Question

Analyze the notion of “risk neutral valuation” in the framework of the black-Scholes model of option pricing

Answer 1

Risk neutral valuation under black schools model:

Under the black schools model, it is assumed that the variance remains constant which is in the case when we look at financial time series data. In this study therefore price of the European call option under the garch model using the risk neutral valuation relationship. The option prices for different spot prices are calculated using simulations.

Since the existence of black & scholes model, it is customary to price options through replication in markets & compute derivative prices in a hypothetical market in which there are risk neutral preferences. Option pricing strategies are therefore done on the basis of risk neutral valuation relationships (RNVR). In the general diffusion there are number of sources of incompleteness introduced through unhedgeable risks. In such cases, the RNVR & girsanov ideas remain, yet some well suited risk premia associated with various sources of unhedgeable risk have been introduced.

The RNVR has the following formula:

¼t[ht+1] = B(t;t + 1)Et[h(Jt; S ¤ t+1 )]

where S ¤ t+1 is a rescaled value of the underlying asset price defined by (t;t + 1)S ¤ t+1 = [St+1=EtSt+1 ]St : In other words, S ¤ t+1 is proportional (given Jt) to St+1 but the mean of its distribution is St=B(t;t+1). The other parameters of the conditional (given Jt) probability distribution of St+1 are identical to the objective ones.