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Questions tagged [black-scholes]

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

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How to price equity options using a Black76 implied volatility surface?

I would like to calculate the fair value of american and european options on various equities and indices using QuantLib C++. Since I do have discrete dividends available for most underlyings, I use <...
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How to interpret CDF($d_1$)/PDF($d_1$) from BS model ?

In my research on put options, I come across the ratio: $\frac{(1-\mathcal{N}(d_1))}{\mathcal{N'}(d_1)}$ where $d_1=\frac{\log(S/X)+(r+\sigma^2/2)t}{\sigma \sqrt{t}}$ and $\mathcal{N}(.)$ is the ...
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Simating a Cppi strategy with stochastic interest rates

I Hope someone can help me despite my english Is not that good. In trying to set up a similation code for a costant proportion portfolio insurance. My portfolio Is composed by risky assets and free ...
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1answer
61 views

Black Scholes on Eurodollar Options

I am trying to replicate the Black Scholes results of CME option calculator for options on Eurodollar Options. (link) I am trying to replicate the implied volatility result by unaltering the spot and ...
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42 views

Effect of mean reverting Volatality in Black and Scholes? [closed]

Can someone please elaborate what would be the effect of a mean reverting volatility (instead of a constant volatility) in pricing options using BS ? Also how would the greeks vary?
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1answer
35 views

Relationship between asset volatility and debt and equity value

So how I understand it, higher asset volatility implies a higher call option price. The Merton Model holds that the value of equity is a call option. This therefore implies that the equity value must ...
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50 views

Pricing collateralised derivatives with stochastic interest rates

I was reading the article by V. Piterbarg, Funding beyond discounting: collateral agreements and derivatives pricing, that you can download on the following link. He states that when the derivative is ...
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1answer
108 views

Positive theta on a long put?

I am trying to hand-price options under the Black-Scholes model. Given the following parameters: Stock price: $12.53$ Strike price: $14.00$ Risk-free rate: $0.03$ Annualized Volatility: $0.10$ Time ...
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94 views

Arbitrage from ATM option trading?

So I was testing out a collar options strategy (long put, short call, and long shares of the underlying stock) in a backtest for a school finance project, and the profits & losses are given by the ...
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114 views

Expectation of option value

Say we are in a BS world where the (conditional on t) price of a call is given by the usual $$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$ Now, what about the ...
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1answer
61 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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2answers
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Proof Black Scholes Theta

I saw the following proof of theta in a paper I read, and I thought it looked pretty neat. Unfortunately I don't understand the step that they do. This is what they do: Now, I don't get how they go ...
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1answer
59 views

Cash deposit in replicating portfolio for BS equation unnecessary?

The book on Option Valuation Methods that I currently study (Higham 2013) constructs a replicating portfolio $\Pi = A(S,t)S + D(S,t)$ for deriving the BS PDE, where $D$ is a cash deposit. $D$ does not ...
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1answer
130 views

Interpretation of IV and its use in stock movement prediction

I would like to validate my understanding of IV as a prediction tool. Black-Scholes model is based on the assumption that rate of return of a stock is a Wiener process: $$ \frac{dS_t}{S_t} =\mu \,...
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1answer
66 views

Why Can I not estimate a CVAR from Heston Model

I fit the parameters of Heston model, using option data for SPX. Now I have the process S and P 500 is expected to follow. I make 100,000 simulations of this process and then calculate the expected ...
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85 views

How accurate are Black-Scholes estimates of Vega, Volga, Vanna

Wikipedia provides analytical formulas for calculating Greeks. I can get Delta, Gamma, Theta all from Bloomberg. I need Vega, Volga, Vanna for my research. Should I use these analytical formulas for ...
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48 views

Cash-or-nothing and Asset-or-nothing price derivation

I was wondering how to derive the price of a cash-or-nothing and asset-or-nothing option by trying to work out the expectation under the risk-neutral measure, while assuming that the underlying ...
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1answer
54 views

Value simple chooser option as a sum of call and put options

There is a well known formula for valuating the chooser's option price: $H_{chooser}=max\{C(S_t, K, T-t), P(S_t, K, T-t)\}=max\{C(S_t, K, T-t), C(S_t, K, T-t)+Ke^{−r(T-t)}−S_t\}=C(S_t, K, T-t) + max\{...
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21 views

Remaining variance and historical variance in Black-Scholes with term structure

When pricing an European vanilla option in a Black-Scholes world with deterministic volatility term structure, what matters is the remaining variance between today $t$ and maturity $T$, i.e. the ...
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2answers
50 views

Why can a deterministic portfolio only grow at risk free rate

In black scholes derivation we assume that portfolio grows at risk free rate because the process is deterministic, my question is why is it riskfree rate? If i have information about some event in the ...
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Which areas of statistical physics do not get enough attention in quantitative finance?

It seems that over the past few decades many ideas from statistical physics have been successfully incorporated into economics and finance to form the sub-discipline of econophysics. However, it is ...
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2answers
158 views

Calculating historical Volatility for the Black Scholes Model [closed]

Below is a problem from the book "Options, Futures, and other Derivatives" by John C. Hull. I did the problem but I am fairly sure that my answer is wrong. I am hoping that somebody can tell me where ...
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1answer
86 views

Expected value of delta-hedged portfolio

Consider portfolio in black-scholes world $\Pi = \Delta S - V$, where $S$ is the stock price and V is the price of the option. I have read that if we set $\Delta = \frac{\partial V}{\partial S} $ ...
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Oscillating errors in finite difference Black Scholes

I am writing an implementation of the explicit finite difference method to price a standard european call option, and comparing the results to the corresponding analytical value to gauge the error ...
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2answers
93 views

P&L Calculation of Option Strategy

I have designed a call writing option strategy, where I am rolling the options upon expiry, i.e., my portfolio consists of one short call position at any given time. I have a time series of the value ...
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0answers
74 views

basic difference between interest rate models

I am reading up on interest rate models, but currently confused about difference in the two types of models: no arb models like ho-lee, vasicek etc. others like nelson siegel, pca models etc. While ...
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2answers
207 views

Interest rates forward implied volatility models

I'm trying to find out which model to use to price a pur forward volatility product named VolBond marketed by structuring desks currently. Let me introduce the products first: Example 1: You pay 100 ...
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2answers
208 views

Does Black Scholes need to assume no arbitrage?

Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ...
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41 views

Unique risk neutral measure in Black Scholes vs Merton Model

I was going through a question on the unique risk neutral measure in the Black Scholes model : Unique risk neutral measure for Brownian Motion One of the answers said it is essentially because there ...
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1answer
130 views

Black-Scholes: Delta/probability of exercise increases with volatility

The delta for an ITM call option with increasing volatility initially decreases, reaches a global minimum, and then increases. If we consider delta as a representation of risk-neutral probability of ...
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0answers
24 views

Pricing options with 0 or negative underlying values

I am trying to calculate the value of an option whose underlying is the calendar spread between two months for a commodity (front month Brent vs 2nd month), usually known as a calendar spread option. ...
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1answer
170 views

Option greeks as dollar P&L

If I write the value of an option as O(S, K, T, V), where S is the underlying price, K is the strike, T is the time to expiry and V the implied volatility, how can I compute the dollar amount that I ...
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Vega of exotic options

I'am wondering if there is a standard definition to the Vega of an exotic product when the underlying model is not Black-Scholes. Let me give some examples : What is the Vega if the price is ...
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1answer
197 views

Black-Scholes pricing of binary options

I'm trying understand something basic about Black-Scholes pricing of binary options. In my example above, the current price is over the strike price. The volatility is extreme but I'm still having ...
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1answer
68 views

Question about calculating asset volatility using Black-Scholes and the Merton Model (Differentiation Question)

I have a problem where I need to relever equity volatility and take into account debt. I'm trying to solve a system of nonlinear equations for $\sigma_v,V$ using $f(\sigma_v,V) = VN(d_1)-De^{-R_ft}N(...
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1answer
167 views

Black-Scholes for Binary Option

Something is wrong with this python code designed to apply Black Scholes to the price of a binary option (all or nothing, 0 or 100 payout). The results I get here is 0.4512780109614. Which I know ...
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3answers
690 views

Forward implied volatility

Can one price accurately by only using vanilla options a derivative that is exposed/sensitive mainly to the forward volatility ? If it is impossible in the real world, what do we mean by saying "...
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4answers
349 views

Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to ...
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Black-Scholes Greeks Calculation with Daily Rates [duplicate]

Background: To simulate a set of closing prices (about a months worth) for a stock, I've been using the average, standard deviation and variance of the daily returns of the stock( LN(S1/S0)) over the ...
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1answer
70 views

Dynamic hedging pnl when pinning

Dynamic hedging, if successfully implemented, should ensure the dynamic hedge earns the exact opposite of the corresponding option position. However, if we buy an otm option, and the stock goes in ...
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2answers
258 views

Pricing Corridor Variance Spreads

Recently in the equity derivatives market there have been some trades on what are known as "Corridor Variance Spreads." The large equity derivative dealers and investment banks have been promoting it ...
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58 views

Implied Volatility of a call plus its delta

I would like to understand if exists a smart way to imply the volatility from a quote that is the sum of a call and its delta: is there any method other than simple iterative minimization?
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1answer
251 views

1y10y vs. 10y1y Swaption

Say you have two identical payer swaptions, exception for their terms and tenors. In other words, suppose you have two payer swaptions: $1y10y$ and $10y1y$. All other things being equal, according ...
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68 views

Best Way of Interpreting Black-Scholes Formula [duplicate]

I'm curious to know the best interpretation of the Black-Scholes formula for a European equity call option: $$C(S,t)=S_tN(d_1)-Ke^{-r(T-t)}N(d_2),$$ where $d_1=\frac{1}{\sigma\sqrt{T-t}}\big[\ln(\...
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0answers
85 views

Beginner question on Black Scholes

Would you please confirm whether my understanding is correct please? (Sorry a lot of questions...) 1) BS is derived based on the assumption that during an infinitesimal time, we can replicate the ...
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2answers
156 views

Heging against stochastic interest rate

I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash. I use the following Black-Scholes formula $$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)]...
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1answer
88 views

Pricing an option with sparse data, high underlying volatility and returns

I'm currently pricing American and European options on an underlying with sparse data (interpolated), high annual volatility and returns over the last year around 300%. The product isn't similar to ...
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42 views

Return / yield of an ATM-Call Option on a zero coupon bond

The zero-coupon bond with unit face value and maturity S for a call option with maturity T and strike K is given by: The bond prices $P(t,T)$ and $P(t,S)$ $$\begin{aligned} ZBC(t,T,S,K) = & P(t,S) ...
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Longstaff Schwartz Algrorithm in R

I recently discovered the LSMonteCarlo library in R which basically determines the price of American options via Longstaff Schwartz method. I tried the ...
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1answer
218 views

derivation of general black-scholes formula

I would like to find a derivation for the Black-Scholes fomrula in the general case (i.e., where the volatility function $\sigma : [0,T] \to \mathbb{R}^+$ and the investment rate $r: [0,T] \to \mathbb{...