Risk-Neutral Pricing of a Claim Paying the Square of a GBM

Summary

The document discusses pricing a security whose payoff is the square of an asset price following geometric Brownian motion. Under risk-neutral valuation, the price is the discounted expected payoff. The questioner applies Itô’s lemma to the squared price but then struggles to integrate the resulting stochastic differential equation.

The answer suggests two routes: calculate the second moment of the risk-neutral GBM directly, or use the log of the squared price and solve the corresponding stochastic differential equation. Under risk-neutral dynamics, the drift uses the risk-free rate, and the second moment produces the stated exponential dependence on the rate, volatility, and time to maturity. The response is a concise hint rather than a full derivation, and assumes the standard GBM model with constant volatility and the risk-neutral measure.

Key ideas

• Risk-neutral pricing discounts the expected terminal payoff under the risk-neutral measure.
• For a squared-price payoff, the relevant expectation is the second moment of the asset price.
• Itô’s lemma can be combined with the log of the squared price to solve the stochastic dynamics.
• The result relies on the standard geometric Brownian motion assumptions and risk-neutral drift.

Tags

# Risk Neutral Pricing Exercise


# Risk Neutral Pricing Exercise












I have the following exercise: A financial security pays off a dollar amount of $S_T^2$. Using Ito`s Lemma, what is the price today $V_t$ of this security? (S follows a Geometric Brownian Motion $dS = \mu Sdt + \sigma S dB$ )

I know that $V_t = e^{-r(T-t)}E^Q[V_T]$. So I tried to calculate $E^Q[V_T]$ with Ito`s Lemma:

- $dV/dt = 0$

- $dV/dS = 2S$

- $d^2V/dS^2 = 2$

So Itos Lemma yields $dV = (2\mu + \sigma^2)S^2 dt + 2\sigma S^2dB $. Now, I have to solve this: $\int_{t}^{T} S^2 = (2\mu + \sigma^2) \int_{t}^{T} S^2 ds + 2\sigma \int_{t}^{T} S^2dB $, but I don`t know how to get the result $V_t = S^2e^{(2r + \sigma^2)(T-t)}$.

Can anyone help me with this exercise?

## Answer by Rylan (score 3)

https://quant.stackexchange.com/a/76177

A few comments:

The question says "using Ito's lemma". We can also do it without using Ito's Lemma by simply calculating $E^Q(S_T^2)$ -- this should give you something that agrees with your form of $V_t$ that you were not sure how to get to.

To use Ito's Lemma in your calculation, I'm not entirely sure what approach the exercise is looking for, but if you recall solving the SDE of GBM, you can do something similar here to get an expression of the form $$\log(S^2_T) - \log(S^2_t) = (2r-\sigma^2)\int_t^Tdu + 2\sigma \int_t^Td\tilde{W_u} $$

which is a few short steps from the solution you're seeking.

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