Hedging and Speculation with Futures and Options

Hedging with S&P 500 Index Futures

SIF hedging: NF = V_F(antiguo) = F_o x Z. NF (Number of contracts) = V_p (portfolio value) / (V_F x B_p) (Beta). With this hedge, risk is removed. If the index goes up, the profit in the portfolio will be offset by losses in SIFs contracts and vice versa.

Example: S&P 500 falls 5%:

• In the portfolio: R_p = B_p x (-5%) = -10% of a portfolio of $10M, resulting in $9M (Final Value).
• In the SIFS contracts: S_i = 980 (S&P 500 value given) x 0.95 (100% - 5%) = 931. F_i = 931 x (1 + 0.04 x 5/12 (next month timeframe)) = 946.52. V_F = NF x (F_i - F_o) x Z (250).
• Gain on futures: V_F(New) / Equity portfolio value.
• Outcome stock portfolio: 2 x (-5%) = -10%.
• Final Value portfolio: $10M - $1M + V_F(New).

Margin Payments

Margin payments are a mechanism to eliminate counterparties' risk. An initial margin payment is required (% of the future value), and profits and losses are recognized daily. If the balance falls below the maintenance margin, additional cash is required to reach the initial margin level again.

Options: Risks and Greeks

Main Risks for Option Sellers

• Calls: The main risk is S_T (stock price at maturity) being greater than the Strike price (K), so that they will be exercised, and the seller will lose money.
• Puts: The main risk is K (strike price) being greater than S_T (market price).

Option Greeks

• Delta: An estimation of how much the portfolio value (calls and puts) may change given a $1 change in the underlying asset.
• Gamma: The change in the option's delta as the underlying asset price (S) changes.
• Vega: The change in the option's price as the volatility of the stock changes.

Combining Options

Long Straddle (Volatility Bet)

A bet on the stock price moving a large amount in either direction. It is done by buying one call and one put on the same underlying asset with the same Strike price and Time to maturity.

Long call (c=5) + long put (p=3). Total cost: 8.

Short Straddle

The opposite of a long straddle.

Bear Spread

A directional bet on the underlying stock price. It is done by buying put options to bet on the stock price decreasing (S_t < K), and the long put also sets a lower limit to your downside losses. But besides, you choose to "cap" your upside potential by selling a put with K₂ < K₁.

Speculation

Long Bull Spread with Calls

A directional bet. It is done by buying call options to gamble on the stock price increasing (S_t > K), and the long call also sets a lower limit to your downside losses. But besides, you choose to "cap" your upside potential by selling a call with K₂ > K₁.

1. S_t < K₁: loss = c₁ - c₂
2. K₁ <= S_t <= K₂: Profit = (S_t - K₁) - (c₁ - c₂). S_BE = K₁ + (c₁ - c₂)
3. S_t > K₂: Profit = (K₂ - K₁) - (c₁ - c₂).

At t=0 you pay c₁ - c₂ > 0 to set up the bull spread.

Hedging with Futures

Hedging a Long Position in Stocks

You hold stock and hedge by shorting futures because you fear a price fall in the next few months. You ensure that any loss in the stock will be offset by a gain on your futures trade. Hence you "lock in" an "effective price" which is close to F_o (Initial futures price). This is possible because S and F move similarly.

Hedging a Short Position in Stocks

You have gone short, so you will need to buy stocks in the future, and you fear a rise in prices. You hedge by buying futures. If stock prices increase, the higher cost of buying them is offset by closing out the future contracts, so the effective cost of buying the stocks is close to F_o (initial futures price).

Beta

Beta is a measure of the response of the return on your stock portfolio to a change in the market return.

Insurance with Long Call Options

Pension funds intend to buy because stocks will rise. If S_t > K, exercise the option.

Interest Rate Swap Valuation Changes

Factors influencing the market value of a 10-year interest rate swap and the fixed leg:

1. Market interest rates
2. Time value of money
3. Credit risk (if the counterpart defaults)
4. Yield curve (changes)
5. Liquidity considerations

The fixed leg of the swap tends to have a more significant impact compared to the floating leg due to its sensitivity to interest rate movements.