Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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62 votes
9 answers
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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 $...
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1 vote
3 answers
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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 ...
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  • 123
20 votes
3 answers
9k 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 ...
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4 votes
2 answers
2k views

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 ...
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46 votes
16 answers
30k 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 ...
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4 votes
3 answers
822 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 ...
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  • 921
16 votes
5 answers
40k 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.
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  • 161
12 votes
3 answers
8k 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 ...
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  • 678
2 votes
3 answers
4k 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 ...
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9 votes
5 answers
4k 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 ...
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6 votes
4 answers
1k 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,...
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35 votes
5 answers
20k 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. ...
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14 votes
1 answer
3k 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 ...
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  • 141
33 votes
11 answers
17k 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 ...
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  • 3,638
17 votes
6 answers
17k 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 ...
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  • 305
20 votes
3 answers
12k 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, ...
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  • 203
34 votes
3 answers
8k 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 ...
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  • 605
20 votes
8 answers
15k 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 ...
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6 votes
2 answers
1k 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 ...
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  • 427
12 votes
4 answers
10k 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, ...
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11 votes
0 answers
423 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 ...
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  • 2,471
10 votes
4 answers
1k 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 ...
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  • 5,191
6 votes
2 answers
1k 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 ...
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  • 123
3 votes
1 answer
355 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-\...
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2 votes
1 answer
934 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 ...
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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 ...
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  • 496
2 votes
3 answers
928 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 ...
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5 votes
1 answer
4k 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 ...
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  • 5,649
4 votes
3 answers
673 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 ...
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  • 2,161
3 votes
2 answers
365 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? ...
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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 ...
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8 votes
2 answers
2k views

Which process is the most commonly used for modeling stock prices?

I'm thinking of writing a master's thesis about pricing options using Levy processes, but I wonder if these processes are actually used for modeling stock prices or not (and which specifically)? And ...
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  • 423
13 votes
2 answers
3k 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. ...
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  • 1,856
8 votes
4 answers
9k views

Basket option pricing: step by step tutorial for beginners

I would like to learn how to price options written on basket of several underlyings. I've never tried to do it and I would appreciate if you can provide some documents, papers, web sites and so on in ...
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  • 2,046
9 votes
2 answers
926 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 ...
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  • 617
17 votes
5 answers
9k 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 ...
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5 votes
1 answer
2k views

How to price this basket option?

Underlying assets are three global stock index : Eurostoxx 50, HSI, KOSPI 200 Maturity: 36 months with advanced redemption date in every 6 months if prices of indexes satisfy given conditions at each ...
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  • 93
14 votes
3 answers
7k 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 ...
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  • 1,822
11 votes
2 answers
4k 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 ...
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  • 5,649
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 ...
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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 ...
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  • 987
4 votes
2 answers
591 views

Ito lemma of Convertible Bond under Two-factor Model Interest Rate

@Behrouz Maleki has provided the PDE of two factor model in other post so could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ...
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  • 61
4 votes
2 answers
399 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 ...
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3 votes
1 answer
2k views

How are Brownian Bridges used in derivatives pricing in practice?

A similar question has already been asked in the past, unfortunately the 2nd question of the OP was never really addressed. Most references found on internet on Brownian Bridge and Monte-Carlo ...
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2 votes
1 answer
1k views

Carr-Madan european contingent claim payoff decomposition formula - application

Looking for some clarification to the values of the parameters used in the Carr-Madan payoff decomposition formula. $$f(S_T)=f(\kappa) + f'(\kappa) (S_T - \kappa) + \int_0^{\kappa} f''(K) (K-S_T)^+ ...
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  • 701
2 votes
1 answer
254 views

Pricing an fx option in the same currency

Let imagine we have an option from EUR to USD priced in EUR, therefore the payoff for a call is: $$\frac{(S - K)^{+}}{S} = K (1/K - 1/S)^{+}$$ This is basically the payoff of a price of a put on 1/S ...
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2 votes
1 answer
5k views

Implied state price density (Question 1 - derivation of the formula)

I came upon the term "implied state price density" in a couple of papers. As far as I understand the concept one basically tries to extract the "pricing density" from the market data. For the sake ...
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  • 3,337
6 votes
1 answer
904 views

How to value a floor when a loan is callable?

Certain bank loans pay a spread above a floating-rate interest rate (typically LIBOR) subject to a floor. I would like to find the value of this floor to the investor. Assume for this example that ...
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  • 13.1k
5 votes
3 answers
907 views

Pricing when arbitrage is possible through Negative Probabilities or something else

Also now asked about here: Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated? Assume that we have a general one-...
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  • 841
2 votes
1 answer
8k views

Pricing of a Forward-start option in a Black-Scholes framework

I have read the pricing procedure of a Forward-start option in a Black-Scholes world in Musiela-Rutkowski, but I don't find their proof clear (pp. 195-6). Let me summarize their argument: Consider ...
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