Option Strategies

Options and Futures

Zhiyu Fu

Financial Engineering

• The Goal of financial engineering: Create financial instruments to meet various investment goals

• The building ground: Uncertain stock price \(S_T\) representing the different realizations of the “state” of the world

• The building blocks:

  	Stock	Bond	Call Option	Put Option
  Price	\(S_0\)	\(e^{-rT}\)	\(C_0\)	\(P_0\)
  Payoff at \(T\)	\(S_T\)	\(e^{rT}\)	\(\max\{\) \(S_T\) \(- K, 0\}\)	\(\max\{K -\) \(S_T\) \(, 0\}\)
  Profit at \(T\)	\(S_T\) \(-S_0 e^{rT}\)	\(0\)	\(\max\{\) \(S_T\) \(- K, 0\} -C_0 e^{rT}\)	\(\max\{K -\) \(S_T\) \(, 0\} -P_0 e^{rT}\)

• The role for engineers:

  • Use the building blocks to create different payoffs;
  • Price the new product based on the building blocks;

Note on Profit

• We define profit as the payoff minus the opportunity cost of money, e.g., \[S_T - S_0 e^{rT},\] with proper time discounting.
  • In other words, the profit is defined as the excess return over the risk-free rate.
• Some references define the profit as the payoff minus the original cost, e.g., \[S_T - S_0,\] without taking into account the time value of money. This profit includes the risk-free return.
  • Economically less meaningful: positive profit can still mean losing money relative to the risk-free benchmark.
  • Especially problematic when there are multiple periods involved.
• When \(r\) is \(0\), these two definitions are equivalent; when not specified, feel free to use \(r = 0\) when asked to compute the profit.

Road Map

• Replicating positions using building blocks
• Create payoff structures
  • Hedge strategies: Covered Call, Protected Put
  • Combinations: Straddle, Strangle
  • Spreads: Bull Spread, Bear Spread, Butterfly Spread
• Your goal:
  • No need to memorize;
  • Given a payoff diagram, understand the purpose of the strategy;
  • Given a payoff diagram, build it from the building blocks and price it;

Long Position in the Stock

• We can create this position \(K\) by:

  1. Buy a stock at \(S_0\) and borrow \(K e^{-rT}\)

     • Payoff at \(T\): \(S_T - K\)

  2. Buy a call and short a put at the strike price \(K\)

     	\(+\)Call	\(-\)Put	Total
     \(S_T < K\)	\(0\)	\(K - S_T\)	\(S_T- K\)
     \(S_T \ge K\)	\(S_T - K\)	\(0\)	\(S_T - K\)

  3. Buy a forward contract with the delivery price \(K\)

     • Payoff at \(T\): \(S_T - K\)
• Identical payoffs, identical prices
  • \(S_0 - K e^{-rT} = C_0 - P_0\)
• A standard future contract choose \(K = F\) such that \(S_0 - F e^{-rT} = 0\) \(\implies F = S_0 e^{rT}\)

Short Position in the Stock

• We can create this position \(K\) by:

  1. Short-sell a stock at \(S_0\) and lend \(K e^{-rT}\)

     • Payoff at \(T\): \(K-S_T\)

  2. Short a call and buy a put at the strike price \(K\)

     	\(-\)Call	\(+\)Put	Total
     \(S_T < K\)	\(0\)	\(K - S_T\)	\(K-S_T\)
     \(S_T \ge K\)	\(-(S_T - K)\)	\(0\)	\(K-S_T\)

  3. Short a forward contract with the delivery price \(K\)

Synthetic Call

• Use Put-Call Parity: \[C = P + S_0 - K e^{-rT}\] we can recreate a call by long stock, short bond, and long put

• Check the payoff:

  	Stock	Bond	Put	Total
  \(S_T < K\)	\(S_T\)	\(-K\)	\(K - S_T\)	\(0\)
  \(S_T \ge K\)	\(S_T\)	\(-K\)	\(0\)	\(S_T - K\)

• Exercise: Create a synthetic put

Covered Call

Covered Call: Sell the call and hedge it with the stock

• Stock traded at $50;
• A call option with \(K\) $60 and maturity 3 months sells for $3.45.
• The cost of the covered call is \(50 - 3.45 = 46.55\).
• The payoff and profit for different stock prices at maturity is:

Stock Price	\(40\)	\(60\)	\(80\)
Long Stock	\(40\)	\(60\)	\(80\)
Short Call	\(0\)	\(0\)	\(-20\)
Payoff	\(40\)	\(60\)	\(60\)
Profit	\(-6.55\)	\(13.45\)	\(13.45\)

Covered Call (II)

• Payoff structure:
  • If \(S_T \le K\): like a stock;
  • If \(S_T > K\): capped at \(K\)

Just like a put. In fact \[S_0 - C = K e^{-rT} - P\] It can be replicated by a put and a bond.

