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Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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6 votes
2 answers
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Garman-Kohlhagen (Black-Scholes) Formula vs. Bloomberg OVML Calculator

I'm trying to price a European call option on USDJPY. We have that $S = 112.79, K = 112.24, \sigma = 6.887\%, r_d = 1.422\%, r_f = -0.519\%, T = 0.25$. My model, based on Black-Scholes, returns the ...
Vladimir Nabokov's user avatar
68 votes
9 answers
92k views

What are some useful approximations to the Black-Scholes formula?

Let the Black-Scholes formula be defined as the function $f(S, X, T, r, v)$. I'm curious about functions that are computationally simpler than the Black-Scholes that yields results that approximate $...
knorv's user avatar
  • 2,119
3 votes
3 answers
4k views

FX Option pricing on Forward vs. Spot

In a GBM world with riskless domestic and foreign interest rates, what would be the correct model for a FX plain vanilla option given the statement that this option is priced on the forward? I guess ...
Tim's user avatar
  • 163
28 votes
3 answers
16k views

Explaining the Risk Neutral Measure

What is the Risk Neutral Measure? I don't believe this has been answered on the internet well and with all the parts connecting. So: What is the risk neutral measure/pricing? Why do we need it? How ...
Trajan's user avatar
  • 2,532
47 votes
16 answers
35k views

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 ...
CuriousMind's user avatar
3 votes
1 answer
2k views

Quantlib: day-by-day evaluation of option value

I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct. I want to calculate the P&L of a certain option trading ...
Wynn's user avatar
  • 105
4 votes
3 answers
1k views

Derivation of BS PDE problem using Delta hedging

I've always been confused with Delta hedging. It is well-known that for a (smooth enough) function of $(S,t)$ we have, due to Ito's lemma, that: \begin{eqnarray*} dC = \left(\frac{\partial C}{\partial ...
Vim's user avatar
  • 903
16 votes
5 answers
49k views

Bachelier model call option pricing formula

Does anybody have the Bachelier model call option pricing formula for $r > 0$? All the references I've read assume $r = 0$. I don't speak French, so I can't read Bachelier's original paper.
Galsunja's user avatar
  • 161
8 votes
4 answers
3k views

Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$

Develop a formula for the price of a derivative paying $$\max(S_T(S_T-K))$$ in the Black Scholes model. Apparently the trick to this question is to compute the expectation under the stock measure. So,...
Trajan's user avatar
  • 2,532
2 votes
1 answer
2k views

Option Pricing for Illiquid case

I am currently studying crypto options trading and have observed that there is often a lack of liquidity for options (such as BTC Options) on various exchanges, including Binance. In many cases, there ...
Starlord22's user avatar
13 votes
3 answers
10k views

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 ...
Egodym's user avatar
  • 698
5 votes
3 answers
2k views

Probability of an Option maturing In-the-money vs. Volatility

How will the probability of an option ending up in the money change if the volatility of the underlying stock increases? Intuitively, I think the answer to this is that if volatility goes up the ...
Trajan's user avatar
  • 2,532
4 votes
1 answer
4k views

Quantlib: Greeks of FX option in Python

I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct. I also want to calculate all the Greeks, and eventually use those ...
Wynn's user avatar
  • 105
21 votes
6 answers
19k views

Risk Neutral Probability

I read that an option prices is the expected value of the payout under the risk neutral probability. Intuitively why is the expectation taken with respect to risk neutral as opposed to the actual ...
Mykie's user avatar
  • 345
20 votes
3 answers
14k views

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, ...
Jeffrey's user avatar
  • 203
9 votes
5 answers
5k views

Estimate probability of limit order execution over a large time frame

I have a negligible amount of money (\$5000) that I would like to invest in a stock. I would like to buy the stock at some point in the next year, and get the lowest possible price. I would like to ...
Kevin Burke's user avatar
3 votes
3 answers
6k views

FX Delta Conventions

I'm currently reading Iain Clark's book Foreign Exchange Option Pricing and I got stuck at one sentence in the beginning of Section 3.3 that I feel is important to understand. He writes: FX ...
DoubleTrouble's user avatar
38 votes
5 answers
22k views

How should I calculate the implied volatility of an American option in a real-time production environment?

There are many models available for calculating the implied volatility of an American option. The most popular method, employed by OptionMetrics and others, is probably the Cox-Ross-Rubinstein model. ...
Tal Fishman's user avatar
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33 votes
11 answers
18k views

Probability of touching

For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
glyphard's user avatar
  • 3,646
14 votes
1 answer
4k 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 ...
Jon's user avatar
  • 141
11 votes
4 answers
2k views

Intuition for Stock Price Numeraire Drift

I would like to ask whether there is an intuition for the drift of price processes under the Stock numeraire. I find it intuitive that the martingale measure under the Money Market numeraire induces ...
Jan Stuller's user avatar
  • 6,178
1 vote
1 answer
624 views

How do trading platforms estimate options pricing P&L graphs?

Using the profit/loss calculator for equity option strategies of a trading platform, it displays estimated P&L curves for some date in the future and across the prices of the underlying with a ...
mentics's user avatar
  • 113
20 votes
8 answers
16k views

Why does implied volatility show an inverse relation with strike price when examining option chains?

When looking at option chains, I often notice that the (broker calculated) implied volatility has an inverse relation to the strike price. This seems true both for calls and puts. As a current ...
Joseph Tanenbaum's user avatar
13 votes
4 answers
11k views

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, ...
Dmitri Nesteruk's user avatar
13 votes
0 answers
505 views

Jim Gatheral's ansatz

In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$ where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
Hans's user avatar
  • 2,806
11 votes
4 answers
8k views

How does an option's time value depend on moneyness?

How does an option's time value (also known as extrinsic or instrumental value) depend on how far it is in the money or out of the money? In other words, how does the time value change as the ...
kamikaze_pilot's user avatar
6 votes
2 answers
2k views

Option Price vs. Implied Volatility

I was doing an exercise on investigating the relationship between European Call option price and its volatility. I was asked to compute $\frac{\partial^2C}{\partial \sigma^2}$ and find out the domain ...
Van Tom's user avatar
  • 143
6 votes
2 answers
2k views

Importance Sampling for pricing options with longstaff and schwartz

I have been asking this similar question before. However, I really want to be concrete and get and concrete explanation. I have been reading the paper by Moreni and try to implement the same ...
Elekko's user avatar
  • 427
5 votes
1 answer
5k views

Option prices in Bates SVJ model?

In this [post] discussed the European put and call price formulas under the Heston Stochastic Volatility model. There exists an important extension of Heston model to include diffusion jumps, known ...
emcor's user avatar
  • 5,795
4 votes
1 answer
3k views

Numerical example of how to calculate local vol surface from IV surface

I'm looking for an excel example (not a copy of Dupire's eqn) of how to convert an IV surface to a local vol surface. If unsuccessful I'll work through Dupire's eqn but would be helpful to look at an ...
Zeus's user avatar
  • 219
4 votes
1 answer
644 views

Bartlett's delta gives wrong signs for calls and puts

There is a paper by Bruce Bartlett introducing a modified delta for SABR model which accounts for the correlation between forward and volatility processes. The main result of the paper is that if $dF$ ...
Hasek's user avatar
  • 814
3 votes
2 answers
2k views

SABR model - beta

In the SABR model, the parameter beta largely controls the back-bond behaviour of the model. How do people estimate beta? One approach is to regress atm vol vs forward, i.e. $$\ln(\textrm{atm vol}) = \...
JohnRoper's user avatar
3 votes
2 answers
483 views

Structuring and Customization

It seems complex derivatives in particular exotic options are not available at any retail broker. Can a regular retail trader get access to these instruments? Maybe through prop firms or banks? ...
user avatar
3 votes
2 answers
1k views

Put-Call relationship for Option on Forward

The forward price of a forward contract maturing at time T on an asset with price St at time t is, $$ F=S_te^{(r-q)(T-t)} $$ where $r$ is the risk free rate and $q$ is the continuous dividend rate ...
Danny's user avatar
  • 514
3 votes
1 answer
533 views

Intuition behind prices modeled by Geometric Brownian Motion

Suppose that we model a price $P_t$ to evolve per $$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$ for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is $$P_t=P_0e^{(\mu-\...
Heatconomics's user avatar
2 votes
3 answers
2k views

Do basket options have a closed form valuation formula?

