- Financial Engineering 209
shows the case whenX′′is the average of the other two strike prices.Reversed
butterflyis the opposite strategy, bringing profits when the stock price changes.
(We have already come across the butterfly in the proof of Proposition 7.8.)
Figure 9.4 ButterflyFinally, we observe that any continuous payoff function consisting of straight
line segments can be manufactured from put and call options. Figure 9.5 out-
lines the step-by-step decomposition of a target profile into a portfolio of op-
tions with various strike price values. The number of options for each strike
price is chosen to match the slopes of the target profile. Such a construction is
sufficient for practical purposes because any continuous payoff function can be
approximated by straight line segments.
9.3.2 Case Study ........................................
We shall combine the portfolio theory techniques with the tools described
above. We have in mind an investor with specific views on the future prices
of assets, who is prepared to accept some risk in order to increase expected
return.
Case 9.2
An investor with $15,000 believes that a certain stock price should rise during
the next month, with expected annualised returnμS= 31%. The current stock
price isS(0) = 60 dollars. Call options expiring in 20 days with strike price $60
are available at $2.112. The effective risk-free rate is 12%.
To analyse this case we shall use the binomial model, assuming that trading
takes place once a day and that the market probabilities are the same for up