# Spreads and Collars

Put-Call Parity Equation:

\$Call [K,T] – Put [K,T] = PV [ F_{0,T} – K ] = {F_{0,T} }^P – Ke^{-rT}\$

Spread: Position consisting of only calls or only puts.

1. Long call [K1,T] + Short call [K2,T]
2. Long put [K1,T] + Short put [K2,T]
3. Perform well when the price of the underlying asset goes up
1. Short call [K1,T] + Long call [K2,T]
2. Short put [K1,T] + Long Put [K2,T]
3. Perform well when the price of the underlying asset goes down
1. Synthetic Long forward + Synthetic short forward
2. Strategy guarantees a cash flow in the future
3. A means of borrowing  or lending money
• Option market-makers have low transaction costs and can sell box spreads which is equivalent to borrowing
• They’re alternatives to buying a bond
4. Costly but no stock price risk
5. Payoff \$=K_{2} – K_{1}\$ – IRRESPECTIVE of the underlying asset
1. m Long call [K1,T] + n Short call [K2,T]  [SAME underlying asset]
2. m Long put [K1,T] + n Short put [K2, T]

Collars:

• Purchase Collar: [AM/OM] Long Put [K1,T] + [OM]Short Call [K2,T] , K2> K1
• Collar width = Diff. of strike prices
• Written Collar: [AM] Short Put [K1,T] + [OM] Long Call [K2,T] , K2> K1
• Zero cost collar:
1. Pick the desired put strike below or at the forward price
2. Find a strike above the forward price such that a call has the same premium
• Collared Stock:
1. This position entails buying the stock, buying a put, and selling a call.
2. It is an insured position because we own the asset and buy a put.
3. The collared stock looks like a bull spread; however, it arises from a different set of transactions. The bull spread is created by buying one option and selling another. The collared stock begins with a position in the underlying asset that is coupled with a collar.

• Non directional speculations – Holder does not care whether the stock price goes up or down, but only how much it moves.
• Long call [K1,T] + Long put [K1,T]
• Advantage: Benefits from stock price movement in either direction
• Bet on High volatility – Investor believes volatility is HIGHER than the market’s assessment
• Cost of straddle will be greater when markets perception is that volatility is greater

• Short call [K1,T] + Short put [K1,T]
• Bet on low volatility – Investor believes volatility is LOWER than the market’s assessment
• Highly risky since a large change in the stock price leads to potentially unlimited loss

Strangle:

• Investor holds a position in both a call and a put with different strike prices but with the same maturity and underlying asset.
• [OM] Long Call [K1,T] + [OM] Long Put [K2,T]
• Reduced premium as compared to the Straddle
• This option strategy is profitable only if there are large movements in the price of the underlying asset.
• Use this strategy when you speculate a large price movement in the near future but unsure which way the movement will be.