Introduction to Financial Options / Lecture 03
Underlying
Call / Put
Strike
t_{\mathrm{exp}} or \sigma
Style
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P_c(K_L)\ge P_c(K_H)
P_p(K_L)\le P_p(K_H)
if K_L<K_H
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P(t_1)\le P(t_2) if t_1<t_2
P(\sigma_1)\le P(\sigma_2) if \sigma_1<\sigma_2
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P_A\ge P_E
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Today’s topic
Introduction to Financial Options / Lecture 03
Suppose we invest $100 today for one year at 10% per year.
As the number of interest payments grows, (1+r/n)^{nT}\longrightarrow e^{rT}.
Introduction to Financial Options / Lecture 03
“If two different things result in the same future outcome, they must be worth the same now.”
The share
Buy it directly today.
Options and cash
Can they replicate holding stock at expiry?
Matching outcomes let us (a) equate prices, (b) identify an arbitrage, and (c) build a hedge.
Introduction to Financial Options / Lecture 03
Same share, same strike K=\$100, same expiry.
Long call: the right to buy the share for $100.
Short put: the obligation to buy it for $100 if assigned.
Move the expiry price to see how the two payoffs meet at the strike.
Below K, the short put carries the position. Above K, the long call does. Their sum is one straight line.
Introduction to Financial Options / Lecture 03
At S_T=\$120, the call is worth $20. Exercise it: receive one share and pay $100.
The short put expires with no value.
Combined payoff: +$20. You own one share and have paid $100.
Introduction to Financial Options / Lecture 03
At S_T=\$80, the call expires with no value.
The short put is worth −$20 to you. The holder exercises: you receive one share and pay $100.
Combined payoff: −$20. You still own one share and have paid $100.
Introduction to Financial Options / Lecture 03
At time T:
\begin{aligned} C_T-P_T &= (S_T-K)^+-(K-S_T)^+ \\ &= (S_T-K)^+ +(S_T-K)^- = S_T-K \end{aligned}
Here x^- = \min(x,0) is the signed negative part.
The share is physical. Think of it as the piece of paper representing one share: you end up long one share.
The strike is future cash. You pay K at expiry, whether through call exercise or put assignment.
Long call minus long put = long share delivered at T, with K due at T.
Introduction to Financial Options / Lecture 03
The share is worth S_0 today. The K payment is due later, so set aside only its present value now.
\operatorname{PV}(K)=Ke^{-rT}
Investing Ke^{-rT} at a continuously compounded risk-free rate r grows to K at T.
Assume European options, no dividends and one common funding rate for now.
Introduction to Financial Options / Lecture 03
Both portfolios have payoff S_T-K at expiry. What does each one cost at time 0?
Call minus put
Buy the call for C_0. Sell the put and receive P_0.
Net cost today: C_0-P_0.
Share minus future debt
Buy one share for S_0. Borrow Ke^{-rT} now and repay K at expiry.
Net cost today: S_0-Ke^{-rT}.
Same payoff at T means the two costs at time 0 must be equal.
Introduction to Financial Options / Lecture 03
Arbitrage Concept
#3
C_0-P_0=S_0-Ke^{-rT}
e^{-rT}\approx 1-rT
For small rT, this gives C_0-P_0\approx S_0-K+KrT.
The KrT term is just the cost of borrowing K for time T at rate r.
Introduction to Financial Options / Lecture 03
What would you do?
The synthetic share is $1 too cheap.
Fair value: C_0-P_0=S_0-K=\$10.
Buy the call, sell the put, short one share, and set aside $100 for the strike.
Assume zero rates and dividends.
Introduction to Financial Options / Lecture 03
Today
Short stock: +$110
Buy call, sell put: −$9
Set aside strike cash: −$100
Net: +$1
At expiry
The options deliver one share for $100. Use the reserved cash to pay the strike, then return the share to close the stock short.
Net: $0, whatever S_T
If the option portfolio were too expensive, reverse all four positions.
Idealized trade: executable prices, no stock-borrow fee, no margins, and no transaction costs.
Introduction to Financial Options / Lecture 03
Stock price
$53.10 (pick’em market)
Fixed carry across strikes
K(1-e^{-rT})\approx KrT=\$0.30
Put-call parity
C-P=S-K+KrT
| Calls | Strike | Puts | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bid size | Bid | Ask | Ask size | Bid size | Bid | Ask | Ask size | |||
| (10) | 18.40 | — | 18.40 | (10) | 35 | (10) | 0.00 | — | No offer | |
| (10) | 13.35 | — | 13.35 | (10) | 40 | (10) | 0.00 | — | No offer | |
| (10) | 8.50 | — | 8.50 | (10) | 45 | (10) | 0.10 | — | 0.10 | (10) |
| (10) | 3.95 | — | 3.95 | (10) | 50 | (10) | 0.55 | — | 0.55 | (10) |
| (10) | 1.00 | — | 1.00 | (10) | 55 | (10) | 2.45 | — | 2.45 | (10) |
| (10) | 0.25 | — | 0.25 | (10) | 60 | (10) | 6.85 | — | 6.85 | (10) |
| (10) | 0.00 | — | No offer | 65 | (10) | 11.60 | — | 11.60 | (10) | |
| (10) | 0.00 | — | No offer | 70 | (10) | 16.50 | — | 16.50 | (10) | |
Can you find all the free money?
Introduction to Financial Options / Lecture 03
A dividend is cash leaving the company. All else equal, the share is worth about D less once it goes ex-dividend, including at expiry.
