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Michael Brenndoerfer

Option Strategies and Payoff Combinations › Volatility Strategies

Michael BrenndoerferNovember 13, 202512 min read

Quantitative FinanceAll chapters

Part 3 of 15Forwards, Futures, and Options

Chapter 19Option Strategies and Payoff Combinations › Volatility Strategies

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You don't have a strong view on direction but do have a view on volatility. You believe the underlying will either move significantly (without knowing which direction) or stay relatively stable. Volatility strategies express these views.

Long StraddleLink Copied

A long straddle involves buying both a call and a put with the same strike price and expiration, typically at-the-money. The strategy profits from large moves in either direction and loses if the underlying stays near the strike.

The combined payoff is:

\text{Long\_Straddle} = \max(S_T - K, 0) + \max(K - S_T, 0) - (C + P)

where:

$S_T$ : stock price at expiration
$K$ : strike price (same for both call and put)
$C$ : premium paid for the call
$P$ : premium paid for the put
$(C + P)$ : total cost to establish the straddle

This formula adds together the payoffs of the long call and long put, then subtracts the total premium paid for both options. The structure guarantees that at least one of the two max terms contributes positive value at expiration. If the stock ends above the strike, the call is in-the-money. If the stock ends below the strike, the put is in-the-money. The only way to lose the full premium is if the stock expires exactly at the strike, where both options are worthless.

The straddle structure ensures that one option is always in-the-money: the call profits when the stock rises above $K$ , while the put profits when the stock falls below $K$ . At any price, at least one option has positive intrinsic value. The total premium paid represents the cost of this bidirectional exposure.

This payoff structure can be expressed more simply by recognizing that exactly one of the two max terms is always positive. When $S_T > K$ , the call is in-the-money with value $(S_T - K)$ and the put is worthless. When $S_T < K$ , the put is in-the-money with value $(K - S_T)$ and the call is worthless. In both cases, the total intrinsic value equals the absolute distance from the strike. This observation simplifies the formula.

This simplifies to:

\text{Profit} = |S_T - K| - (C + P)

where:

$|S_T - K|$ : absolute distance of the stock price from the strike (always positive)
$(C + P)$ : total premium paid (the breakeven threshold)

This elegant form shows that profit equals the absolute price movement away from the strike, minus the cost of the straddle. The straddle profits when the stock moves more than $(C + P)$ in either direction from the strike. The absolute value function captures the directional indifference of the straddle: you don't care which way the stock moves, only how far it moves. This is why straddles are called volatility strategies rather than directional strategies.

The two breakeven points occur where profit equals zero. Setting $|S_T - K| - (C + P) = 0$ and solving:

\begin{aligned} |S_T - K| &= C + P \\ S_T - K &= \pm(C + P) \\ S_T &= K \pm (C + P) \end{aligned}

where:

$K$ : strike price of both options
$C + P$ : total premium paid for the straddle
$K - (C + P)$ : lower breakeven point
$K + (C + P)$ : upper breakeven point

The derivation above shows why the straddle has two symmetric breakeven points equidistant from the strike. The absolute value equation has two solutions because the stock can move either up or down by the required amount. The distance from the strike to each breakeven equals the total premium paid, which represents the market's expectation of how much the stock might move.

In[19]:

Code

# Long straddle
K_straddle = 100  # At-the-money strike
call_prem = 6  # Call premium
put_prem = (
    5.5  # Put premium (slightly cheaper due to put-call parity adjustments)
)
total_premium = call_prem + put_prem

S = np.linspace(70, 130, 200)

# Component payoffs
straddle_call = call_payoff(S, K_straddle, call_prem, "long")
straddle_put = put_payoff(S, K_straddle, put_prem, "long")

# Combined straddle
long_straddle = straddle_call + straddle_put

# Calculate key metrics
lower_be = K_straddle - total_premium
upper_be = K_straddle + total_premium

Out[20]:

Visualization

Long straddle V-shaped profit and loss diagram showing a loss trough at the strike price and rising profits in both directions, with annotations for the lower breakeven, upper breakeven, and maximum loss. — Long straddle payoff profile involving a long call and long put at the same strike price. The V-shaped profit curve demonstrates that the strategy profits from significant price movement in either direction, with losses limited to the total premium paid if the stock remains near the strike. This volatility strategy requires a move exceeding the total premium to reach breakeven.

