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

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Option pricing within the Black Scholes model

Have a question regarding regular option pricing. In the standard Black-Scholes model, with interest r and volatility $\sigma$. Determine the arbitrage free price at t of an option which at $T>t$ ...
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Black Scholes vs Binomial Model

I'm trying to confirm my understanding of the 2 models. It is my understanding that the black-scholes is a special case of a binomial model with infinite steps. Does this mean that if I were to start ...
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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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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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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 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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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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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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what is the vol in the BS formula?

I need to compute the delta of an option for which I know a) the time to maturity, b) the price of the option, c) the price of the underlying asset. what is the formula to get this delta It seems ...
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109 views

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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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 ...
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106 views

Lower bound of ITM Calls when computing Implied Volatility

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 ...
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what's the relationship between forecasted stock volatility and implied volatility?(option)

what's the relationship between forecasted stock volatility and implied volatility? I know that implied volatility is the volatility calculated by BS formula, is there any relationship between implied ...
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111 views

How does Vega of a call/put behave under the Black-Scholes model?

I have two questions. I would prefer a reference if possible. Is the value of vega bounded for $\sigma\in [0,\infty)$? (I assume so, I imagine it goes to 0 as $\sigma$ go to infinity.) Are there any ...
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143 views

Time-zero price of two specific contingent claims

I am unsure how to start with the following problem. I have two contingent claims where contingent claim (1) pays $\int_0^T S_u du$ and contingent claim (2) pays $(\log S_T)^2$ at time $T$ Now I ...
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129 views

Lookback option explicit formula using Black Scholes

I would like to compute the time-0-price for a lookback option using Black Scholes formula, the explicit formula is given by ...
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224 views

Price of a down-and-out call in terms of European call

If $EC(S_0, K, \sigma, r, T)$ represents the price of a European call option with strike $K$, expiry $T$, initial price $S_0$, volatility $\sigma$ and where the constant interest rate is $r$, then I ...
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278 views

Expected value of Black-Scholes

(Apologies for any formatting mistakes) Within the Black Scholes model, given that you are estimating the volatility from historical data - and all other parameters assumed exact - one usually ...
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Error message in calculation Implied Volatility

I am unsuccessfully trying to find the Implied Volatilities for the SPX on a given date using information of the CBOE, as well as Open Interest, but as I run the code I am getting and error message ...
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161 views

Arbirtage free price process question in Bjork's Arbitrage Theory in Continuous Time

I am currently working through questions in Bjork's Arbitrage Theory in Continuous Time. However, I am unable to solve the following question, 7.2 in the book. A solution would be greatly appreciated. ...
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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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411 views

Drawbacks of Black-Scholes option pricing model

Will highly appreciate if anybody can provide logical financial proof why the Black-Scholes option pricing model overestimates the value for long-term options as described in this paper "Warren ...
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224 views

How to prove price of Asian option under geometric averaging is cheaper than a European call?

This was an exam question at Cambridge University. Let $S_t = S_0\exp(\sigma W_t + (r-\dfrac{1}{2}\sigma^2)$ and a bank account returns a continuously-compounded rate of interest $r$. Consider the ...
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Black 76 for Options on Interest Rate Futures

This is my first time using Black76 to value options on IR futures and I have a question on $F$ and $K$. I understand the price for an IR future is usually quoted as $100 - r$. Do I use this price ...
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149 views

Trouble arriving at Black-Scholes Formula

I am attempting to arrive at the Black-Scholes formula for my own understanding. I can accept one can use the risk-free distribution & rate, so I am attempting to use the distrution to arrive at ...
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285 views

Black-Scholes: Why the focus on volatility?

We know Black-Scholes is an imperfect model for options pricing. Why is so much of the analysis of its defects focused on implied volatility? The fact that IV varies for the same stock at the same ...
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Physical Option Implied Distribuition

So I got risk neutral probabilities from stock option prices. How can I then map them to a physical measure?
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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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169 views

Can option prices be characterised by an ODE?

If a stock price, $S(t)$, is governed by a geometric brownian motion. Is it possible to characterise the value of an option $V(S,t)$ as an ODE rather than a PDE (given $S$ is itself a function of ...
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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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285 views

Question about option theta

I have a question about a option theta. When I evaluate the option theta of near expiry put option using Black-Scholes formula given the data as follow: Index Level = 20,500 Strike Price = 20,000 ...
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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) ...
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212 views

What is the difference between the methods (listed in content) in pricing convertible bond?

To price the convertible bond, one of the models is the bond plus equity option method. That is, the value of convertible bonds is evaluated by finding the value of the straight bond and the value of ...
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230 views

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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195 views

Black model - volatility estimation

In the Black (1976) model: We should use the settlement prices of the underlying futures contract in order to estimate the volatility, right? Or can we also use the spot prices? Because the ...
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Black-Scholes in Delphi [closed]

when trying to implement the Black-Scholes formula in Delphi, I've found this: http://www.espenhaug.com/black_scholes.html I've checked the results against option-price.com and found they are ...
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options pricing using vwap

This is a question about why options prices do not take volume into account. The popular option valuation formula "black-scholes" certainly does not account for this and I don't suggest that it does. ...
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Black--Scholes hedging argument

I'm trying to understand the standard hedging argument to derive the Black--Scholes PDE. There's one aspect of the derivation which I can't get passed and I'd be very grateful for some clarification ...
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543 views

How to calculate return rates with negative prices?

I'm dealing with electricity options and I'm considering the possibilty of negative prices. I want two estimate the historic volatility. However, an arithmetic mean doesn't feel appropriate and ...
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Am I reading this correctly? probability way too small with BS model

For a stock trading at $27, $28 strike, 0% interest, 15% annual vol, and one day until expiration there is about a 1 in 17000 chance of it being exercised? $d_2 = ...
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Formula for variance of European call/put in Black Scholes

I have a quite basic question, but I can't find a reference with it. Recall that we can use the Black-Scholes formula to price a European call or put for a market consisting when: the underlying ...
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264 views

Black-Scholes fastest computation method

What is the fastest way to numerically compute Black-Scholes-Merton option prices? I'm trying to find fastest and still precise method. Currently I'm using numerical approximation of Normal cdf with ...
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Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
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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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Is the price of European put option monotone in volatility if we replace BM in Black-Scholes with a general Levy process?

Under the Black-Scholes model, we have the European put option is $\mathbb{E} [e^{-rt}(K-S_t)]$, where we take $\log(S_t)=X_t$ and $dX_t= \sigma dW_t - \dfrac{1}{2}\sigma^2 dt + rdt$. Here the option ...
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198 views

Change option B&S pricing

Consider a market composed by two stocks whose prices $X$ and $Y$ are given by B&S diffusion $$dX_t= \mu X_t dt+ \sigma X_tdW_t$$ $$dY_t= \mu Y_t dt+ \sigma Y_tdB_t$$ Supposing the market is ...
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Replicating strategy in the Black-Scholes model

I have a two-asset Black-Scholes model for a financial market: $dB_t=B_t r dt$ $dS_t=S_t(\mu dt+\sigma dW_t)$ I introduce a European claim $\xi=max(K,S_T)$ with maturity $T$, for some fixed $K$. I ...
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Relationship between European, American options volatility

Suppose, if the price of a European option (say a put) can be shown to be monotone in volatility (say for any maturity), does it follow that American options has to be monotone in volatility? ...
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Is vega of Black-Scholes European type option always positive?

We assume we work in the risk-neural measure with a stock which pays no dividend and a continuous discount rate. For PUT and CALL only: can someone please clarify if what I said is correct? The ...
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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?