Black-Scholes is a mathematical model used for pricing options.

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Option greeks: sensitivity to 1% move

In a Black&Scholes framework how can I compute the following sensitivities: to 1% move in the underlying price to 1% move in implied volatility I would like the greeks to tell me how many ...
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94 views

Black-Scholes formula with deterministic discrete dividend (Musiela approach)

For deterministic discrete dividend, there are two approach Musiela approach, works when every dividend are paid at maturity of the option. Hull approach, works when every dividend are paid ...
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57 views

Why gamma and theta have opposite signs?

I saw some textbooks use B-S equation to explain why gamma and theta have opposite signs in most of the cases. For example, John Hull's classic book. The explanation is, first write B-S equation in ...
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76 views

Bivariate Black-Sholes Model

Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market. $r_{f}$ is risk-free rate. Under real-world measure $P$: $dS_{1} (t)=S_{1} (t) ...
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112 views

Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
3
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159 views

Black-Scholes PDE to heat equation, nonconstant coefficients

Can someone provide me with details or a reference on how to transform the Black-Scholes PDE with nonconstant coefficients (i.e. $r=r\left(S,t\right)$, $\sigma=\sigma\left(S,t\right)$) to the heat ...
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76 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
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51 views

What is the main reasons to use Miltersen & Schartz (1998) model for commodity futures options

versus a standard Generalised Black and Scholes model (if there are any?) I have read the paper but I am not to sure about its practical implications as would people with more experience using this ...
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71 views

At-the-money Call Spread approximation

In a trading manual I got during a course, the value of the ATM Call-Spread is approximated by $CS_{ATM}=\frac{1}{2}StrD+(F-m)\times\Delta CS$ The lecturer skipped the part where he derived this ...
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57 views

Calculating the error of a Trinomial Model

I've been trying to find a formula to obtain the maximum relative error a trinomial model with n timesteps will incur given all other inputs as compared to the standard BSM model. I'm concerned mostly ...
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80 views

Changes to option valuation for dollar-pegged underlying

In Russia, options on futures on the RTS index are priced in points instead of currency, with points being directly related to the value of the US dollar such that, for example, if the dollar rises, ...
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28 views

Use of Black-Scholes Model on Guaranteed Fund Investment

I am stuck with a revision question at home on Black-Scholes pricing model. The question is on a fund manager selling one unit of the fund to a customer for S(0) at time 0 and then guaranteeing at ...
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22 views

How to apply the chain rule for partial derivatives to transformations?

I'm currently working to solve the Black-Scholes model partial differential equation (it's a model for a.o. stock option prices). The Black-Scholes equation for a calloption C(S,t) is given by $ ...
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139 views

Black Scholes - how to calculate delta with a vol skew

I am trying to calculate the delta of an option at different strike prices where the underlying has a pronounced implied volatility skew in order to correctly hedge an options strategy. Researching ...
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96 views

PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
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726 views

Black Scholes well coded Python

I have some trouble with the following code. Some jump and a decentered path are present but it's not the case, normally for Black Scholes diffusion ! Is anyone see a problem in my code ? ...
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493 views

Question about Merton model to estimate default probability and recovery rate of the company

I recently come across Merton's model to estimate the default probability and recovery rate of the company. Here is the inputs ...
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188 views

Pricing a Power Contract derivative security

I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is ...
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The role of Gamma in replicating a put

I am analyzing portfolio protection by replication of a put. Having my portfolio with value $V$ I could buy put giving me the payoff $P$ resulting in a call like pay-off scenario $C=V+P$. Say, I ...
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33 views

Implied Vola from historical option prices

I have daily Close data of ODAX-options, obtained from ivolatility.com. One third of the daily data shows premiums that are just above the inner value. Even when inserting an implied vola of almost ...
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42 views

Any Simple Way to Prove Black Scholes Type Identies?

A certain complicated option pricing formula results in products of Black Scholes $N$ components like this: $-p_1N(d_1)N(d_6)+p_sN(d_2)N(d_5)>?0$ where $p_s>p_1$ Trying to find a simple way ...
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Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation

Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation and show that the transformed payoff is g(x,τ) = be^[(1/2)((k+1)^2)τ+(1/2)(k−1)x]H(x),  ...
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77 views

America option early exercice boundary via Monte Carlo simulation

I am trying to calculate an american option price via the simulation of the early exercise boundary using the method presented in this document: Monte Carlo Method For pricing a put Option. I have ...
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287 views

Java Implied Volatility Solving

After using RQuantLib and RCaller from Java I am desiring a bit more speed on my implied volatility calculations (for anyone who has used this knows it is quite slow). I need to price a large number ...
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Explain $1Gamma vs %1 Gamma

What is $1 Gamma and what is 1% Gamma? please describe the difference? I understand Gamma but cant make the diff between the two.
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BS Implied Volatility under Normal returns

If I use theoretical prices under a normal valuation model, and I estimate their implied volatility using BLACK SCHOLES implied volatility, do I'll get corresponding log normal volatility?
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105 views

Black (1976) model: boundary conditions with non-convergence of spot and forward prices

Let's suppose we have a futures contract F in a market where the relation $$F(t,T)=S(t)e^{r(T−t)}$$ doesn't hold. What are the the boundary conditions for the derivation of the Black (1976) ...