Out[21]:

Console

Lower Breakeven: $88.50
Upper Breakeven: $111.50
Maximum Loss: $11.50

The straddle requires a significant move in either direction from the strike to reach the breakeven points. Maximum loss occurs only when the stock expires exactly at the strike price, as both options expire worthless. The total premium paid reflects the market's implied volatility expectations. Profit equals the absolute distance from the strike minus the total cost, creating symmetric payoff in both directions once breakeven is achieved.

The long straddle is a bet that the underlying will move more than the market expects. The total premium paid represents the market's implied expectation of volatility. If the actual move exceeds this expectation, the straddle profits. This is why straddles are often described as "buying volatility."

A critical consideration: buying straddles before major announcements (earnings, FDA decisions, elections) is typically not profitable. The market anticipates these events and prices options accordingly. The implied volatility is already elevated, making the straddles expensive. After the announcement, implied volatility collapses, often overwhelming any gains from the directional move.

Long StrangleLink Copied

A long strangle is similar to a straddle but uses different strike prices: you buy an out-of-the-money call and an out-of-the-money put. This reduces the premium paid but requires a larger move to become profitable.

The strangle structure places the call strike above the current price and the put strike below the current price. Both options are out-of-the-money at initiation, making them cheaper than the at-the-money options used in a straddle. However, this lower cost comes with a tradeoff: the stock must move further before either option gains intrinsic value.

In[22]:

Code

# Long strangle
K_call_strangle = 105  # OTM call strike
K_put_strangle = 95  # OTM put strike
call_prem_str = 3.5  # Lower premium because OTM
put_prem_str = 3  # Lower premium because OTM
total_premium_str = call_prem_str + put_prem_str

S = np.linspace(70, 130, 200)

# Component payoffs
strangle_call = call_payoff(S, K_call_strangle, call_prem_str, "long")
strangle_put = put_payoff(S, K_put_strangle, put_prem_str, "long")

# Combined strangle
long_strangle = strangle_call + strangle_put

# Calculate key metrics
lower_be_str = K_put_strangle - total_premium_str
upper_be_str = K_call_strangle + total_premium_str
cost_savings = total_premium - total_premium_str

Out[23]:

Visualization

Long strangle profit and loss diagram showing a flat loss region between the two out-of-the-money strikes and rising profits outside each strike, with annotations for the lower and upper breakeven points. — Long strangle payoff profile using out-of-the-money options to lower the entry cost compared to a straddle. The flat region between strikes represents the zone of maximum loss where both options expire worthless. While cheaper to establish, the stock must move significantly further to overcome the wider breakeven points.

Out[24]:

Console

Strangle Cost: $6.50
Cost Savings vs Straddle: $5.00
Lower Breakeven: $88.50
Upper Breakeven: $111.50
Range Between Strikes: $10.00

The strangle costs less than the straddle but has wider breakeven points. The probability of maximum loss is higher because any price within the range between the strikes results in both options expiring worthless. The cost savings come at the expense of requiring a larger absolute move to reach profitability. The dead zone between strikes increases the likelihood of maximum loss. This tradeoff makes sense when you expect a very large move and want to minimize capital at risk while accepting lower probability of profit.

Out[25]:

Visualization

Overlay profit and loss chart comparing a long straddle and a long strangle, with the straddle showing a deeper but narrower loss trough and the strangle showing a shallower but wider loss region between its outer strikes. — Overlay comparison of long straddle and long strangle profitability. The straddle offers a higher probability of profit with narrower breakeven points but requires a larger capital outlay (higher max loss). The strangle reduces the maximum risk and cost but requires a larger price swing to achieve profitability due to the gap between strikes.

Short Straddles and StranglesLink Copied

Selling straddles or strangles reverses the payoff profile: you profit from stability and lose from large moves. Short volatility positions collect premium upfront but have unlimited risk if the underlying makes a significant move.

These strategies are popular among traders who believe implied volatility is overstated relative to realized volatility. However, they require careful risk management because the potential losses are theoretically unlimited on the upside (for the short call component) and substantial on the downside (limited only by the stock going to zero).

Out[26]:

Visualization

Short straddle inverted V-shaped profit and loss diagram with maximum profit at the strike price and mounting losses in both directions, annotated with the max profit level and upper and lower breakeven points. — Short straddle payoff profile illustrating the high risk of selling volatility. The strategy generates maximum profit if the stock remains exactly at the strike price at expiration but faces theoretically unlimited losses if the price moves significantly in either direction. This inverted V-shape highlights the danger of selling volatility without protection as losses accelerate with larger price movements.