# 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?

• I remember that I have seen this a long time ago in a note by Peter Carr. Not sure if he ever published it. A bit of care is needed when you do the decomposition that leads to a $E_t$ around the BS closed form solution because at this stage the BM driving $\sigma$ and the one driving $S$ should be independent. If they are not you end up with an adjustment to $S(t)$. I have not checked your formulas in detail but it seems you took this into account. May 2, 2022 at 16:58
• @KurtG. In Carr and Lee section 6 theorem 1 and formula 6.1 (see link below) they discuss asymmetric dynamics, but never really apply to a stochastic vol model with correlation. Furthermore their formula 6.1 is rather abstract and pertains to PCS, not a closed-form Hull White type formula with nonzero correlation. Not sure if this is the paper you meant. math.uchicago.edu/~rl/PCSR22.pdf
– user34971
May 2, 2022 at 18:55
• I'll need to work through above mentioned theorem and formula 6.1 of Carr-Lee to see if it is really equivalent or encompasses the formula I wrote down. I don't think they equivalent are actually - Carr & Lee discuss displaced diffusions, non zero interest rate and local volatility using their theorem 6.1, but not stoch vol model with correlation.
– user34971
May 2, 2022 at 19:06
• I like your approach a lot. It would be nice to find also an approximation for the local vol price and then we could mix them like an LSV. For interest, Yuan and I have a formula for the SV price using a completely different approach. It fits to the smile rather than to a specific SV. Would be v interesting to compare papers.ssrn.com/sol3/papers.cfm?abstract_id=2735702 May 4, 2022 at 19:01
• @PeterA Thank you vm, also for the link to your paper - I'll read it and indeed could be interesting to compare.
– user34971
May 4, 2022 at 20:03