# Tagged Questions

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

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### Merton model riskless self-financing derivation

Suppose $dA_t = A_t[\mu dt+\sigma dW_t]$ (assets' value) under the physical measure, plus the other assumptions of the Merton model. Suppose further that debt and equity are tradeable assets that ...
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### What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S: $dS=\mu S dt + \sigma S dX$ Clearly $S$ ...
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### Black-Scholes Equation - Riskless portfolio derivation

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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### American Swaption Pricing with Monte-Carlo method

I want to price an American swaption but I am not sure about what I am doing. Tree methods and PDE discretization seem difficult to adapt to a swaption. I am trying a Monte-Carlo approach. (in ...
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### Black-Scholes derivation assumption contradiction

In many books and derivations of the Black-Scholes PDE one sees that $$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF$$ which implicitly assumes that $d\Delta=0$. Somewhere down the road one then ...
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### Black (1976) model: relationship between spot and forward prices

Does the Black (1976) model require the existence of the relation $F(t,T)=S(t)e^{r(T−t)}$? I studied the derivation of the Black-Scholes formula. However, although I know the Black formula, I've ...
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### Trinomial model converges to Black-Scholes weakly

Consider risk-neutral trinomial model with $N$ periods presented by $$S_{(k+1)\delta}H_{k+1}, \ \ \text{for} \ \ k=0,\ldots,N-1$$ where $\delta:=\frac{T}{N}$ and $\{H_k\}_{1}^{N}$ is a sequence of i....
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### Black-Scholes Equation with dividend

Consider a European option with payoff $$g(S_T) = S_T^{-5}e^{10S_T}$$ Assume that the interest rate is $r = .1$ and the underlying asset satisfies $S_0 = 2, \sigma = .2$, an pays dividend at ...
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### What should be the sign of greek letter $\rho$?

I'm reading the book Risk Management and Shareholders Value in Banking by Resti & Sironi. I quote a paragraph from the book (Chapter 5, appendix): The derivative of an option’s value with ...
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### Calculate volatility from call option price

Given call option price, what is the simplest formula to get the volatility value ? Test Data: ...
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### Black Scholes Formula for Collar Option

I am wondering if there exists a Black Scholes pricing formula for a collar option?
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### Black-Scholes PDE & Terminal Condition

Just a quick question I was hoping someone could shed light on. So far I am familiar with the Black-Scholes PDE with the terminal condition at time $T$ been $V(t=T,S)=(S-K)^+$. I also understand ...
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### Result linked to Black-Scholes evaluation

Why does this $$Se^{-D(T-t)}e^{-d_1^2/2} - Ee^{-r(T-t)}e^{-d_2^2/2}$$ equal to $0$? (Where $E$ is a strike)
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### How to compute the volatility for the Merton's Model for Private firm?

After one day of research i did not figured how to compute the input volatility for PRIVATE COMPANY in order to calculate the PD. My goal is to compute the PD of each of my company in my portfolio, ...
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### Pricing call option

Question: The price of a stock is 100. With equal probabilities, it either goes up to 130 or down to 70. What is the price of a 1 year call option with exercise price 100. Risk free rate is 5%. ...
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### Implied Volatility Calculation

I want to calculate the implied volatility from the option data that I took from Bloomberg (call Option written on S&P500 index with the maturity of 19-Dec-2009 and strike of 1300), but volatility ...
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### How market making in Index options is done?

I have been thinking about this one for last couple of days. With options on share we hedge on cash and the underlying equity as per Black-Scholes formulation. But I am confused on Index options. ...
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### Is this arbitrage?

Assume the stockprice as in the Black-Scholes model (Geometric Brownian Motion): $$S_t=S_0e^{(\mu-\sigma^2/2)\cdot t+\sigma W_t}$$ Wouldn't there be an immediate arbitrage opportunity, to just buy ...
### Find the parameter $d$ of the Affine Option Pricing Model in Duffie, Pan and Singleton (2000)
According to Duffie, Pan and Singleton (2000) for any real number $y$ and any $a$ and $b \in \mathbb{R}^n$, the price of a security that pays $\exp(aX_t)$ at time $T$ in the event that $bX_t \leq y$ ...