Risk-Neutral Prices for a Squared-Stock Payoff and a Digital Put
Summary
The problem asks for Black–Scholes prices of two terminal payoffs: the square of the stock price and a unit payment when the stock finishes below a strike. It applies risk-neutral valuation by discounting each payoff’s expected value. The squared payoff uses the lognormal stock distribution’s second moment, while the digital payoff reduces to the discounted risk-neutral probability that the stock ends below the strike.
The attempted squared-payoff result follows from the stated geometric Brownian motion assumptions. The digital calculation, however, contains a distribution error: the log of the stock-price ratio is normally distributed, but the ratio itself is lognormal, not normal. The final probability expression should consequently use the standard normal cumulative distribution function evaluated at the standardized log-strike threshold. The setup illustrates moment pricing and digital-option valuation, while also showing why transforming a lognormal variable requires care. It assumes the standard Black–Scholes framework and does not discuss dividends or alternative rate assumptions.
Key ideas
- Risk-neutral pricing discounts the expected terminal payoff at the risk-free rate.
- The squared-stock payoff is priced using the second moment of the lognormal stock price.
- A unit payoff below a strike is valued as a discounted risk-neutral probability of finishing below that strike.
- The logarithm of the stock-price ratio is normally distributed under the stated model.
- The attempted digital-payoff derivation incorrectly treats the ratio itself as normally distributed.
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Full text
# Price of financial assets at $t=0$ in Black-Scholes framework
# Price of financial assets at $t=0$ in Black-Scholes framework
> Given the share price equation $$ dS_t=rS_tdt+\sigma S_tdW_t $$ working in the framework of Black-Scholes model, find the price at $t=0$ of the following two financial assets: (a) The asset pays at $t=T$ exactly $S_T^2$. (b) The asset pays at $t=T$ exactly $1$ if $S_T<K$, where $K$ is constant specified in the contract.
My attempt at solution.
(a) Since the solution of share price equation is given by $$ S_t=S_0\,\text{exp}\left[\left(r-\tfrac12\sigma^2\right)t+\sigma W_t\right] $$ we calculate using risk-neutral pricing \begin{align} C_0&=e^{-rT}\mathbb{E}_Q\left[C_T\right]\\ &=e^{-rT}\mathbb{E}_Q\left[S_T^2\right]\\ &=e^{(r-\sigma^2)T}S_0^2\,\mathbb{E}_Q\left[e^{2\sigma W_T}\right]\\ &=e^{(r-\sigma^2)T}S_0^2\,e^{2\sigma^2T} \\ &=e^{(r+\sigma^2)T}S_0^2\,\quad \text{(Answer)} \end{align}
(b) Denoting by $\Theta(x)=\begin{cases}1, x\ge 0\\ 0, x<0\end{cases}$ the unit step function we find \begin{align} C_0&=e^{-rT}\mathbb{E}_Q\left[C_T\right]\\ &=e^{-rT}\mathbb{E}_Q\left[\Theta(K-S_T)\right]\\ &=e^{-rT}\,\mathbb{Q}\left\{S_T<K\right\}\\ &=e^{-rT}\,\mathbb{Q}\left\{\frac{S_T}{S_0}<\frac{K}{S_0}\right\}. \end{align} Denote $a=r-\tfrac12\sigma^2$, $b=\sigma\sqrt{T}$, $x=\frac{K}{S_0}$. Then $S_T/S_0=X$, $X\sim N(a,b^2)$ and \begin{align} \mathbb{Q}\left\{e^X<x\right\}&=\mathbb{Q}\left\{X<\ln(x)\right\}\\ &=\mathbb{Q}\left\{\frac{X-a}{b}<\frac{\ln(x)-a}{b}\right\}\\ &=\Phi\left(-\frac{\ln(1/x)+a}{b}\right) \end{align} where $$ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^xe^{-x^2/2}dx. $$ Thus $$ C_0=e^{-rT}\Phi\left(-\frac{\ln(S_0/K)+r-\tfrac12\sigma^2}{\sigma \sqrt{T}}\right)\quad \text{(Answer)} $$
> Question: Is this solution correct?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.