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  Lower & Upper Bounds for Options

 

Option

Minimum Value

Maximum Value

European call (c)

ct ≥ Max(0,St-(X/(1+RFR)t)

St

American Call (C)

Ct ≥ Max(0, St-(X/(1+RFR)t)

St

European put (p)

pt ≥Max(0,(X/(1+RFR)t)-St)

X/(1+RFR)t

American put (P)

Pt ≥ Max(0, (X-St))

X

Where t is the time to expiration

 

If the asset has an underlying stream of cash flows, the minimum value for European Options change to:
 

c0≥Max{0,[S0-PV(CF,0,T)]–(X/(1+r)T)}

p0≥Max{0,(X/(1+r)T)- [S0-PV(CF,0,T)]}

 
 

Example

What are the minimum values of an American-style and a European-style 3-month call option with a strike price of $80 on a non-dividend-paying stock trading at $86 if the risk-free rate is 3%?

Solution

American: $6.59;   European: $6.59

 





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