flow
2. the contract has the same size as the position to be hedged
3. the cash flow being hedged is linearly related to the futures’

In the event of a mismatch between the position to be hedged and the futures
contract, the
hedge may not be perfect.
Options
Introduction to Options

Call Option: Right to buy an asset at a specified price before expiration

Put Option: Right to sell an asset at a specified price before expiration

Exercise Styles:
● European: Can only be exercised on the expiration date
● American: Can be exercised any time before expiration

Key elements in defining an option:

● Underlying asset and its price S
● Exercise price (strike price) K
● Expiration date (maturity date) T (today is 0)
● European or American

Option Payoff
 long call: make money
 short call: lose money

 long put: lose money

 short put: make money
Graphical payoffs
Corporate Securities as Options

Many corporate securities can be viewed as options:

Common Stock: A call option on the assets of the firm with the exercise price being
its bond’s redemption value.

Bond: A portfolio combining the firm’s assets and a short position in the call with
exercise price equal bond redemption value.

Warrant: Call options on the stock issued by the firm.

Convertible bond: A portfolio combining straight bonds and a call option on the firm’s
stock
with the exercise price related to the conversion ratio.

Callable bond: A portfolio combining straight bonds and a call written on the bonds.
Use of Options

Hedging: Protect downside while keeping upside (eg. using put options)

Options’ Notation

Price Bounds of Options

1. C ≥ 0 (Call option price is never negative)
- Since a call option gives the right but not the obligation to buy the
stock, the worst possible outcome for the option holder is that the option
expires worthless.

2. C ≤ S (The payoff of stock dominates that of a call)

- a call option only gives the right to buy the stock, which means it cannot
be more valuable than owning the stock outright.
3. C ≥ S − KB (assuming no dividends)
4. Combining the above, we have max[S − KB, 0] ≤ C ≤S.

 The lower bound ensures the call option price is at least the intrinsic value
(S−KB) or zero.
 The upper bound ensures that the option price never exceeds the stock
price.
Put-Call Parity

The Put-Call Parity theorem is a fundamental relationship in options pricing that

links the prices of European call and put options with the underlying stock and a
risk-free bond. It ensures no arbitrage in the market.

American Options and Early Exercise

Dividends & Asset Volatility

Black-Scholes Formula

 price European options

EXAMPLE
At-the-money Pricing Shortcut

The Greeks
Delta
Gamma

Vega

Theta
Rho

Implied Volatility

Volatility Smile

If the Black-Scholes assumptions held, all options on an asset would have

the same implied volatility. In practice, implied volatilities vary with strike
prices. This pattern is known as a volatility smile, which is typically U-
shaped, meaning deep ITM and OTM options have higher implied volatility
than ATM options.

Causes:
● Skewness and fat tails in asset return distributions
● Market demand/supply imbalances for different strike options
● Cash risk perception (eg. higher demand for puts as protection)
Implications:
● Violates Black-Scholes assumption of constant volatility
● Traders adjust models using stochastic volatility or jump diffusion
The volatility smile suggests that the market does not assume the
underlying asset follows a lognormal distribution (as assumed by Black-
Scholes Model). Instead, it reflects the market’s view of the true probability
distribution of the underlying asset price at expiration.

Key Implications:

1. Fat Tails (Higher IV for ITM & OTM Options)

 The implied volatility (IV) curve for options often forms a smile shape, meaning that
in-the-money (ITM) and out-of-the-money (OTM) options have higher IV than at-the-
money (ATM) options.
 This implies that the market expects extreme price movements (both big jumps up
and crashes) to occur more frequently than what a lognormal distribution (assumed
in Black-Scholes) would predict.
 Black-Scholes assumes that asset returns follow a normal (lognormal in price)
distribution, but real markets tend to have "fat tails"—more extreme moves than a
normal distribution would suggest.

2. Skewness (When Smile is Asymmetric, aka "Skew")

 If the IV curve is not symmetric, it forms a "skew" instead of a perfect smile.
 In equity markets, OTM put options tend to have higher IV than OTM call options.
This suggests that market participants fear large downside moves more than large
upward moves.
 This leads to negative skewness in the risk-neutral distribution, meaning the
probability of a sharp decline (crash) is higher than a sharp rise.
 This is because investors often buy put options for portfolio protection, increasing
their demand (and price), which raises IV.

3. Implied Risk-Neutral Distribution

 The Black-Scholes model assumes a lognormal distribution, meaning asset prices
have a symmetrical probability of moving up or down in percentage terms.
 In practice, we can invert the Black-Scholes formula to extract the market-implied
probability distribution of future prices.
 If IV varies across strikes (forming a smile or skew), the implied distribution is not
lognormal—instead, it often shows:
o Fat tails (suggesting higher chances of extreme moves).
o Skewness (if IV is higher for puts, the distribution is negatively skewed,
meaning greater downside risk).
 This means that the risk-neutral world, in which options are priced, embeds the
market's expectations of jumps, crashes, and non-normal behavior.