Options Trading: A Comprehensive Guide to Options Strategies and Pricing Models

Posted on Apr 27, 2024 in Economy

Minimum and Maximum Values on American Calls

• Minimum: C_a(S₀,T,X) ≥ Max(0, S₀ – X)
• Maximum: C_a(S₀,T,X) ≤ S₀
• The maximum value of an American call is the stock price.  Hint: the intrinsic value for an in-the-money call is S₀ – X.  Think of the situation where X = 0.  This would imply an intrinsic value of S_0.

Lower Bound for European Calls

• Because they can only be exercised at expiration, the lower bound for a European option on stock with no dividend is
• C_e(S₀,T,X) ≥ Max[0,S₀ – X(1+r)^-T]. 
• Adjustment for dividends
• Adjustment for foreign currency calls

American Call versus European Call

• As an American call allows for more flexibility in exercise than an European call, its value must be greater than or equal to the value of the European call
• Thus, the lower bound for an American call on no-dividend stock becomes
• C_a(S₀,T,X) ≥ Max[0,S₀ – X(1+r)^-T]

Minimum and Maximum Values on American Puts

• Minimum:  P_a(S₀,T,X) ≥ Max(0, X – S₀)
• The minimum value of an American put is its exercise or intrinsic value.
• Maximum: P_a(S₀,T,X) ≤ X
• The maximum value of an American put is the exercise price.  Hint: the most one can gain from a put is X – S₀, and think of the situation where S₀ = 0.

Lower Bound and Maximum Value for a European Put

• Lower bound of a European put on no-dividend stock:
• P_e(S₀,T,X) ≥ Max(0, X(1+r)^-T – S₀)
• Dividend adjustment
• Adjustment for a currency put
• Maximum value of a European put:
• P_e(S₀,T,X) ≤ X(1+r)^-T

The BSM Option Pricing Model: Call Options

Put Pricing Model

Option Greeks:

Delta

• An option’s delta measures the sensitivity of the option’s price to a change in the price of the underlying.
• Delta can be computed using the following formulas:
• Because N(d₁) is a probability, 0 ₁)

Gamma

• Gamma measures the sensitivity of the option’s delta to a change in the price of the underlying. It is largest when the price of the underlying is near the exercise price.
• Gamma is computed the same way for calls and puts and is always positive.

Rho

• Rho measures the sensitivity of the option’s price to a change in the risk-free rate.
• Rho for a call is always positive so an increase in the risk-free rate results in an increase in the call premium. 
• Rho for a put is always negative so an increase in the risk-free rate results in a decrease in the put premium. 
• Neither the call price nor the put price is extremely sensitive to changes in the risk-free rate and this is particularly true with short-lived options

Vega

• Volatility is the most critical variable in the BSM-OPM because: (1) the option price is very sensitive to changes in volatility; and, (2) it is the only variable that is not directly observable.
• Vega is computed the same way for calls and puts and is always positive:
• Because all of other inputs to the option pricing model are directly observable, differences in option premium estimates must be caused by different forecasts for the volatility of the underlying. 
• It follows that many trades are volatility-based trades and traders are effectively betting that their estimate of future volatility is better than that of other traders.

Theta

• An option’s theta measures the sensitivity of the option’s price to a change in time.
• The characteristic of option prices declining (other factors held constant) purely as a result of the passage of time is known as time decay or time value decay.
• Time decay is greatest for at-the-money options.
• Time decay increases as an option approaches maturity.

Implied Volatility

• Another approach to estimating volatility is to assume the option’s current price correctly reflects the underlying’s volatility. 
• The implied volatility is obtained by finding the standard deviation that when used in the BSM-OPM makes the price predicted by the model equal to the market price of the option.
• Because the BSM-OPM formula cannot be arranged so that the standard deviation can be solved for directly, determining the implied volatility is an iterative or trial and error process.

Uses of Volatility

• Volatility data is useful when limit orders are being made and the trader needs a volatility estimate to help determine the limit price.
• Many traders will be sellers/writers of options when current volatility is greater than average past volatility and buyers of options when current volatility is less than average past volatility.

Volatility Term Structure

• Volatility tends to be an increasing function of maturity when the volatilities for short-date or nearby contracts are historically low.  This is because there is an expectation that implied volatilities will be increasing. 
• Implied volatility tends to be a decreasing function of maturity when short-date volatilities are historically high.  This is because there is the expectation that implied volatilities will be decreasing.

Volatility Smile and Skew

• The term volatility smile refers to the variation of implied volatility with the strike or exercise price. 
• The reason the term “smile” is used is that the relationship between implied volatility and exercise price is u-shaped indicating that the implied volatility for at-the-money options is frequently lower than that for out-of-the-money or in-the-money options. 
• For equity options, the smile is frequently skewed and the resulting pattern is called volatility skew: implied volatility is the highest for deep in-the-money options and then is decreasing as it moves towards out-of-the money options.

