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#                                                   #
#        Chapter 8: Stochastic Calculus             #
#               and the Ito-Formula                 #
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Summary: Recall the main result of the first chapter 
on trading strategies: Hold delta_k stocks at the end 
of day t_k and close the position on day t_N. Suppose 
V_0 is your initial money. Then at time t_N this stra-
tegy has generated the (positive or negative) amount 

 
V_N = V_0 + sum_{k=1}^N delta_{k-1}*(S_k - S_{k-1})     (1)


if S_k denotes the closing price of the traded stock 
on day t_k. Here we assumed zero interest rates. For 
non-zero interest rates, the quantities V and S in (1) 
above have to be substituted by discounted quantities. 

Now assume that the underlying price is modelled by a 
Black-Scholes model in continuous time. Then the sum 
on the right hand side of (1) is a sum over stochastic 
quantities. If we consider the continuous time limit 
of (1), then an obvious notation for (1) would be 


      V_T = V_0 + int_0^T delta_t * dS_t                (2)


with a stochastic differential dS_t. Apparently, in 
equation (1) it makes a difference whether we have

    delta_{k-1} = delta_{k-1}(s_0,...,S_{k-1})         (3a)

meaning we can use past or current information, or

    delta_{k-1} = delta_{k-1}(s_0,...,S_{k-1},S_k)     (3b)

which would mean that we can use future information:
The case (3b) would mean that when we adjust our stock 
position at the end of day t_{k-1}, we already know 
which underlying price S_k will realize tomorrow and 
of course this information then can be used to make 
unlimited profit. Thus, mathematically, it is a crucial 
requirement that the delta's showing up in (1) or then 
in the continuous time version (2), fulfill the require-
ment (3a), they cannot depend on future information. 

In the more mathematical framework of stochastic inte-
grals, the condition (3a) leads to the notion of Ito-
integrals, which are, at least in the context of option 
pricing, apparently the right ones to work with, whereas 
a more general condition (3b) may lead to so called 
Stratonovich integrals, and, in the continuous time limit, 
these objects have indeed different limits. From a prac-
tical point of view this sounds quite obvious, but from 
a more mathematical perspective, for someone who is used 
to numerically calculate standard (that is, non-stocha-
stic) integrals just by discretizing to Riemannian sums, 
this sounds a bit surprising. Thus, in this chapter 
these issues are considered more closely.

 

pdf-file: Chapter 8: Stochastic Calculus and the Ito-Formula


Option Pricing
Time Series Models















Hochschule RheinMain,
Applied Mathematics:


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