• Why sell covered call?
  • Earn a premium from selling the call
  • Limit the potential loss
• Alternatively, sell a put and save in risk-free bonds

Break-even Point

• Break-even means the profit (net of interest) is zero

\[ \text{Profit} = \underset{\text{Payoff}}{\underbrace{\min\{S_T, K\}}} - \underset{\text{Cost}}{\underbrace{(S_0 - C_0) e^{rT}}} = 0 \]

• Statement: The break-even point is between \(0\) and \(K\) \[0 < (S_0 - C_0) e^{rT} < K\]
• Why?
  • Use the upper/lower bounds on a call options;
  • Or, use the put-call parity

The General Statement on the Break-even Point

Implication of No-Arbitrage Conditions

Suppose a portfolio have the price \(X\) and the payoff \(Y_T\) that is a random variable that have an upper bound \(\bar Y\) and a lower bound \(\underline Y\). Given the risk-free rate \(e^{rT}\), then we must have:

\[ \underline Y < X e^{rT} < \bar Y \]

• Reasoning:
  • Recall the no-arbitrage condition requries that if one portfolio offers a higher payoff under any circumstance than the other, then it must be more expensive than the other portfolio.
  • Corollary: A zero-cost portfolio cannot have a payoff that is strictly positive or negative.
• Apply to this scenario:
  • Suppose \(X e^{rT} < \underline Y\), then we borrow \(X\) to buy the portfolio (zero cost), and have strictly positive payoffs after paying back the loan.
  • Suppose \(X e^{rT} > \bar Y\), then I can short-sell the portfolio and lend out money to gurantee to buy it back with extra savings;

Protective Put

cover a stock position with a put

• Example: A stock traded at $50. A put option with strike $40 and maturity 3 months sells for $1.28.
• Payoff structure:

Stock Price	\(30\)	\(50\)	\(70\)
Stock Payoff	\(30\)	\(50\)	\(70\)
Stock profit	\(-20\)	\(0\)	\(20\)
Payoff w/ Put	\(40\)	\(50\)	\(70\)
Total Profit	\(-11.28\)	\(-1.28\)	\(18.72\)

• Can also be replicated by a call option and a bond

Straddle

Straddle: Long call + Long put at the same \(K\)

• Why: bet on volatility
  • Consider a stock with expected price to be \(K\).
  • If the stock does not move, no payoff from the straddle;
  • With volatility in either direction, the straddle will have a payoff.

Strangle

• Strangle: Long call + long put at different \(K\)
• Why: bet on a even larger volatilty
  • Consider a stock with expected price to be \(K\).
  • If the stock moves withina range, no payoff from the straddle;
  • With a large volatility in either direction, the straddle will have a payoff.

Spreads

Bull Spreads

• An option spread is an option strategy in which the payoff is limited;
• Bull spread: Long call with a low \(K_l\) + short call with a high \(K_h\)
• Payoffs:
  • Maximum: \(K_h - K_l\)
  • Minimum: \(0\)
  • Bullish between \(K_l\) and \(K_h\)
• Why using bull spread?
  • Limit both the upside and the downside

Bull Spreads (II)

• The cost between the bounds of payoffs: \[0 < (C_l - C_h) < (K_h - K_l) e^{-rT}\]
• \(C_l > C_h\): the lower the strike, the more likely to be exercised, and the higher is the payoff;
• \((C_l - C_h) < (K_h - K_l) e^{-rT}\): The maximum benefit from a lower strike price is the difference in payoffs, assuming the the lower strike can always be exercised;

Bear Spread

• Bear spread: Long put with a high \(K_h\) + short put with a low \(K_l\)
• Why using bear spread?
  • Limit both the upside and the downside
• Cost between the bounds of payoffs: \[0 < (P_h - P_l) < (K_h - K_l) e^{-rT}\]
  • \(P_h > P_l\): the higher the strike, the more likely to be exercised, and the higher is the payoff;
  • \((P_h - P_l) < (K_h - K_l) e^{-rT}\): The maximum benefit from a higher strike price is the difference in payoffs;

Butterfly Spread

• Butterfly spread: Long call with a low \(K_l\) + Long call with a high \(K_h\), and short two calls with a middle strike \(K_m\equiv \frac{K_l + K_h}{2}\)
• Payoffs: Maximum profit at \(K_m\) is \(\frac{K_h - K_l}{2}\)
• Why using butterfly spread?
  • Betting on the price to be close to \(K_m\)
• Cost: \(0 < (C_l - 2C_m + C_h) e^{rT} < (K_h - K_l)\)
• Option price is convex in the strike price: \[ C_m < \frac{C_l + C_h}{2} \]
• Exercise: Construct a butterfly spread using puts

A Philosophical Note on the Butterfly and the Complete Market

First Welfare Theorem

Under complete markets, complete information and perfect competition, the market allocation is Pareto optimal.

• Complete markets: Every asset in every possible state of the world can be freely traded;
• The role of financial market: Allow resources to be traded across time and states;
  • Debt: Trade resources across time;
  • Stocks: Trade the future risky income with current capital investment;
• The role of financial derivatives: make the market more complete
• Butterfly spreads: pays off only between \((K_l, K_h)\)
• As \(K_l\) and \(K_h\) get closer, theoretically we are able to trade the resource in every possible state of the world \(S_T\)!