Practice Pricing Options By Hand

 
Think of this like a homework assignment to help you do basic reasoning about option prices. It requires no more than elementary arithmetic and the concept of expected value.
 
Assume 0% cost of carry for these questions.
 
1️⃣
A stock is fairly priced at $100. In one year, it will either be $50 or $125. What are the probabilities that make the stock fairly priced?
 
Answer
The expected value method
Just use algebra
 
If you are long the stock:
let p = probability of winning
If the stock is fairly priced, the expected value of winning and losing sum to zero.
p * $25 + (1 - p) * -$50 = 0
$25p + $50p - $50 = 0
$75p = $50
p = 2/3
The odds method
The intuitive way to see this proposition is you are laying 2-1 odds when you buy the stock. In other words, you are risking $50 to win $25. Mechanically, to convert odds to probabilities you sum both the 2 and the 1 and that becomes the denominator. The 2, the first number, is the numerator. So the probability of winning is 2/3. This balances the expected value. You win $25 2x and lose $50 one time if you think of playing the game 3x. If you were short the stock you are getting 2-1 odds. So you put the second number in the numerator and the sum in the denominator: 1/3
 
This is quite mechanical and best to apply after you understand the intuition — the sum of the numerator and denominator are a count of all the possible ways to reach the outcome and the components of the odds are the frequency of the win or loss depending on your perspective.
 
 
2️⃣
Fill in the table for log moneyness for the put and call strikes
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Answer
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3️⃣
Fill in the table for max payoff of the options
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Answer
The max payoff of the options where K is strike:
Puts = K-lower price bound
Calls = upper bound price - K
 
In this case we have defined the lower and upper bounds to be $50 and $125 respectively. In reality the lower bound of a stock is $0 and the upper bound of the stock is infinite.
 
However, the upper bound of the call is NOT infinite. [Extra credit: What is the upper bound of call option price…hint all calls for a given stock have the same upper bound! You can ask me on Twitter for the answer. ]
 
This table shows the answer for the max payoffs in this question.
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4️⃣
Fill in the table for fair value of the options
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Answer
This is another expected value question. We’ll do an example for a put and a call.
 
The 90 put has:
  • 1/3 chance of being worth $40
  • 2/3 chance of being worth $0
90 put = 33% * $40 + 67% * 0 = $13.33
 
The 120 call has:
  • 2/3 chance of being worth $5
  • 1/3 chance of being worth $0
 
120 call = 67% * $5 + 33% * $0 = $3.33
This table shows the answer for the option fair values in this question.
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5️⃣
Fill in the table for delta of the options
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Answer
The key to this question is remembering that delta is a hedge ratio.
 
Delta = Change in option price for change in stock price
 
Let’s work through an example.
 

The 60 put

Stock up case
  • Option price changes from $3.33 to $0
  • Stock price changes from $100 to $125
Delta in up case:
($0 - $3.33) / ($125-$100) = -13.33
 
Stock down case
  • Option price changes from $3.33 to $10
  • Stock price changes from $100 to $50
Delta in down case:
($10 - $3.33) / ($50 - $100) = -13.33
 
Weighting:
 
2/3 of the time the option has a -13.33 delta
1/3 of the time the option has a -13.33 delta
 
The weighted average between these scenarios is the option’s delta.
2/3 * -13.33 + 1/3 * -13.33 = -13.33
 
Follow that logic for all the options and the table is:
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6️⃣
Look back at the deltas. What do you notice?
Answer
The presence of steep negative skew in the distribution increases the option’s delta significantly.
 
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  • Look at the moneyness of the options with equivalent deltas on the call and put side.
    • The 26.7% delta call is about 5% out of the money
    • The 26.7% delta put is a whopping 36% out of the money. It’s much further away but has a much higher delta.
 
This is a reminder that delta and probability of finishing in the money can diverge in many scenarios (high skew, long time to expiration, or “high volatility”).
 
The delta of these puts is being driven by how far the stock can fall more so than it’s probability and expected value is the product of both those ideas.
 
But critically, the key observation is that deltas more than anything are hedge ratios.
Proof
Imagine selling 100 of the 110 calls and buying 2,000 shares of stock for $100.
 
Up scenario P/L
  • Call P/L = collected option premium - loss at expiration
    • $10 - $15 = -$5
      Options have a 100 share multiplier and you sold 100 of them so net call p/l = $50,000
  • Stock p/l = 2,000 x $25 = $50,000 Total up P/L =$0
Down scenario P/L
  • Call P/L = collected option premium - loss at expiration
    • $10 - $0 = $10
      Options have a 100 share multiplier and you sold 100 of them so net call p/l = $100,000
  • Stock p/l = 2,000 x -50 = -$100,000 Total down p/l = $0
In both the up and down scenarios, your net p/l is zero. Perfectly hedged!
 
If you looked at the simple “proof” or demonstration, you can do the same proofs for all the options.
 
This chart shows just how far out of the money calls and puts are of equivalent delta in such a sharply skewed binary distribution.
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