A Conditional Black–Scholes Formula for Up-and-In Puts in Stochastic Volatility

Summary

The note sketches a pricing expression for an up-and-in put under a stochastic volatility model in which the asset return and volatility share a Brownian driver, with an additional independent return component. For a barrier at or above the current asset price and strike, the proposed value is an expectation of a transformed Black–Scholes call price. The transformation uses a random factor built from integrated variance and the correlated volatility driver, while the remaining volatility input is based on average integrated variance scaled by the uncorrelated component.

The stated computational advantage is a reduction of the pricing problem’s dimensionality by one, in the spirit of Hull–White conditioning. The note presents the formula as a derivation and asks whether related work has appeared in the literature. It does not provide a numerical example, calibration, proof details, or validation against another pricing method. The expression is therefore best read as a proposed analytical result under the specified model assumptions, rather than evidence of accuracy for arbitrary stochastic volatility dynamics.

Key ideas

• The note proposes a conditional pricing expression for an up-and-in put under correlated stochastic volatility.
• The expression converts the barrier claim into an expectation involving a transformed Black–Scholes call price.
• Integrated variance and the correlation-driven volatility component enter through a random scaling factor.
• The conditional formulation is presented as reducing the dimensionality of the pricing problem by one.
• The document offers no numerical validation and leaves the relationship to prior literature as an open question.

Tags

# Single barrier options in stochastic volatility models


# Single barrier options in stochastic volatility models












In this note/sketch, I derive among others a closed-form formula for an up and in put (UIP) in stochastic volatility models of the form $$ dS(t) = \sigma(t) S(t) \left[ \rho dW(t) + \sqrt{1-\rho^2} dZ \right],\\ d\sigma(t) = a(\sigma(t)) dt + b(\sigma(t)) dW(t) $$ where $dW(t)dZ(t) = 0$.

I argue that the price of an UIP with barrier $B \geq S(t), K$ is $$ UIP(t) = E_t \left[ \frac{K}{BM_{t,T}} C^{BS} \left(S(t)M_{t,T}, \frac{B^2M^2_{t,T}}{K},\sqrt{1-\rho^2} \, \sigma_{t,T} \right) \right] $$ with $C^{BS}(\cdots)$ denoting the Black-Scholes price, and $$ \sigma_{t,T} = \left( \frac{1}{T-t} \int_t^T \sigma^2(u) \, du \right)^{\frac12}, \\ M_{t,T} = \exp\left\{ -\frac{\rho^2}{2} \int_t^T \sigma^2(u) \, du + \rho \int_t^T \sigma(u) dW(u)\right\}. $$

The advantage of such a Hull and White type closed-form formula is that the dimensionality of the pricing problem is reduced by one.

I've been going through various papers on barrier options pricing and have not come across the above closed-form expression yet for SV models with nonzero correlation. Have I missed a paper where this (or something similar) has been discussed?

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