sparse grid method in the libor market model. option valuation and the

Delta. In order to set up a riskfree hedge, as in (2.17), one needs to have the knowledge
of the process for ∆ t . Recall, that ∆ t shows how much of the underlying has to
be held at a timepoint t as a part of the hedge portfolio. On the other hand, ∆ t can
be interpreted as the sensitivity of the option’s price to the movement in its underlying
and, therefore, is equivalent to the first derivative of the option value with respect to
the price shifts in the asset. Note, that Black-Scholes PDE, given by (2.20), implicitly
provides us with such information.
Considering its utmost importance for replication of a claim, ∆ t will reappear in
the focus of our attention later in this text.
2.3 Pricing with Martingales and Measures
Let’s try to fit the main ideas of derivative pricing developed so far into a couple of
sentences. The risk of selling a derivative security can be hedged away by establishing
a replicating, self-financing portfolio of market assets. The cost of setting up a strategy
that eliminates the risk for a short side is, by no-arbitrage, the only correct price for
a product being replicated. According to no-arbitrage, any riskless asset or a porfolio
must earn a risk-free rate.
We have just seen that an option value can satisfy a partial differential equation,
therefore it can be approximated numerically with Finite Difference or Finite Volume
Methods. Another approach that has been discussed is risk-neutral valuation. It involves
taking expectations of a future payoff with respect to a unique risk-neutral probability
measure, possibly resulting in a closed form expression.
Complementing each other, both approaches are used intensively and ultimately
arrive at the same result. In this section we will introduce some mathematical tools
that are employed by risk-neutral valuation in continuous time, namely, martingale
processes and the change of measure result.
Some definitions
• Numeraire. A Numeraire is any asset B, whose price is strictly positive at any
moment in time. For example, S t
B t
is a price of a discounted stock price S t . The
price of the cash bond B t serves as a numeraire for the stock price.
• Martingales. A martingale is a F t -adapted stochastic process X t such that under
a certain measure, e.g. Q:
E Q = [ X t |F t
]
= Xs , ∀s < t
In words, Q-expectation of the future value of the process is always a current
value of the process. As time evolves, the process does not drift away from
its starting point, but fluctuates in its neighbourhood. Again we have already
encountered such processes previously in the text. The process for discounted
stock price e rt S t is a martingale under the risk-neutral measure, see (2.2).
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