Lecture 4
• The option price should always be above the lower bound, and
• The option price should always be below the upper bound.
• If an option ever traded at a price outside this range, it presents an arbitrage
opportunity.
Formulas:
Call upperbound:
,
Call lowerbound:
Put upperbound:
Lecture 5
Put lowerbound:
,
Compare actual with calculated:
Overpriced call: Ce <= S0, but Ce > S0, overpriced -> sell -> short call
T=0
St < 10
St > 10
Short call 9
0
-(St – 10)
Buy share -8.5
St
St
Invest -0.5
0.53
0.53
0
St + 0.53
10.53
Underpriced call: -> buy -> long call:
T=0
St > 20 (K)
St < 20 (K)
Long call -3.8(given in
St -20
0
question
Short sell stock 23
0
0
Invest -19.2
21.01
21.01
0
St + 1.01
21.01
Overpriced put: -> sell -> short put:
T=0
St < 50 (K)
St > 50 (K)
Short put 49.5 (given in -(50 – St)
0
question)
Invest -49.5
50.50
50.50
0
St + 0.5
50.50
Underpriced put: -> buy -> long put:
T=0
St > 40
St < 40
Long put -0.8
0
40 – St
Buy share -37
St
St
Borrow 37.8
-39.74
-39.74
0
St – 39.74
0.26
Lecture 6
Once we understand the basics of call and put options, we can start combining
multiple “vanilla” options to form sophisticated option trading strategies.
– Risk management: to re-engineer the payoff and risk of an existing position, or
– Speculation: to take a specific position to profit from a predicted price
movement.
Put-call parity:
– There is a relationship between the price of calls and puts which are written on
the same stock (and have same strike and time to expiry).
– Put-call parity says that, if we know the price of a call, we can derive the price of
a put.
– And vice versa: if we know the price of a put, we can derive the price of a call.
Lecture 8
Valuation of options
– Options must trade with a specific range:
Lecture 9 and 10
European-style payoffs: – A derivative is “European” style if its payoff depends
only on the finishing share price (ST ). – European derivatives can be valued very
quickly using: • Pascal’s triangle (which tells how many paths lead to a specific
ending node) • Path probabilities to each ending node
Long call: max(0, ST- X)
Long put: max(0, X - ST)
Lecture 11
Many exotic derivatives have payoffs that depend in some way on the path
travelled by the underlying asset over the life of the derivative: path-dependent
payoffs.
– Floating lookback call: max (0,ST - Smin)
– Floating lookback put: max (0,Smax - ST )
– Fixed lookback call: max (0,Smax - X)
– Fixed lookback put: max (0,X - Smin).
– Asian call: max(0,Savg - X)
– Asian put : max(0,X - Savg)
Fixed Lookback call:
Floating Lookback call: (follow each path and take min ignore initial):
Asian call: