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I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ...


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 ...


3

Let the risk-neutral dynamics under your LV model be given by $$ \frac{d S_t }{S_t } = \mu_t dt + \sigma(t,S_t) dW_t $$ Let's drop the drift contribution (not relevant here) and apply Itô's lemma to obtain: $$ d \ln(S_t) = -\frac{1}{2}\sigma^2(t,S_t) dt + \sigma(t,S_t) dW_t $$ In order to simulate from this SDE, you need to choose a particular discretisation ...


3

You can convert the implied volatility to local volatility using this formula: $\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$ Where y is ...


2

For the first question, you can just plug in t for T and S for K: $\sigma^2 \left(t, S \right)=\left. \sigma^2 \left(T,K\right) \right|_{T=t,K=S}$ For the Monte Carlo part, the barrier would apply to the history of the stock price over some window (which could be from today to the option maturity, but other variations are possible) instead of just the ...


2

The market will quote Call and Put options prices within a bid-ask spread. In order to imply the volatility, one may choose to use the bid, the ask, or the mid. Although the mid is a better idea in general, there is no right choice. The point is that there is always a spread in the implied volatility. Now, the Put-Call parity only holds within the a spread. ...


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

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,...


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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), ...


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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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