Description
1. Be a Financial Engineer!
Here we will study mixes of European calls and puts to craft di§erent types of payouts.
Here assume maturity date T, with underlying security price ST at T. The payout for
a European call and put, respectively, at time T with strike price K is, respectively:
c (T; T) = (ST K)
+
p (T; T) = (K ST )
+
DeÖne a digital call or put option to have respective payout as follows:
dC (T; T) = 1 if ST > K, 0 otherwise
dp (T; T) = 1 if ST < K 0 otherwise
A long position in one unit of a derivative with payout DT means the value of the asset
in the portfolio is DT . A short position in one unit means the value is DT .
Combining various amounts of long and short positions in call and put options can yield
a general piecewise linear but overall continuous value V (ST ) at the time of maturity.
Including digital options allows discontinuities. Here we will see several important
combinations.
Use a computer to generate plots by assigning reasonable values to the parameters K,
etc. You just need to produce one example of each. In all cases, horizontal axis is
ST > 0 and the vertical axis is V (ST ).
Note: As you graph these, keep in mind you are looking at the ìpayoutî V (ST ). If
you purchase one of these derivatives at a price say Vt at t, then your net ìproÖtîor
ìloss” at time T would be V (ST ) Vt=Z (t; T) (i.e., subtract o§ the money you would
have had in your pocket if you never purchased the derivative). This would be a more
useful graph, perhaps, e.g., to determine the range of ST for which you end up with a
net positive, but that would require determining the price at t, Vt
, which we are not
dealing with here. One thing to look for in any of these instruments: is the potential
loss unbounded? In other words, is min V (ST ) Önite?
Also, in what follows below, except for one case Iím not asking you to derive an analytic
formula for V (ST ). If you write core code to compute the payout for the call, put,
digital call, digital put, then just write code to combine these as suggested to obtain
the graphs. Your code should be general purpose, e.g., with parameters such as K
adjustable. When you generate graphs, plug in reasonable values, e.g., K = 1 or
whatever. Scale your horizontal and vertical axes accordingly.
(a) Graph V (ST ) for each of the following: long call, long put, short call, short put
(b) A straddle is a combination of one unit of a call and one unit of a put for the
same security at the same strike price K. Verify V (ST ) = jST Kj, and graph
this for one choice of K.
(c) A call-put spread is to long a call at K1 and short a call at K2 with K1 < K2.
Graph V (ST ) for one choice of the pair K1; K2.
(d) A butteráy is a combination of the following. Let K1 < K2, and 0 < < 1. Let
K = K1 + (1 ) K2. Then:
long calls at strike price K1
long (1 ) calls at strike price K2.
short 1 call at strike price K
.
Fix one choice for K1 < K2. Graph a butteráy V (ST ) for each of the following
cases: = 1=3; 1=2, 2=3.
(e) Call ladder: with K1 < K2 < K3, long one K1 call, short one K2 call, short one
K3 call.
(f) A digital call spread: with K1 < K2, a long a digital call with strike K1 and short
a digital call with strike K2.
