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$C= S_0 N(d_1) - K e^{-rT} N(d_2)$ $C$, $S_0$ and $K$ have units of currency (e.g. USD). $N(d1)$ and $N(d_2)$ are unit-less (dimensionless), the formula is dimensionally correct. Considering, $d1 = \frac {ln{\frac {S_0} K} + r T + \frac {\sigma^2} {2} T} {\sigma \sqrt T }$ $r$ and $\sigma^2$ have units of "per year", as they are stated on an annualized ...

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The solution to the above SDE is (this is will known and can be seen by applying Ito's lemma) $$S_t = S_0 \exp\left( (u-\sigma^2/2) t + \sigma B_t \right),$$ Thus the log-return is given by $$\log(S_t/S_0) = (u-\sigma^2/2) t + \sigma B_t$$ and is normally distributed as $B_t$, Brownian motion at time $t$, is normally distributed. In fact the distribution ...

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You can't really derive or prove boundary conditions. You impose them and try to economically motivate them. Let's consider a European-style call option and go through the boundary conditions step by step. $S=0$ When the underlying asset's value is zero, then the option to buy this asset is worthless. Thus, $$C(t,S=0,v)=0.$$ $S\to\infty$ As the underlying ...

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A very interesting topic ! Black-Scholes originally did not make use of the Girsanov theorem and arrived at the equation the way you described. Later theoretical work on arbitrage pricing uncovered the concepts of the risk-neutral measure and derivatives pricing as an expectation under that measure. That work relies on stochastic calculus far more and one ...

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Approximating implied volatility of European options can be done in a few ways--this is just one. Below is a python implementation that uses Newton Raphson. You can use the implied_volatility function to find the approximate implied volatility. You can then check it by plugging the output from that back into the option_price function. import numpy as np ...

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In general you have don't have an exact solution for the non linear equation - so you have to use numerical methods. Have you seen this Non linear option pricing book. It covers the topics that you mentioned. People generally do compare the prices produced by non traditional models to the traditional ones - the purpose could be to benchmark the model ...

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From the GBM we have: $$(S_{t}-S_{t-1}) \approxeq dS_t = \mu \cdot S_t \: dt + \sigma \cdot S_t \: dW_t,$$ where $W_t$ denotes a Brownian motion. Here, the first part, $(S_{t}-S_{t-1})$, can be seen as a discretization of $dS_t$. Now, squaring the GBM, $dS_t^2$, is an informal way of denoting the quadratic variation of the process, that is $[dS_t,dS_t]$. A ...

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You got to be careful with $\mathbb{P}$ and $\mathbb{Q}$. Indeed, $N(d_2)$ is the probability of the event $\{S_T\geq K\}$ in the risk-neutral world. Note that $r$ (or $r-q$) is the drift in the risk-neutral world and hence this variable occurs in $d_2$. Since time to maturity and volatility are typically small numbers, i.e. $d_1=d_2+\sigma\sqrt{T-t}\approx ... 3 Insurers do use derivative pricing models such as Black-Scholes to price the sort of guarantees you describe. As far as I know, this used to be known as the "replication method" in the industry jargon, and it allows insurers to price guarantees in a market-consistent manner, hence enabling them to efficiently hedge them with traded instruments. In particular,... 3 The first equality is a bit tedious to derive but straight-forward. As commented by vanguard2k, the notation$N'(x)$is meant to denote the density function of the standard normal distribution: $$N'(x) = n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$$. Simply insert the$d_{i}$terms, $$d_1 = \frac{\log\left(\frac{S}{K}\right) + (r-q+\frac{1}{2}\sigma^... 3 Accounting data won't work for what you are looking for. The only way to do it is to look to public firms on same industry, similar growth stage, same regulatory/legal challenges and compute the volatility of those and use it as a proxy for your firm. It is the best you will be able to get, and it will be a bad approximation. The Merton and KMV models ... 2 The original Merton model takes a simplified view of the debt structure in assuming the total value of outstanding debt (or some portion thereof) D matures at a specified time T. Shareholders are long a European call option on the firm value struck at the face value of debt and bondholders are long a risk-free zero coupon bond and short a European put ... 2 The jumps are modeled as an independent Compound Poisson Process. Here's a paper (Leopold Simar: Maximum Likelihood Estimation of a Compound Poisson Process in The Annals of Statistics 4(6) November 1976) that describes how to get MLE for such process 2 Regarding your first question, you actually have:$$ d\Pi_t=-\left(\frac{\partial f_t}{\partial t}+\frac{1}{2}\frac{\partial^2f_t}{\partial S_t^2}\sigma^2S_t^2\right)dt$$The equation represents the portfolio evolution in an infinitesimal timespan dt (i.e. from t to t+dt). Note that the term S_t is the stock price at time t hence it is already ... 2 The PDE, in its original form, has got variable coefficients -depend on S and t - e.g., co-efficient of \frac{\partial^2 V}{\partial S^2} has got \sigma(t) and S. They are hard to solve and analyse, and if one can find some transformations of variables, that reduce it to constant coefficient, then it gets a lot easier. P Wilmott has chosen the three ... 2 Expected volatility in the underlying price over the life of the option is a major component of the BSM option pricing model. When you calculate the volatility based on the current market price, you're figuring out what the market thinks the volatility would be, that's why it's called implied volatility. So to answer your question, you can either assume a ... 2 It seems you have not done any basic research. Reading this wikipedia page should answer a lot of your questions. American is not related to U.S (no geographical relation). It is a naming convention that may have some historical explanations. By large, equity stocks are listed products - not OTC traded (though this obviously also exists). Implied vol is not ... 1 It's the current spot price, or S_t. The "next chapter" might show if/why they drop the subscript for this approximation, but in the end the variable in the black-scholes equation will be the current spot price. 1 Your option has exposure to interest rates for two different reasons: The discounting of (expected) terminal payoff. The forward (~cost of financing of the delta hedging). Mathematically, if you note your call as a function of rates and of the forward (itself a function of rates) C(r, F(r)), you have by chain rule:$$\frac{dC}{dr} = \frac{\partial C}{\... 1 You need to hedge dynamically (ie, with changes in IV/delta) to accurately hedge your position. This also winds up being a potential risk for a covered option position if there are big moves or general lack of liquidity, since it can be more difficult and also more important to hedge at those times given large market moves. 1 Here is an answer from the ASX for anyone interested: You might want to consider using the Black 76 model. https://en.wikipedia.org/wiki/Black_model XJO options are over the XJO index however the market prices them using the SPI future (Futures contract over the S&P/ASX 200 index) as the underlying. You can use Black Scholes but will need to create ... 1 If$\sigma=0$there is no randomness: the spot follows a single deterministic path. That is, the measure consists of a point mass at that path. Any equivalent measure can again only give a point mass at that same path, with the same drift. So in this case we must have$\mu = r\$ to have an equivalent martingale measure. This is arbitrage free, but there ...

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Although I, admittedly, did not go hunting through your code for an error, I have seen this phenomenon before using this model. This model (like all other models) isn't perfect. This is especially true when you can only observe those parameters that come from the balance sheet quarterly. There are scenarios where no asset vol can imply the current market ...

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