Questions tagged [black-scholes]

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

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52 views

Pricing with-profit/smoothed bonus annuity using Black-Scholes

Would this be possible? Subsequently, would the pricing of such an annuity be somewhat similar to pricing a lookback option?
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175 views

School project about Black Scholes with stochastic volatility

In a university project I am looking at Black Scholes model with a stochastic volatility. I’m still not quite sure about my focus (I am in the beginning 'Idea phase'). I want to explain the theory ...
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620 views

Black-Scholes formula with deterministic interest rate and dividend yield

Does any one have the Black-Scholes formula for a European call with time-dependent but deterministic interest rate and dividend yield ?
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152 views

B-S Put Option Formula: Derivation using expected value under Q

I have been working on an old problem in one of my finance classes and, since no solution has been provided and I won't be able to contact my teacher anytime soon, I was hoping I could ask you guys to ...
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60 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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233 views

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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933 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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1answer
269 views

Distribution of realized volatility for stock prices from a GBM

If you generate random stock price paths according to a GBM with daily increments, what will be the distribution of the realized volatility? Assume that the realized volatility is measured over daily ...
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12 views

Option Delta Calculation - Local Vol Model vs Black-Scholes Model

I am looking to get the greeks for option chain. Which model does work better for greeks calculation especially the delta. I am having issue with the Black-Scholes Model Delta since it always ...
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18 views

Options delta as a percentage of option price

I'm dissatisfied with the usefulness of delta and would like to get your feedback on a slight tweak on it. Example Consider two options for a made-up stock at \$5 with IVs around 120%. Option A: ...
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1answer
41 views

Black-Scholes and solving for both $r$ and $\sigma$ ; Do I have a unique solution?

Below is a problem that I am working on. I believe that my incomplete solution is correct as far as it goes. I would like to know if my solution is incorrect. I plan to solve the system of two ...
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73 views
+50

Optimal portfolio balancing based on past performance

Suppose you start with some money and invest for a fixed amount of time T. You have one risky asset and one risk free asset to choose from. You can change the ratio of the two assets any time ...
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45 views

Black-Scholes Theory vs Actual Market Price

I have a question of which I am uncertain on how to answer, that is: Assume the Black and Scholes differential equation for option pricing with constant risk free rate, $ r $ and constant volatility $...
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30 views

Derivation of Black-Scholes for a derivative on a stock that pays continuous dividends, and the derivative pays continuous cashflows

I need help with the derivation of Black-Scholes PDE. The condition is that the derivative is written on a stock that pays dividends continuously (dividend yield D). Additionally, the derivative pays ...
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30 views

Calculating the risk free interest rate, or the continuously compounded yield on a T-bill, at any given time

I'm working on a program using the Black-Scholes model to price options over time. I need to be able to derive the risk free interest rate, and found this while researching: In theory, r is a ...
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44 views

Delta hedging an option with earlier expiry

The answer here states: For instance a volatility product that would expire at 10:42 am on a random day would be off term. One that expires at the same time than a major listed contract would ...
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1answer
120 views

Black Scholes Separable Solutions

I want to find all the solutions of the Black Scholes PDE that are of the form f(x,t)=theta(x) or f(x,t)=phi(t). Can someone explain and help with this? I know the PDE formula is $f_{t}(t, x)=-\...
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36 views

Risk-neutral Simple Return Moment Log-return Moment

I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns. Specifically, by making the same standard assumptions of the Black-Scholes model with the ...
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1answer
87 views

Historical volatility calculation to price options with the Black-Scholes formula

I'm looking for a reference algorithm for calculating historical volatility to price options. I know there are several volatility calculation models that use the time series of the underlying's ...
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31 views

How to mathematically calculate the probability of GBM generating difference of less than some value

I have a custom index that follows Geometric Brownian Motion (GBM) with volatility v. I started this index at 10k with 4 decimal places i.e the starting price of ...
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27 views

Market model for european/american options on underlying paying discrete cash (and maybe proportional) dividends

Black Scholes is the market model for european and american options on an underlying paying no dividends. What is the standard market model for european or american options of underlyings paying ...
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1answer
72 views

Estimation of volatility into Black-76 formula

I am trying to estimate the (annualized) volatility that should go into an European Swaption (such as 2y5y). Given we take the black76-formula, where the discounting is the term outside the ...
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47 views

Black-Scholes model - Calibration of the risk-free rate

I know there is a lot of content about this topic, but I have not seen a post which gives a satisfying answer to my problem. I am trying to hedge a European call option with real market data under ...
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40 views

Is my derivation of Black-Scholes equation correct or am I missing something (eg assumption)?

