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To simplify the problem, let us consider normal local volatilities $\sigma \left ( S_t, t \right)$ and implied volatilities $\sigma_i \left ( K, T \right)$ such that the model is: $$dS_t = \sigma \left ( S_t, t \right) dW$$ (no rate, repo, dividends, etc.) and $\sigma_i \left ( K, T \right)$ is the normal volatility input into Bachelier's formula ...

8

Some points below as food for thought: Suppose you possess an implied volatility surface over a continuous strike cross time to expiry domain (how to get there from the discrete market specification is another question). Further assume that you have to price a path-dependent option, e.g. a Barrier or an Asian. If you are using Black-Scholes, what implied ...

4

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3

3

I worked on a single name Equity Derivatives trading desk. Implied volatility is remarked at least once per day, but that depends also on market movements, volatility movements, volumes, etc. For this reason it is usually marked at the end of the day and re-checked after opening. About intra-day risk management, they mostly use a prior end-of-day ...

2

Note that with $H(\cdot)$ the Heaviside function $$\frac{d}{ds} H(s-K) = \delta(s-K)$$ but $$\frac{d}{dK} H(s-K) = \color{red}{-}\delta(s-K)$$ You can also use the Leibniz integral rule to write that $$\frac{d}{dK} \int_K^\infty \phi_{S_T}(T,s) ds = -\phi(S_T,K)$$

2

As spot goes to infinity, the transition density goes to zero, and hence the result. Underlying assumption being that it goes to zero faster than quadratic($s^2$). Ps: there seems to be a typo in your derivative but does not matter for the purpose here.

2

Let us denote $\mathcal{C}$ the European call prices and consider the map $K\mapsto\mathcal{C}(t,T,S_t,K)$ the market price of calls maturing at $T$. We can obtain a link between the risk-neutral probability distribution of the stock price and current call prices via the Breeden-Litzenberger formula: $$f^S_T(K)=f_{S_T|S_t}(S_T=K)=e^{r(T-t)}\frac{\partial^2\... 1 Reposting comments as an answer: If you are doing Monte Carlo you'd have a new volatility function to use (rather than a constant vol like in black scholes) for each time and stock price from the local volatility model (It's usually written \sigma (S, t)). So you could use this local volatility function during a simulation regardless of the type of payoff.... 1 Just the chain rule; \frac{d}{dK} H \left (S-K\right)=\delta \left (S-K\right) \frac{d}{dK} \left (S-K\right)=-\delta \left (S-K\right)  1 Although a local volatility model$$ dS_t = \sigma(S_t,t) S_t dW_t  is able to fit exactly quoted market prices of vanilla options, the concept of vega in a local volatility model is at best ill-defined, even for vanilla options. However, if you insist on obtaining vega for an option in a pure local volatility model, then you could bump the functional ...

1

In his article, Dupire (1994) developed the local volatility approach under the assumption that options are traded for a continuum of maturities and strikes. In reality, only a finite number of options generating a grid of strikes and maturities is traded. Then the reconstruction of the local volatility function is obtained by interpolation methods. However,...

1

Unfortunately not written in Python, but in R. If you have experience with R this real life example posted on an underground quant blog has step by step what you may be looking for: (Scroll down to conclusion) https://quantipy.wordpress.com/2017/08/21/implementation-of-dupires-formula-for-local-volatilities/ (I do not take credit for this persons work), ...

1

I first want to clarify one statement. You write "the local volatility of the option ...". A local volatility is, unlike an implied volatility, not a property of an option but instead a function of time and the underlying assets price. I.e. an option does not have an implied volatility but the local volatility function describes the dynamics of the ...

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In equity options a no-arbitrage argument shows that implied volatility has to be continuous along the forward line The "forward line" is the path of forward levels as a function of the maturity T conditional on a certain value of the spot at T=0 or conditional on the terminal value $F_T$ at maturity. The forward line that terminates at $F_T = K$ is the ...

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The benefit of Dupire formula is that you can find a local volatility function from the market price of vanilla options, then you can use these local volatilities (by constructing a surface) to value exotic options. Also, this single (theoretically) unique volatility function will value all the vanilla options in line with their market prices rather than ...

1

From a cursory look, the FdBlackScholesBarrierEngine seems to do what you want; when the localVol parameter is set to true, it will use the local volatility contained in the passed process. I'd suggest you to check the code, though. As a further note: the GeneralizedBlackScholesProcess class converts the Black volatility to the local one internally (see the ...

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