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9 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 ...
bhutes's user avatar
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7 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 ...
amdopt's user avatar
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6 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-...
Kevin's user avatar
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6 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 ...
Kevin's user avatar
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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 ...
Ivan's user avatar
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5 votes
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Black and scholes option pricing

Let us start by considering a bear spread strategy, consisting on long a European put with strike $K_2$ and short another European put with strike $K_1$. Then the payoff of this portfolio at expiry $T$...
Daneel Olivaw's user avatar
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,...
Daneel Olivaw's user avatar
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 ...
AKdemy's user avatar
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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 ...
Pleb's user avatar
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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 ...
Magic is in the chain's user avatar
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 ...
Magic is in the chain's user avatar
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 ...
Daneel Olivaw's user avatar
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 ...
RRL's user avatar
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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 ...
Will Gu's user avatar
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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'...
Mehdi Zare's user avatar
1 vote

interpret payoff

Note that $$min(max(S-K_1,0),K_2)=max(S-K_1,0)-max(S-(K_1+K_2),0)$$ so this is the payoff of a call spread (you're long the $K_1$ strike and short a $K_1+K_2$ strike). Then you can use standard Black-...
user35980's user avatar
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1 vote
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Calculating the PnL of a delta-hedged option at a point in time

If we denote the value of a BS call option by $C(vol, exp)$ then we know that at any time $t$ the total expected p/l including the option value and the expected realized delta hedging p/l is $$C(0.27, ...
dm63's user avatar
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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)...
Kurt G.'s user avatar
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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 ...
D Stanley's user avatar
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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 ...
Soumirai's user avatar
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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 ...
Chris's user avatar
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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 ...
Zeus's user avatar
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