Introduction to Financial Options / Lecture 02
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Oral final exam
Assignments become available in the week their material is taught. They are due before the next class.
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Introduction to Financial Options / Lecture 02
We defined the right. Now we need to value it.
Introduction to Financial Options / Lecture 02
You can take the good outcomes.
You can walk away from the bad ones.
The right
Keep the upside if it happens.
No obligation
Reject the downside if it happens.
Key question for today:
What determines the value of that asymmetry?
Introduction to Financial Options / Lecture 02
Share price today
$100
A regulated utility with predictable cash flows.
Very little uncertainty about the outcome.
Introduction to Financial Options / Lecture 02
Share price today: $100 · Drug-trial result arrives in three months.
Possible outcomes:
Drug fails
$50
Drug works
$400
What probabilities of success and failure are implied by today’s $100 price?
100=q(400)+(1-q)(50) \quad\Longrightarrow\quad q=\frac{1}{7}=14.3\%
Success: 14.3% · Failure: 85.7%
These are toy pricing probabilities, not a forecast of the trial. Assume zero rates, no dividends and exactly two expiry outcomes.
Introduction to Financial Options / Lecture 02
Three-month call · strike K=\$100
Boring Utility Co.
Almost surely S_T=\$100:
C_0 \approx \max(100-100,0)=\$0
No upside surprise to capture.
Binary Biotech
Using the implied probabilities:
C_0=\frac{1}{7}(400-100)+\frac{6}{7}(0)
C_0=\$42.86
Same spot. Same strike. Same average future stock price. Different uncertainty.
Undiscounted toy model. Zero rates and dividends.
Introduction to Financial Options / Lecture 02
Keep the option strike and average expiry share price at $100. Move the outcomes apart.
The stock’s average is unchanged. Calls and puts gain from opposite tails.
Equal toy pricing probabilities for each two-state case. Zero rates and dividends.
Introduction to Financial Options / Lecture 02
Both stocks trade at $100 today. One finishes at $100; the other finishes at $80 or $120. Give both calls the same strike.
Values are undiscounted expected call payoffs under toy pricing probabilities.
Introduction to Financial Options / Lecture 02
A wider distribution has a larger standard deviation, \sigma. That extra uncertainty can make a call more valuable even when the mean is unchanged.
Tight distribution
Most outcomes land near $100.
Wide distribution
More probability reaches large payoffs.
This naturally leads us from thinking about outcomes to thinking about distributions.
Introduction to Financial Options / Lecture 02
Options are bets on volatility. More generally, they are bets on the shape of probability distributions.
C_T=(S_T-K)^+
Convex payoff
This non-linearity makes the distribution of outcomes matter, not only their average.
Introduction to Financial Options / Lecture 02
Stock · linear payoff
S_0=e^{-rT}\int_0^\infty s\,f_Q(s)\,ds
Only the mean matters.
Call · non-linear payoff
C_0=e^{-rT}\int_K^\infty (s-K)f_Q(s)\,ds
The shape of the distribution matters.
Here f_Q is a pricing density, not a forecast. Lecture 04 will explain the distinction.
Introduction to Financial Options / Lecture 02
Both calls have strike $100. Both distributions have an average expiry price of $100.
Distribution A
$80 or $120, each with 50% probability.
What is the expected call payoff?
Distribution B
$80 with 25%, $100 with 50%, or $120 with 25% probability.
What is the expected call payoff?
A: $10. B: $5. The range is the same, but A puts more probability on the upside payoff.
Toy pricing probabilities and zero rates. Compare values, not just the highest possible payoff.
Introduction to Financial Options / Lecture 02
So far, we have valued what options may pay at expiry. An American call also lets you exercise before expiry. Should you?
Suppose the stock is $110 and the call’s strike is $100.
Exercise now
Receive a share worth $110 and pay $100.
Value now: $10
Keep the choice
At expiry, the share is either $80 or $140, each equally likely.
Call payoffs: $0 / $40
Waiting is worth $20 in this toy model. Exercising now gives up the remaining $10 of time value.
Toy pricing probabilities, zero rates, no dividends and no trading frictions.
Introduction to Financial Options / Lecture 02
An unexpired option contains two things:
Intrinsic value
What exercise gives you today.
+
Time value
The remaining chance that uncertainty creates something better.
Exercise keeps the intrinsic value but destroys the remaining time value.
If you want to exit, selling the option is usually better than exercising it.
Important exceptions and complications: dividends, stock borrow, interest rates, puts, transaction costs and contract details.
Introduction to Financial Options / Lecture 02
Arbitrage Concept
#2
Two otherwise identical American options expire at T_1 and T_2, with T_2>T_1.
Shorter expiry · T_1
Exercise at any time through T_1.
\subset
Longer expiry · T_2
Every earlier exercise date, plus more.
V_A(T_2) \ge V_A(T_1)
The longer-dated American option contains all the rights of the shorter-dated option.
European options: more time usually adds value, but this is not pure dominance. The later contract cannot be exercised at T_1, so dividends, rates and carry can affect the comparison.
Introduction to Financial Options / Lecture 02
Same spot, strike, rates, dividends, and call style. Only daily volatility and expiry differ. Near the money, value roughly scales with total volatility; variances add.
4 months of A ≈ 1 month of B: same total volatility, same toy-model call value.
Toy model: independent daily moves, constant volatility, matched calls, zero rates/dividends, and equal trading days per month.
Introduction to Financial Options / Lecture 02
In a simple model, the scale of a typical percentage move grows like:
\sigma\sqrt{T}
More volatility can offset less time. In this simple model, the move scale is \sigma\sqrt{T}.
A preview, not a universal pricing rule. Events, jumps and changing volatility complicate this scaling.
Introduction to Financial Options / Lecture 02
Uncertainty creates value when you have the right to choose.