- Financial Engineering 213
We shall build a simple model reflecting the analysts’s point of view. Addi-
tional information obtained by the analyst will result in modified probabilities
as compared to the market probabilityP. (We assume, as before, that the
market probability of up and down movements is 1/2 in each step.) Namely,
the probabilityQassigned by the analyst will be the market probabilityP
conditioned by the event 58≤S(20)≤62. In particular,
Q(S(20) =x)=P(S(20) =x| 58 ≤S(20)≤62)={ P(S(20)=x)
P(58≤S(20)≤62) if 58≤x≤62,
0ifx<50 orx>62.As a result, the analyst will arrive at the following values:
1.Stock.Under the modified probabilityQ
μS∼= 2 .6788%,σS∼= 3 .9257%.
2.Call Options. Take a call with strike price X = 58 dollars. Hence
CE∼= 3 .2923 dollars (this price is found in the binomial model without any
restriction on the range of stock prices after 20 days). For an investment in
options we find that
μC∼= 8 .8816%,σC∼= 71 .095%.
3.Bull Spread.Construct the spread by purchasing a call with strike $58
and selling a call with strike $60. The premium received for the latter is
$2.10, hence a single spread costs $1.18. The expected return and risk are
μbull∼= 38 .4094%,σbull∼= 52 .3997%.
4.Bull Spread Combined with Risk-Free Asset.Investing 94.58% of
the capital in the risk-free asset and the remainder in a bull spread, we can
construct a portfolioPwith the same expected return as stock, but lower
risk,
μP∼= 2 .6788%,σP∼= 2 .8396%.
From the point of view of VaR, we consider the worst case scenario (among
those admitted by the investor) whenS(20)∼= 58 .59 dollars, which may happen
with conditional probability 0.2597. In this scenario each of the above invest-
ments will bring a loss, which can be regarded as VaR at 74.03% confidence
level. The values of VaR and the market price of risk are collected below:
Investment Stock Call options Bull spread Bull spread with
risk-free asset
Market price
of risk^0.^50.^10.^70.^7
VaR
at 74.03%