Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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36 views

Which is more Appropriate way to calculate Leverage with Options Contracts? [closed]

the connecting of words and bold colors was because of something here in the posting feature, it would not let me take it away when originally posted. i have re-copied and tried correct those errors. ...
6 votes
1 answer
188 views

Importance sampling for Monte Carlo with local volatility in practice

I am given a diffusion with a local volatility to price barrier options: $$dX(t)=X(t)\mu dt+X(t)\sigma(t,X)dW_t$$ I want to use Importance Sampling to price barrier options "far" out of the ...
0 votes
0 answers
24 views

Some questions about the pricing and the construction of Fixed Coupon Notes (FCN) [closed]

I'm currently studying Fixed Coupon Notes (FCN) out of my own curiosity. I've already read some articles and watched a video about it. One of the articles about FCN: https://cegafi.medium.com/...
1 vote
3 answers
2k 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 ...
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 ...
0 votes
0 answers
33 views

Monte carlo pricing on zero coupon bon under the Vasicek model [closed]

I would like to price an European call on zero coupon bond under the Vasicek model. I am planning to follow the Excercise 33 (hereby) from Lamberton Lapeyre (Introduction au calcul stochastique ...
0 votes
0 answers
46 views

Stopping times, question on exercise

I'm completing exercises from Steve Shreve - Stochastic Calculus for Finance I and I'm stuck on one subtask for which I can't find missing element for 11 stopping ...
3 votes
0 answers
109 views

Very close local volatility and implied volatility using Dupire's equation

I used Dupire's equation to calculate the local volatility as in https://www.frouah.com/finance%20notes/Dupire%20Local%20Volatility.pdf and Numerical example of how to calculate local vol surface from ...
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 ...
-1 votes
0 answers
97 views

How to determine break-even price on a delta hedge?

Suppose there is a portfolio of short $x$ shares and long 1 call option. This call option has a strike and premium. If the stock moves up, you loss money on the short position but gain on the option ...
2 votes
1 answer
223 views

Black and scholes option pricing

I have to solve the following problem in the Black and scholes model: find the price at anty $t\in[0,T)$ for an option whose payoff at the maturity is: \begin{equation} 0 \ \ \ \text{if} \ S_T<K_1\\...
0 votes
1 answer
56 views

In the derivation of the Black-Scholes PDE, using delta hedging, how is this linked to the risk neutral valuation? [closed]

I was reading this paper: http://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf I don't understand the paragraph here: "The most interesting feature of the Black-Scholes PDE (8) is that ...
0 votes
1 answer
245 views

Configuring barrier option in Quantlib-Python

Is there a possibility to configure the period the barrier is active, using Quantlib for python? Namely to set up the start and the end dates we compare the spot vs the barrier. If we look at quantlib-...
1 vote
0 answers
56 views

Why should delta-neutral backspread always result in credit?

Natenberg mentions in chapter titled "Volatility Spreads" : under the assumptions of a traditional theoretical pricing model, a delta-neutral ratio spread where more options are purchased ...
1 vote
2 answers
138 views

How to calculate the local volatility from implied volatility in practice

The local volatility can be derived from the implied volatility. But in practice how we deal with the first-order and second-order derivatives? I have seen this formula $$ \sigma_{\mathrm{Dup}}(T, K)^{...
2 votes
0 answers
84 views

Any innovations in mathematical processes behind option pricing models?

I am working on my thesis about option pricing models beyond classical Black-Scholes Model by looking for some recent innovations on mathematical processes behind the pricing structures. By that I ...
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 ...
1 vote
1 answer
144 views

American option under Ornstein-Uhlenbeck stock price

I came across with the following problem: For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial condition $X_0 = x$, find the stopping time $\tau$ that maximizes $\mathbb{E}[e^{-r\...
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 ...
0 votes
0 answers
54 views

Option pricing Greeks in Python - incorrect Gamma with MC option pricing (Black) using AAD autograd / JAX libraries - but works with closed form?

I am attempting to use AAD (Adjoint Algorithmic Differentiation) with a simple Black MC pricer, and found that the Gamma is incorrect. The output was compared to Black analytical Greeks, as well as ...
-2 votes
1 answer
72 views

Special Exotic Option Pricing Approach [closed]

I am currently stuck with the following problem: You need to price the following exotic option, where the share price of Stock ABC is the underlying: • Time to maturity: 2 years • Right to exercise: ...
1 vote
0 answers
31 views

Cash balance sign in hedging portfolio

Consider a derivative which depends on $n$ assets with price vector $X=(X^1,\dots,X^n)$. The derivative value $V_t$ is given by the function $v(t,X)$, so that the hedge ratios for the hedging ...
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 ...
3 votes
1 answer
126 views

Reduced volatility in local stochastic volatility model

in Local Stochastic Volatility models I always read or hear "first the stochastic volatility model is calibrated to reduced vols and then the local volatility model corrects it" also I head ...
9 votes
1 answer
180 views

What is the market standard for IR option pricing when moving to SOFR

From books it looks like market standards to price IR options, like swaptions, are SABR, LMM or mix of the two (SABR-LMM). But LMM models the forward LIBOR rate. What will happen to it once LIBOR ...
0 votes
0 answers
46 views

Why is call option delta increasing in below setup? [duplicate]

I have a question. I really appreciate if someone can reply. Enter the same strike and stock price to an options calculator. Set the expiration days to say 20 and calculate delta. It comes out around ...
0 votes
0 answers
95 views

How to price american barrier with Local-Stochastic Volatility

I have attended a conference where one speaker mentioned that the market standard to price FX and Equity derivatives is now the Local-Stochastic volatility model. I understand this class of model is a ...
3 votes
1 answer
371 views

Simulation scheme for SABR beside the standard Euler discretization

QUESTION: Beside Euler Scheme, is there another more robust (and preferably easy to implement) way to simulate asset path with SABR dynamics? Simulation that will withstand even for high volatilities....
2 votes
1 answer
153 views

Interpreting Implied Volatility in Commodities Options

I understand that implied volatility is the expected volatility of an underlying contract in the Black option pricing model. This is easy to interpret for assets delivered at a point in time. But how ...
0 votes
0 answers
106 views

Does Put-Call parity have influence over American Option pricing in practice?