Suppose I'm simulating a European call option on a basket consisting of N stocks with slightly varying volatilities but all other parameters remain the same. From the perspective of an estimate, it ...
John1942's user avatar
2 votes
1 answer
987 views

FX Euro-American Knockout Option Pricing

this is my first time asking questions here. I want to look for some calculation method to price a very exotic option. The FX Euro-American Knockout Option (EAKO) is an option that has an American ...
Fangy's user avatar
  • 21
2 votes
1 answer
1k views

Probability of exercise in the Black-Scholes Model

What's the intuition behind the fact that the limit of $\mathcal{N}(d_2)$, i.e. the (risk-neutral) probability of exercise, in the Black-Scholes Model tends to $0$ when the volatility tends to ...
Xavi Hernandez's user avatar
1 vote
1 answer
805 views

The greeks, vanillas and digitals

Question 1: I know website’s like: https://optioncreator.com/ display the pricing and payoff graphs of regular plain vanilla puts and calls. I would like to know if there is any website that displays ...
user avatar
48 votes
9 answers
5k 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 ...
Piers Myers's user avatar
34 votes
3 answers
9k 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 ...
nkhuyu's user avatar
  • 605
20 votes
4 answers
6k views

From Fourier Transforms to Option Values

I am trying to understand how Fourier transforms & Characteristics functions can be used to calculate option values. However, I am having difficulty following the process that is used in several ...
sets's user avatar
  • 1,471
17 votes
5 answers
10k views

How to get greeks using Monte-Carlo for arbitrary option?

Let's assume I have an arbitrary option that I can price using Monte-Carlo simulation. What is the general approach (i.e. without relying on specific option type) to calculating the greeks in this ...
Alexey Kalmykov's user avatar
16 votes
4 answers
8k views

How does volatility affect the price of binary options?

In theory, how should volatility affect the price of a binary option? A typical out the money option has more extrinsic value and therefore volatility plays a much more noticeable factor. Now let's ...
CQM's user avatar
  • 1,862
13 votes
2 answers
4k views

What are the main flaws behind Ross Recovery Theorem?

Stephen Ross’ new paper claims that it is possible to separate risk aversions and historical probabilities if the Stochastic Discount Factor is transition independent using Perron-Frobenius Theorem. ...
fni's user avatar
  • 1,896
12 votes
2 answers
7k views

Heston Model Option Price Formula

What is the formula for the vanilla option (Call/Put) price in the Heston model? I only found the bi-variate system of stochastic differential equations of Heston model but no expression for the ...
emcor's user avatar
  • 5,795
11 votes
3 answers
1k views

How to choose a risk-neutral measure when the market is incomplete?

I am more of a probabilist than a financial mathematician. I am currently working on the features of American put options under a particular stochastic volatility model. Like most stochastic ...
Lost1's user avatar
  • 1,023
10 votes
2 answers
2k views

Implying risk-free rates using Put/Call parity

I recently purchased SPX options data from the CBOE. Normally, if the data is OK and the Put-Call parity holds, one should expect to correctly imply ZC (Zero Coupon bond) prices and forwards by ...
BS.'s user avatar
  • 165
9 votes
2 answers
1k views

How to price an option allowing to change a call into a put?

A recruiter asked me this question: Suppose you have the following contract: a call option with maturity $T$ = 2 years the possibility to change this call into a put at $t$ = 1 year What is the ...
glork's user avatar
  • 617
9 votes
1 answer
2k views

Option pricing and mean reversion

In different books one can find a formula for option pricing when we assume that $\ln(S)$ follows a mean reversion process $$ dS_t/S_t=\kappa(\theta-\ln(S_t))dt+\sigma dZ$$ If we calculate an ...
JoergVanAken's user avatar

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