Limitation of the BSM

• One of the assumptions of the BS-OPM is that volatility is constant over the life of the option.  Obviously this is not the case; see VIX website.
• The existence of the volatility smile and skew indicates that implied volatility typically varies with the exercise price.  But why should the same underlying have differing levels of risk for the same exercise date?  Isn’t there really only a single “true” volatility for a given underlying?

Delta Hedging/Delta Neutral

• Delta hedging refers to holding shares of stock and selling calls (buying puts) in an effort to maintain a risk-free portfolio.
• The position must be adjusted as delta changes.

Delta-Gamma Neutral

• To insulate the value of an option portfolio from small changes in the price of the underlying, a trader would construct an option portfolio whose delta is zero.
• A trader who wishes to protect an option portfolio from large changes in the price of the underlying would construct a portfolio whose delta and gamma are both zero.

One-Period Binomial Model

• C_u = Max(0,uS – X) if the stock goes up in price, or
• C_d = Max(0,dS – X) if the stock goes down in price.

Hedge Portfolio

• Construct a hedge portfolio of h shares of stock and one short call.  Current value of portfolio:
• V = hS – C
• At expiration the hedge portfolio will be worth either:
• V_u = huS – C_u, or
• V_d = hdS – C_d
• If the portfolio is “hedged” it must be risk free and V_u and V_d must be equal.
• Setting V_u = V_d and solving for h gives:
• These values are all observable so h is easily determined.
• Since the portfolio is riskless, it will earn the risk-free rate.  Thus:
• V(1+r) = V_u = V_d

Two-Period Binomial Model

Pricing Put Options

The Value of an Option

• Premium = Intrinsic or Exercise Value + Time Value

Moneyness –                       Call                      Put

• Intrinsic Value = St – X
• Time Value = Call price – Intrinsic Value

Determinants of Option Value

• The longer the maturity of a call option, the greater its premium (value).
• The greater the volatility in the value of the underlying, the greater the call or put premium.
• There is a direct relationship between the value of the underlying and the call premium and an inverse relationship between the value of the underlying and the put premium.

Initial Margin Requirements-Selling or Writing Options on Individual Stocks

Stock Transactions

Buy Stock

Sell Short Stock

Call Options Transactions

Buy a Call – Right to Buy

• Buying a call is a bullish strategy that has a limited loss (the call premium) and an unlimited potential gain.
• Buying a call with a lower exercise price has a greater maximum loss but greater upside gains.
• For a given stock price, the longer a call is held, the more time value it loses and the lower the profit.

Profit Eqn:

• Break Even= S_t ^*­= X + C

Write a Call

• Selling a call is a bearish strategy that has a limited gain (the premium) and an unlimited loss.
• Selling a call with a lower exercise price has a greater maximum gain but greater upside losses.
• For a given stock price, the longer a short call is maintained, the more time value it loses and the greater the profit.
• Profit Formula

Put Options Transactions

Buy a Put:  Right to Sell

• Buying a put is a bearish strategy that has a limited possible loss (the put premium) and a large, but limited (stock price cannot fall below $0) potential gain.
• Buying a put with a higher exercise price increases both the maximum possible loss and the potential gain.
• Assuming all other factors are held constant, the longer a long put position is held, the more time value it loses and the greater the loss (or smaller the profit) on an offsetting trade.
• Profit

• Break- Even

Write a Put

• Writing a put is a bullish strategy that has a limited possible gain (the put premium) and a large, but limited (stock price cannot fall below $0), potential loss.
• Writing a put with a higher exercise price increases both the maximum gain and the potential loss.
• Assuming all other factors are held constant, the longer a short put position is held, the more time value it loses and the greater the profit (or smaller the loss) on an offsetting trade.
• Profit formula same as buy only N_C0>
• Break-even= X-P; Max. Profit = P

Covered Calls

• Strategy consists of buying the stock and writing a call (NS = -NC). Frequently, calls are written on previously purchased shares.
• Maximum profit = X – S₀ + C, minimum = -S₀ + C
• Breakeven stock price found by setting profit equation to zero and solving:  S_T^* = S₀ – C
• Profit

• A covered call reduces downside losses on the stock at the expense of upside gains
• A covered call with a lower exercise price provides greater downside protection but a lower maximum gain on upside
• Assuming all other factors are held constant, the longer a covered call is held, the more time value it loses and the greater the profit (or smaller the loss) on an offsetting trade.
• Writing the call introduces the risk that the trader may have to sell the stock at an unsuitable time.

Protective Puts

• A protective put sets a maximum downside loss at the expense of some of the upside gain. It is equivalent to an insurance policy on the asset.
• Strategy consists of buying the stock and buying a put (N_S = N_P).
• Choosing an exercise price is equivalent to the insurance problem of deciding on the deductible.
• For a given stock price, the longer a protective put is held, the more time value is lost and the lower the profit.
• Profit:

• Maximum profit = ¥, minimum = X – S₀ – P
• Breakeven stock price found by setting profit equation to zero and solving:  S_T^* = P + S₀

Synthetic Puts and Calls

• Using put-call parity (will be discussed later in the course):
  • This implies long put = long call, short stock, and long risk-free bond with a face value of X.
  • The right-hand side of the equation is a “synthetic” put.

