Setting Up a Derivative Pricing PDE for State-Dependent Volatility

Summary

The document poses a derivative-pricing problem for an asset whose price follows a stochastic differential equation. Its drift depends nonlinearly on the price through a linear term and a sine term, while its diffusion coefficient is proportional to price divided by the sum of squared price and one. The random shock is described as standard normal.

The requested task is to derive a Black–Scholes-style partial differential equation for pricing a derivative on this asset. The equation itself is not supplied, and the prompt does not specify a risk-free rate, dividend yield, or other market assumptions needed to complete a pricing model. As a result, the material frames a modeling question about how state-dependent drift and volatility enter derivative pricing, but gives no derivation, solution, numerical example, or validation.

Key ideas

• The asset model specifies a nonlinear, price-dependent drift.
• The diffusion coefficient also varies with the asset price.
• The prompt asks for a Black–Scholes-style pricing PDE for derivatives on the modeled asset.
• No PDE solution or derivation is included.
• Further market assumptions are not stated in the prompt.

Tags

# Derivative Pricing of an Asset


# Derivative Pricing of an Asset












The Stochastic Differential Equation that models the change in an asset price is

$$ dS = (12S-sin(S))dt+\frac{\sigma S}{S^2+1}dX $$

where dX's are random variables drawn from standard normal distribution. What is the Black –Scholes-ish partial differential equation that must be solved in order to price derivatives of this asset

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.