# Questions tagged [black-scholes]

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

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### Black Scholes differential

I'm studying a BS derivation and I don't understand one part .We have a portfolio consisting of $\Delta(t)S(t)+B(t)$ where the first term is risky and the second is a riskless bond. The part i don't ...
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### Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
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### Why $N(d_1)$ and $N(d_2)$ are different in Black & Scholes

I'm struggling to understand the meaning of $d_1$ and $d_2$ in Black & Scholes formula and why they're different from each other. As per the formula, $$C = SN(d_1) - e^{-rT}XN(d_2)$$ which ...
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### Calculate strike from Black Scholes delta

I have a list of deltas and their corresponding volatilities in an FX market but I want to go from delta to strike price. In this Question similar problem is being discussed How can I calculate the ...
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### What causes the call and put volatility surface to differ?

I currently have a local volatility model that uses the standard Black Scholes assumptions. When calculating the volatility surface, what causes the difference between the call volatility surface, ...
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### Ways of treating time in the BS formula

The Black-scholes formula typically has time as $\sqrt{T-t}$ or some such. My questions: What is the granularity of this? If we treat $t$ as the number of days, then logically on the day of expiry, ...
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The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\... 2answers 608 views ### Verifying an identity of an equation for Black Scholes formula I just started working on the Black Scholes formula with help of the book Financial option valuation by Higham. Apparently you are possible to derive the following function: \log(\frac{SN'(d_1)}{e^{-... 9answers 4k views ### Are there any new Option pricing models? Back in the mid 90's I used the Black-Scholes Model and the Cox-Ross-Rubenstein (Binomial) Model's to price Options. That was nearly 15 years ago and I was wondering if there are any new models being ... 6answers 18k views ### How do you explain the volatility smile in the Black-Scholes framework? Does anyone have an explanation for the currently naturally forming volatility smile (and the variations) in the market? 2answers 7k views ### How do we use option price models (like Black-Scholes Model) to make money in practice? In quantitative finance, we know we have a lot of option price models such as geometric Brownian motion model (Black-Scholes models), stochastic volatility model (Heston), jump diffusion models and so ... 1answer 2k views ### How do different models impact option Greeks? If I trade an option using delta, vega, Prob OTM, etc. these are derived from a model. How do leading models impact valuations in terms of the Greeks? I suppose to form a baseline it would have to be ... 3answers 7k views ### Is there an all Java options-pricing library (preferably open source) besides jquantlib? I am looking for an all-java implementation of black scholes, preferably open source. I found jquantlib and quantlib (C++). Any other recommendations? The jquantlib site seems to be down. I'd prefer ... 9answers 6k views ### Why the expected return rate of a stock has nothing to do with its option price? OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ... 2answers 5k views ### How to extrapolate implied volatility for out of the money options? Estimation of model-free implied volatility is highly dependent upon the extrapolation procedure for non-traded options at extreme out-of-the-money points. Jiang and Tian (2007) propose that the ... 2answers 626 views ### Expectation of Gamma times S^2 in Black-Scholes model Can somebody prove that:$$E[S_t^2 \times \Gamma(t,S_t)] = S_0^2 \times \Gamma(0,S_0)$$where S_t follows a lognormal process as in the Black-Scholes model, and Gamma is the second derivative \... 4answers 10k views ### Which risk-free interest rate to use in Black-Scholes equation Sorry but i'm new in quantitative finance. According to BS derivation the risk-free interest rate is the rate to wich the rate of a particular investment tends when the risk tends to zero. Suppose i ... 2answers 319 views ### Option Valuation Can Black-Scholes option values be derived via the Capital Asset Pricing Model, without resort to the use of a risk-free portfolio being created from the option and a Delta determined quantity of the ... 1answer 1k views ### derivation of the hedging error in a black scholes setup I'm reading the following short paper by Davis. In section 2.6 he wants to derive an expression for the hedging error. Assume we have Black scholes setup:$$ dS_t = S_t(r dt + \sigma dW_t) dB_t =...
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The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value, by Peter P. Carr and Robert A. Jarrow, in The Review of Financial Studies, Volume 3, Issue 3, ...
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### Why must the risk free rate be free from risk in risk neutral valuation?

I am reading through documentation related to Funding Valuation Adjustments (FVA) which discuss risk free rate and funding matters and the following question came to my mind: in risk neutral valuation ...
Assuming the Black Scholes model and pricing formula of a European call option. Then, if the call is ITM, i.e. if $ln(\frac{S}{K})>0$, the $d_1$-term will go towards infinity as $\sigma$ goes to ...