Option Spreads: Basic Concepts

• Spread: is a purchase of one option and sale of another
• Two Types of Spreads
  • Verticle, Strike and Money Spread: purchase of an option with a particular exercise price nd the sale of another option differing only by exercise price.
  • Horizontal, Time or Calendar: purchases an option with an expiration of a given month and sells an otherwise identical option with a different expiration month.

Call Bull Spread{Bullish}

• The call bull spread is created by both buying and selling calls.  It is created by purchasing a call with a lower exercise price and selling a call with a higher exercise price.  (X1
• At expiration of the options, the strategy has a limited gain if the stock increases in price and a limited loss if the stock decreases in price.
• On a per-share basis:
• = Max(0,S_T – X₁) – C₁ – Max(0,S_T – X₂) + C₂
• Maximum profit = X₂ – X₁ – C₁ + C₂
• Minimum = – C₁ + C₂
• Breakeven:  S_T^* = X₁ + C₁ – C₂
• The profit (loss) on a call bull spread increases as expiration approaches if the stock price is closer to X2 (X1) and decreases if the stock price is closer to X1 (X2).  This is due to the time value of an option being greatest for near-the-money options.

Put Bear Spread {Bearish}

• Buy put with strike X2, sell put with strike X1, where X2 > X1.
• At expiration of the options, the strategy has a limited gain if the stock decreases in price and a limited loss if the stock increases in price.
• On a per-share basis:
• = – Max(0,X₁ – S_T) + P₁ + Max(0,X₂ – S_T) -P₂
• Maximum profit = X₂ – X₁ + P₁ – P₂
• Minimum = P₁ – P₂
• Breakeven:  S_T^* = X₂ + P₁ – P₂
• The profit (loss) of a put bear spread increases as expiration approaches if the stock price is closer to X1 (X2) and decreases if the stock price is closer to X2 (X1).  This is due to the time value of an option being greatest for near-the-money options.

(Equity) Collar {Bullish}

• Buy stock, S0, buy put with strike X1, sell call with strike X2 where X1
• A common type of collar is a zero-cost collar.  The call strike is set such that the call premium received offsets the put premium paid so that there is no initial outlay for the options.
• A collar is a risk-reducing strategy and offers the potential for a small profit while limiting risk.
• On a per-share basis:

P = S_T – S₀ + Max (0,X₁ – S_T) – P₁ – Max(0,S_T – X₂) + C₂

• Maximum profit = X₂ – S₀ – P₁ + C₂
• Minimum = X₁ – S₀ – P₁ + C₂
• Breakeven:  S_T^* = S₀
• The collar is equivalent to a bull spread plus a risk-free bond paying X1 at expiration.
• A collar can also be thought of as the combination of buying a protective put and writing a covered call on the same long stock position.
• The profit (loss) on a collar increases as expiration approaches if the stock price is closer to X2 (X1) and decreases if the stock price is closer to X1 (X2).  This is due to the time value of an option being greatest for near-the-money options.

Straddle {Neutral}

• A straddle is the purchase of a call and put that have the same exercise price and expiration date. Typically, the exercise price on the two options is close to the current stock price.
• Note a straddle is not a spread because you are buying both options!  By definition, a spread involves both the purchase and sale of options.
• A straddle is a volatility-based strategy designed to profit if there is a large up or down move in the stock.  It has the potential for large gains for either increases or decreases in the stock price and also has a limited loss (the sum of the premiums paid for the two options).
• On a per-share basis:

P = Max(0,S_T – X) – C + Max(0,X – S_T) – P

• Breakeven:  S_T^* = X – C – P and S_T^* = X + C + P
• Maximum profit: ¥
• Minimum = – C – P

Time Spread (Calendar Spread)

Example:                 XYZ stock is trading at $50 

Outlook:                  You are neutral on XYZ stock short term.

Possible strategy:   

Buy Time Spread to take advantage of short term time decay.

Sell one July 50 strike call         $2.50

Buy one August 50 strike call    $3.75

                                  Net Debit  $1.25  

At Expiration:

• Maximum Profit = Long call value at near term expiration – Net Debit
• Maximum Loss = Net Debit paid for spread $1.25

IN SUMMARY: This spread will be profitable if XYZ stabilizes to the at-the-money strike at July expiration. Time decay is in your favor because time erosion accelerates as July expiration approaches. Maximum profit depends on the value of the long call at near term expiration.  Maximum loss is equal to the debit paid for the spread.

Stock Dividends: 10% @ X= $60

No. of Shares = 100 x 1.10 = 110 shares

X = $60/ 1.10 = 54.55

Stock Splits: 4 for 3 stocks

No. of Shares = 100 x (4/3) = 133 shares

X = $60/ (3/4) =

• Bullish – Short puts & long calls
• bid price is the maximum price the market maker will pay for the option.  (You Sell At)
• ask price is the minimum price the market maker will accept for the option (You buy at)