212 Mathematics for Finance
4.Options Combined with Risk-Free Investment.The risk can be ad-
justed to an arbitrary level if options are combined with a risk-free asset.
Suppose that the investor is willing to accept similar risk to the stock in-
vestment. Investing 94.77% of the capital without risk and the remainder in
options, the investor can construct a portfolio with the same standard devi-
ation as that for the stock. The expected return on the portfolio is slightly
lower than that on stock,μP∼= 1 .3457%,σP∼= 8 .0962%.Remark 9.2
The slope of the line connecting the risk-free assetFwith any other portfolioA
on the (σ, μ) plane is given byμAσ−ArF,called themarket price of risk.Itcan
be used to compare different portfolios: Those with steep slope are preferable.
We can see that the above risky investments have similar values of the market
price of risk, about 0.1 in each case. (These values would in fact be identical if
the Black–Scholes model were used for stock prices.)
The advantage of the investment in a portfolio of options and risk-free
assets can be seen if we consider VaR, given in the table below for two different
confidence levels (chosen to be compatible with the probabilities in the binomial
model). On the other hand, VaR would be disastrously high if the whole amount
were invested only in options: The investor could lose everything at the given
confidence levels.
Investment Stock Call options Forwards Calls with
risk-free asset
Market price
of risk0. 1087 0. 0882 0. 0931 0. 0882
VaR at 94.23% $1, 931. 78 $15, 000. 00 $9, 753. 63 $798. 73
VaR at 99.41% $2, 836. 84 $15, 000. 00 $14, 278. 95 $798. 73Case 9.3
An analyst researching the company has come to the conclusion that the stock
price after 20 days will not fall below $58 or raise above $66. All market pa-
rameters remain as in Case 9.2. From the point of view of the analyst, compare
the expected return and risk for stock, options and a bull spread with strike
prices $58 and $62.