Derivatives and Risk Management

Put-Call Parity

Stock + Put = Call + Present Value of Strike Price (Long Bond)

Put-Call-Forward Parity

Forward = Spot Price * rate, therefore you can replace S0 with the equivalent formula

Delta

change in option’s price for a change in the underlying

positive for calls, negative for puts

Δ > 0 → bullish → benefit from price increase

Δ < 0 → bearish → benefit from price decrease

ATM options have a Δ of ~0.5

Gamma

change in option’s delta for a change in the underlying

Always positive for long options

Vega

Change in option’s price for a change in volatility

Always positive for long options

Theta

change in option’s price as time passes (time decay)

Negative for long options

Value at Expiration (Call and Put)

Profit at Expiration (Call and Put)

Value = profit minus initial net cost (or plus initial net proceeds)

Bull spread

increases in value as the underlying stock price rises

• Bull Call Spread: Buy a lower-strike call and sell a higher-strike call. You pay a net premium (debit).

• Bull Put Spread: Sell a higher-strike put and buy a lower-strike put. You receive a net premium (credit).

Purpose: Profit from an increase in the underlying stock price while limiting risk.

Bear Spread

bearish options strategy that increases in value as the underlying stock price falls. It involves buying one option and selling another of the same type and expiration, but with different strike prices

• Bear Put Spread: Buy a higher-strike put and sell a lower-strike put. You pay a net premium (debit).

• Bear Call Spread: Sell a lower-strike call and buy a higher-strike call. You receive a net premium (credit).

Purpose: Profit from a decline in the underlying stock price while limiting risk.

Straddle

Buy a call and a put (long straddle) at the same strike (typically as close to ATM as possible)

The idea is you know there is a new event coming up but do not know if it is good or bad.

• You want the buy the straddle as close to the news event as possible and try to profit of movements to either side

• Straddles rarely make money, and the potential movement is likely priced in

Collar

Long stock + long protective put + short covered call, typically with the same expiration date

• Long put: Protects against downside losses (sets a minimum selling price).

• Short call: Generates premium to offset the put's cost but caps upside gains.

A collar limits both potential losses and potential gains

You could have bought the stock awhile ago at a cheaper price and lock in the collar later

Calendar Spreads

Buy and sell the same type of options (same strike, same underlying) but with different expiration dates

sell near-term, buy long-dated = long calendar spread

buy near-term, sell long-dated = short calendar spread

Both can be done with calls or puts

theta play = near term option TV decays faster

Historical Volatility

Derived from past return data of the underlying

Historical Volatility Formula

Implied Volatility

Derived from the option pricing model

• option price vs. strike, exp date, rf, S0 ← all observable

• therefore, you can get the implied volatility

Formula is to get from monthly to annual implied volatility. Will not need to calculate implied volatility.

Covered call

Own the shares sell the call

Covered put

Short the stock and sell a put option (short put)

profits from a declining stock price, but the short put caps your gains below the strike price, while losses remain unlimited if the stock rises

Protective Put

Protective put = long stock + long put

Protective Call

Protective call = short stock + long call

Adjusting Portfolio Duration Formula

Interest Rate Forwards (FRAs) and Futures — Payoff Formula

Forwards (FRAs)

OTC

Customizable

Counter part risk

Futures

Market-to-market

Fixed-Income-Futures

Preferred way to hedge bond positions

BPV-Based Hedge Ratio (BPVHR)

Currency Swap

At initiation and expiration, notionals are exchanged at a fixed fx-rate

Periodic interest payments are exchange

Cross-currency basis swap

allows two parties to exchange principal and floating interest payments in different currencies

Equity Swaps

Can be for a single stock, basket of stocks or stock index

Total Return vs Price Return

Total return includes dividends

Cash vs Physical Settlement

Cash Settlement: At expiration, the parties exchange cash equal to the contract's gain or loss, rather than delivering the underlying asset.

Physical Settlement: At expiration, the underlying asset is delivered in exchange for the agreed payment.

Changing the Beta of a Portfolio

BT - BS < 0 = short position

BT - BS > 0 = long position

Cash Equitization

Using equity index futures to give a portfolio's cash holdings exposure to the equity market without immediately purchasing stocks.

Formula simplifies can cash has a beta of 0

Variance Swaps

Directional bets on implied versus actual volatility

Probability of Change in the Federal Funds Rate