8 votes

Why does Black Scholes formula give inconsistent dimensional analysis result?

$C= S_0 N(d_1) - K e^{-rT} N(d_2)$ $C$, $S_0$ 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 ...
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  • 938
7 votes

Why is rate of return on the stock normally distributed under GBM?

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(...
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  • 13.2k
5 votes
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Boundary conditions Heston's stochastic volatility model

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 ...
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  • 13.8k
5 votes

Black Scholes model without using Girsanov's theorem? It might happen?

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 ...
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  • 1,346
4 votes
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Deriving implied volatility programmatically

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 ...
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  • 3,715
3 votes

Black Sholes Options Pricing Clarification Questions

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 ...
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  • 4,333
3 votes

Black Scholes informal derivation - question about a term in the equation

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 ...
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  • 3,298
3 votes

Nonlinear Black-Scholes model Vs linear Black-Scholes

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 ...
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3 votes
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Option and probability of finishing in the money?

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-...
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3 votes

Risk-neutral pricing the "un"guaranteed benefits of an insurance policy

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,...
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3 votes
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derive black scholes greeks

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'(...
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  • 2,310
3 votes
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How to compute the volatility for the Merton's Model for Private firm?

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 ...
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  • 6,800
2 votes

Proper maturity in the Merton's model

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 ...
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  • 3,250
2 votes

How to estimate Black Scholes parameters using Maximum Likelihood estimate method

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 ...
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  • 682
2 votes
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Problems in understanding BSM formula

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 ...
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2 votes
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Modifying Basic Black Scholes Equation For Time Dependent Variables - Per Wilmott?

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 ...
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2 votes

Deriving implied volatility programmatically

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'...
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1 vote

Why is call option value same as portfolio value at all times in Black Scholes model?

It is fairly standard to hedge a sold option as follows: at any time $t$ buy $\alpha(t)=\frac{\partial}{\partial S}c(t,S(t))$ amounts of stock $S(t)\,,$ and invest $\beta(t)=\frac{c(t,S(t))-\alpha(t)...
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  • 1,419
1 vote

Black Scholes informal derivation - question about a term in the equation

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 ...
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  • 1,073
1 vote

Hull's book - Futures option's rho

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 ...
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  • 604
1 vote

In literature, is IV constantly adjusted during option delta hedging?

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 ...
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  • 1,578
1 vote
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Dividend yield on ASX 200 (XJO) index options

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 ...
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  • 219
1 vote
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Why is the black-scholes model arbitrage free when σ>0?

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 ...
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  • 1,855
1 vote

How to interpret negative asset volatility numerical results in Merton model?

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 ...
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