Final answer:
The student is seeking the derivation of the value of a cash-or-nothing digital put option in the Black-Scholes framework, which involves integrating the risk-neutral probability function and multiplying by the discount factor (rT).
Step-by-step explanation:
The student is asking for a derivation of the valuation formula for a cash-or-nothing digital put option within the Black-Scholes model. Specifically, the student is interested in the expected value under the risk-neutral measure, designated by ℓ, of a binary indicator that pays off if the asset price S is less than or equal to the strike price K. In the Black-Scholes framework, this can be represented as ℓ[1{S ≤ K}], with the present value factor given by (rT), where r is the risk-free rate, and T is the time to maturity of the option.
The formula for the value of a cash-or-nothing put option can be found by integrating the risk-neutral probability density function of the asset's price at time T, discounted by (rT), and evaluating it at the strike price K. The Black-Scholes model uses a logarithmic normal distribution for the asset price under the risk-neutral measure, which leads to using the cumulative distribution function of a standard normal to determine this probability.
In summary, the valuation involves calculating this integral and applying the Black-Scholes assumptions about the statistical behavior of the asset's price. Such a valuation is used by financial practitioners in the pricing and hedging of options.