Explain why margin accounts are only required when clients write options but not when they buy options?
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Chapter9-ParityandOtherOptionRelationships-Students.pptxChapter 9 Parity and Other Option Relationships
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Notation
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| c: | European call option price |
| p: | European put option price |
| S0: | Stock price today |
| K: | Strike price |
| T: | Life of option |
| s: σ: | Volatility of stock price |
| C: | American call option price |
| P: | American put option price |
| ST: | Stock price at option maturity |
| D: | PV of dividends paid during life of option |
| r | Risk-free rate for maturity T with cont. comp. |
Assumptions
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Assume that there are some participants for which the following are true
Face no transaction costs
All trading profits (net of trading losses) are subject to the same tax rate
Can borrow and lend at the same risk-free interest rate
Are prepared to take advantage of arbitrage opportunities as they arise
Therefore it is reasonable to assume there are no arbitrage opportunities
r > 0 and r is the nominal interest rate
American vs European Options
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An American option is worth at least as much as the corresponding European option
C c
P p
Upper Bounds on Options
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Upper Bound
C ≤ S0 & c ≤ S0
P ≤ K & p ≤ Ke-rT
Lower Bound on a Non-Dividend Paying Stock
c ≥ max(S0 – Ke-rT, 0)
p ≥ max(Ke-rT – S0, 0)
Put-Call Parity: No Dividends
Consider the following 2 portfolios:
Portfolio A: European call on a stock + zero-coupon bond that pays K at time T
c + Ke-rT,
Portfolio C: European put on the stock + the stock
p + S0
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Values of Portfolios A & C
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| ST > K | ST < K | ||
| Portfolio A | Call option | ST − K | 0 |
| Zero-coupon bond | K | K | |
| Total | ST | K | |
| Portfolio C | Put Option | 0 | K− ST |
| Share | ST | ST | |
| Total | ST | K |
The Put-Call Parity Result
Both are worth max (ST , K) at the maturity of the options
They must therefore be worth the same today. This means that
c + Ke -rT = p + S0
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The American Put-Call Parity
S0 – K ≤ C – P ≤ S0 – Ke -rT
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Early Exercise
Usually there is some chance that an American option will be exercised early
An exception is an American call on a non-dividend paying stock
This should never be exercised early
C = c
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Bounds for European or American Call Options (No Dividends)
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Should Puts Be Exercised Early ?
Are there any advantages to exercising an American put when
S0 = 60; T = 0.25; r=10%
K = 100; D = 0
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Bounds for European and American Put Options (No Dividends)
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American Calls
If you exercise investor receives:
Only exercise if
Thus:
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Put-Call Parity with or w/o Dividends
European Option; D = 0
c + Ke -rT = p + S0
American options; D = 0
S0 − K ≤ C − P ≤ S0 − Ke−rT
European options; D > 0
c + D + Ke −rT = p + S0
American options; D > 0
S0 − D − K ≤ C − P ≤ S0 − Ke −rT
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Put-Call Parity with Stocks
c + Ke-rT = p + S0
If K = F0
c = p as (c – p) is another way to create a synthetic forward
If S0 = K
Non-dividend paying stock c – p = S0 – S0e-rT
Dividend paying stock c – p = S0 – S0e-rT – D
Put-Call Parity allows us to see how to create synthetic stock, option, or US T-Bill (i.e. zero-coupon bill)
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Extensions of Put-Call Parity
European options with Discrete Dividends
c + D + Ke−rT = p + S0
European options with Dividend Yield
c + Ke−rT = p + S0e-qT
European Option; Forwards
c + Ke-rT = p + Fe-rT
European Option; Currency
c + Ke-rT = p + S0e-rfT S0=DC/FC
European options; Bond
c + Ke−rT = p + B0 – I
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Currency Options
A Call option on € means you pay $
This can easily be interpreted as a Put option on $ and one pays €
“EUR Call USD Put, AMT: EUR 100 mil USD 120 mil”
Can interpret as:
Call on € with right to pay $120 mil for €100 mil with a K = $1.20
Put on $ with right to sell $120 mil for €100 mil with K = €0.8
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Effect of Variables on Option Pricing
| Variable | c | p | C | P |
| S0 | + | − | + | − |
| K | − | + | − | + |
| T | ? | ? | + | + |
| s | + | + | + | + |
| r | + | − | + | − |
| D | − | + | − | + |
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