Share before payout− D cash→Ex-dividend share
Own the share
Buy it for S_0 today. Collect the dividend before expiry, then still own the share at T.
Use the options
The call-minus-put delivers the share at T for K, but does not collect the intervening dividend.
For a known dividend worth D_0 today: C_0-P_0=S_0-D_0-Ke^{-rT}.
Introduction to Financial Options / Lecture 03
Just before the ex-dividend date, an in-the-money call holder has a choice.
Exercise now
Pay K now, own the share, and receive the dividend D. Give up the call’s remaining optionality.
Keep the call
Keep the option’s time value and postpone paying K, but miss the dividend.
Exercise just before ex-dividend when dividend D exceeds the remaining optionality plus the benefit of delaying K.
With negligible interest, the rule is simply D greater than the remaining time value. If you can sell the call above intrinsic value, sell it and buy the share instead of exercising.
Introduction to Financial Options / Lecture 03
For European options, set dividends aside for a moment. In reality there are different rates for cash you borrow and cash you lend.
r_{\mathrm{short}} = rate paid to borrow cash. r_{\mathrm{long}} = rate earned when lending cash. Usually r_{\mathrm{short}}>r_{\mathrm{long}}.
Borrow cash: r_{\mathrm{short}}
To replicate C-P, buy the share and borrow the cash that will become K at expiry.
C-P\le S_0-Ke^{-r_{\mathrm{short}}T}
Lend cash: r_{\mathrm{long}}
To replicate P-C, short the share and invest cash that will become K at expiry.
P-C\le Ke^{-r_{\mathrm{long}}T}-S_0
S_0-Ke^{-r_{\mathrm{long}}T}\le C-P\le S_0-Ke^{-r_{\mathrm{short}}T}
This gap is the funding spread. In this class, we ignore it for simplicity.
Introduction to Financial Options / Lecture 03
A deep in-the-money put can be exercised before expiry. The attraction is receiving K now, not later.
Exercise now
Sell the share for K today. Invest the proceeds and earn interest until expiry.
Keep the put
Keep the choice to sell later. If the share rebounds, you need not exercise.
Exercise when the interest gained by receiving K early exceeds the value of keeping that choice.
Higher rates and less remaining optionality favor exercise. A coming dividend favors waiting, since the ex-dividend share-price drop helps the put. If the put can be sold above its exercise value, sell it instead.
Introduction to Financial Options / Lecture 03
Return to our S_0=\$110, K=\$100, C-P=\$9 trade. Its clean parity value is \$10, so the apparent edge is \$1.
Borrow the share
The trade shorts one share for six months. At a fixed 2% annual borrow fee on \$110:
B_0\approx 110\times 0.02\times 0.5=\$1.10
Adjust the parity test
The cheap-synthetic side has a lower bound, not the clean equality:
C-P\ge S_0-Ke^{-rT}-B_0
Here, with r=0, that lower bound is \$8.90. The quoted \$9 is not an arbitrage.
The \$1 clean edge is smaller than the \$1.10 borrow bill. A changing fee or recall makes the short even less certain.
Introduction to Financial Options / Lecture 03
Stock price
$53.10 (pick’em market)
Fixed carry across strikes
K(1-e^{-rT})\approx KrT=\$0.30
Put-call parity
C-P=S-K+KrT
| Calls | Strike | Puts | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bid size | Bid | Ask | Ask size | Bid size | Bid | Ask | Ask size | |||
| (10) | 18.25 | — | 18.35 | (10) | 35 | (10) | 0.00 | — | No offer | |
| (10) | 13.35 | — | 13.45 | (10) | 40 | (10) | 0.00 | — | No offer | |
| (10) | 8.45 | — | 8.55 | (10) | 45 | (10) | 0.05 | — | 0.10 | (10) |
| (10) | 4.05 | — | 4.15 | (10) | 50 | (10) | 0.50 | — | 0.55 | (10) |
| (10) | 0.90 | — | 0.95 | (10) | 55 | (10) | 2.40 | — | 2.50 | (10) |
| (10) | 0.20 | — | 0.25 | (10) | 60 | (10) | 6.75 | — | 6.85 | (10) |
| (10) | 0.00 | — | No offer | 65 | (10) | 11.45 | — | 11.55 | (10) | |
| (10) | 0.00 | — | No offer | 70 | (10) | 16.50 | — | 16.60 | (10) | |
Can you find all the free money?
Introduction to Financial Options / Lecture 03
C_0=P_0+S_0-Ke^{-rT}
Long put
Own the right to sell at K.
Long share
Add the share’s directional exposure.
Borrow strike cash
Borrow Ke^{-rT} now. Repay K at expiry.
Strip away stock and financing. A call and put contain the same optionality.
Matching European options. Idealized funding and stock trades, with no dividends or trading costs.
Introduction to Financial Options / Lecture 03
Choose an option, then move the stock hedge. The stock changes the slopes, but not the kink.
K=S_0=\$100. Expiry values per share, before option premium, funding and trading costs.
Introduction to Financial Options / Lecture 03
My view is mainly directional
“I think the share will rise.”
The share or a forward expresses that view more directly.
My view is about the distribution
“I think the market is underpricing large moves.”
An option buys exposure to the tails. A stock hedge can remove much of its direction.
The width and shape of returns matter, not just their mean.
An unhedged call or put still has direction. The distinct thing you buy or sell is optionality.
Introduction to Financial Options / Lecture 03
We will start answering that question in Lecture 04. Finally.
Introduction to Financial Options / Lecture 03
Calls and puts differ in direction, not in the kind of optionality they contain.