Question: The following is my derivation of the Black-Scholes equation. Is it correct or am I missing some details (eg assumption)? Let $V$ be value of an option. Suppose value $\Pi$ of a portfolio ...
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33 views

OTC equity option under foreign currency CSA

What adjustment do I need to make to the Black-Scholes equation when the CSA of an OTC equity option is in a different currency than the underlying in order to get the correct price? For instance, ...
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65 views

Reference for pricing geometric-mean basket option

Let $(Z_1,\ldots,Z_N)$ be an $N$-dimensional Brownian motion with correlation matrix $\rho$ and consider the multivariate Black-Scholes model \begin{align} dS_i(t) \ = \ (r-q_i)\, S_i(t) \, dt \, + \,...
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41 views

Put-call parity for equity share and debt share

Considering Merton's structural approach" for credit risk modeling, we arrive to prove that the pricing formules are $S_t=V_t\phi(d_{T,1})-Fe^{-r(T-t)}\phi(d_{T,2})$ for equity share and $F_t=FP_0(t,T)...
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1k views

Swaption : Bloomberg Black implied volatility quotes and pricing in the Black model

I used a lot Bloomberg's VCUB for data, but never used its integrated swaption pricer "Quick Pricer for Swaptions", nor Bloomberg's "full" swaption pricer from "SWPM -OV". I am retrospectively quite ...
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43 views

How can I graph futures options profit/loss when the options have different underlyings?

Consider a portfolio of vanilla SPX monthly options that consists of two components, a SEP 2019 3000 Call and a DEC 2019 3000 Call. It's easy to graph these as they both share the same independent ...
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46 views

Calculation of Conditional Expected Value and Pay-Off Diagram

I have a stock with mu 6% and sigma 20% following a random walk and I would like to to calculate the Conditional expected Value of the stock in 10 states with equal probability (10%). Meaning, I would ...
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63 views

How is a LIBOR Market Model volatility skew determined?

LIBOR based interest rates are derived from the prices (supply / demand) of swaptions, caps and floors. These prices are generally quoted in yield vols. Their prices are given by the Black formula. ...
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131 views

Different scaling conventions for greeks

I have been following this tutorial (http://gouthamanbalaraman.com/blog/value-options-commodity-futures-black-formula-quantlib-python.html). It says in the conclusion and I quote:It is worth pointing ...
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193 views

Interpretation of drift parameter $\mu$ in GBM

Currently studying Ito's calculus. Looking on the GBM model: $ \frac{d S_t}{S_t} = μ dt + \sigma d B_t$ we end up on the expected stock price at time t: $E[S_t]=s_0 e^{\mu t}$.What does actually $\mu$ ...
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1k views

What is Dual Delta?

I understand that it is the partial derivative of option price with respect to strike. What is it used for though? What does your dual delta signify?
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141 views

Isn't this modified stop-loss strategy an arbitrage?

In John Hull's The Book, section 18.3 he briefly discussed a stop-loss strategy for writing a call option: buy one share of stock whenever $S_t>K$ and sell it otherwise (except at time $0$: if $S_0\...
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171 views

Black's formula for a call option on a non-tradable underlying

I am looking for an explanation of the following fact, which seems to be rather simple yet I am missing something. Say that $S_t$ is a stock following GBM $$ dS_t = r S_td_t + \sigma S_t dW_t,$$ and ...
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308 views

Are there any papers measure the accuracy of various option pricing models against real market price?

There are many stochastic volatility option models not only require significant more computation/simulation comparing to the standard BSM model but also introdue large source of possible problems at ...
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447 views

What is the intuition behind the equivalent martingale measure result?

"Suppose that f and g are the prices of traded securities dependent on a single source of uncertainty and define phi = f/g. The equivalent martingale measure shows that, when there are no arbitrage ...
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136 views

zero curvature boundary condition

Assume I am solving numerically Black Scholes PDE $$u_t+0.5\sigma^2s^2u_{ss}+rsu_s-ru=0$$ and I decided to have boundary condition on the right boundary as $u_{ss}=0$. One way is to write the discrete ...
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139 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) formula?...
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2answers
121 views

Implied volatility is returning infinity

I am trying to calculate implied volatility using javascript , I have following code ...
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1answer
55 views

Black Scholes Replication If Underlying Does Not Move?

Let's say you are long a call and want to replicate that call buy being short underlying and long bonds. If the underlying moves up in the next period but not enough to cover theta, the option ...
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1answer
110 views

Black Scholes: two assets, same $W$-process

Consider a Black Scholes model with two risky assets that are driven by the same $W$-process, and then 1 risk-free asset. When is this model arbitrage-free and complete? We have only 1 driving ...
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1answer
64 views

Cash-or-Nothing Call Option

I am trying to price a cash or nothing call option and I know know that the Cash or Nothing formula for a call option is $C(t,s)=Xe^{-r(T-t)}*N(d)$ If I have payoff X=100 r=0.03 T=2 $\sigma=0.3$ I ...
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1answer
72 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
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1answer
177 views

SPY American option Greeks and Premium

I am trying to replicate Ivolatility.com's option calculator for a client. Here's the example Using standard Black Scholes model, I can replicate the exact calculations if there is no dividend. With ...
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1answer
290 views

How to derive the Black Scholes partial differential equation from a stock log-normal distribution?

Is there a way to go from this $$\ln S_t=\ln S_0+(\mu-\sigma^2/2)t+\sigma W_t $$ $$\ln S_t\sim N[\ln S_0+(\mu-\sigma^2/2)t, (\sigma^2)t]$$ To the Black-Scholes partial differential equation?
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1answer
77 views

How to compute Pr(S>100) when S follows Geometric Brownian Motion?

I have been trying to resolve this problem, under (b), but I cannot find the correct answer. For i=1, my ultimate answer (P=1) deviates from the correct answer (P=0.7580). Please let me know whether ...

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