I am learning my options and from what I read it seems that put-call parity is regarded as only being applicable to European options because the time to exercise is known. American options, on the ...
2 votes
0 answers
28 views

What are some good books to get started with option theory? [duplicate]

Recently graduated in econometrics but starting to realize my knowledge is limited. Any and all tips are welcome!
62 votes
9 answers
78k 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 $...
1 vote
2 answers
124 views

Pricing FX options on pegged currencies

I'm wondering what's the standard (if any) for practitioners to trade volatility on pegged currencies. Is there any specific convention? I'm thinking situations like EURCHF before the unpeg, how were ...
0 votes
0 answers
22 views

Show that $S_0$ is the smallest value of a super hedging strategy for a Call option in a arbitrage free market

I attempt a proof : We want to show $S_0=\inf\{V_0 : \exists\theta\;s.t\;V_T(\theta)\geq H\}$ Suppose this is not the case. There exists $V_0^{*}$ such that $V_0^{*}< S_0$ and $V_T^{*}(\theta)\geq ...
0 votes
0 answers
21 views

The case of the complete Trinomial model

In my journey to hope for a better understanding of incomplete markets I have decided to focus on the Trinomial model (maybe some of you have seen my previous questions). I have decided to consider ...
1 vote
3 answers
1k views

How does a Delta Hedged portfolio yield the Risk-free?

Here I'm considering the simple case of a dealer writing call options on a stock and hedging the short position with a "textbook" Delta Hedge, i.e. goes long on $N_c \times Delta$ stocks (where $N_c$ ...
0 votes
0 answers
46 views

Hedging possibility in a market with more state of the world than asset (discrete time)

For a European Call option, by proposing the initial price of the underlying asset I am sure to be able to meet my commitments, however this result is not true for a Put option. However, by proposing ...
0 votes
0 answers
55 views

Black Scholes and out of the money Index Options

I understand that the Black-Scholes model is not very effective when modeling call options that are deep out of the money. I found a paper on the web by Song-Ping Zhu and Xin-Jiang He related to this ...
0 votes
1 answer
191 views

Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated? [closed]

Re close votes: I believe this is a fair kind of opinion-based question because it's like those ethics questions in academia se or workplace se or because it's pedagogical. Context: I'm actually ...
1 vote
1 answer
67 views

Difference between closed form binomial option value and monte carlo simulation

I am trying to calculate the price of a European call option using both the the closed form expression and a monte carlo simulation. But the value's I get from both these methods are not the same: ...
0 votes
0 answers
30 views

Trinomial model option pricing

If I have well understood, in the trinomial model we have a kind of risk neutral pricing formula that depends on a parameter. This means thaht as in the binomial model, we could use directly this ...
0 votes
0 answers
31 views

Trinomial model

What is the aim of having the price of a self financing portfolio in the trinomial model if we know that the option we are considering is not duplicable ? Do we have to assume that the payoff of the ...
1 vote
1 answer
155 views

What's the price of a lookback call option in the arbitrage-free CRR-model?

If we consider the CRR-model in two periods, i.e. T=2. Let $S^1$ be the risky asset with $S_0^1=100$ and $S^0$ the bond with $S_0^0=1$. Furthermore, we assume the model is arbitrage-free with $y_b=-0....
1 vote
0 answers
57 views

Superhedging in Cox-Ross-Rubinstein model revisited

I am doing the following exercise from a math finance textbook but I got stuck at the end of the part 2. I found nothing on the internet concerning solutions of exercises from this textbook (called ...
0 votes
0 answers
44 views

Can the Feynman-Kac formula be used for asset classes that don’t have options?

So rather than a call option C(S_t,t) we have some type of asset with asset price is given by S(x,t) where x is any type of variable that the asset price depends on. I.e Price of wooden desks, W(x,t) ...
1 vote
1 answer
73 views

Pricing & hedging vanilla interest rate options with SABR LMM

Are there any advantages of pricing and hedging plain vanilla interest rate options with more complex SABR LMM instead of simpler SABR model? Should one always go with the SABR LMM as a universal ...
2 votes
0 answers
60 views

Perpetual Option Paying Chooser Option

A perpetual option solves the ODE $$rSV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$ The general solution is $$V(S)=aS+bS^{\gamma}$$ where $\gamma=-\frac{2r}{\sigma^2}<0$. For an American put option with ...
0 votes
1 answer
67 views

Extension of CRR model

I'm considering an extension of the binomial model where the risky asset can take three values at each node, that is $ S_{t+1}=\left\{ \begin{array}{ll} S_t\cdot u\\\nonumber ...
1 vote
1 answer
72 views

SABR LMM vs no-arbitrage term structure of SABR parameters

There exists a LIBOR Market Model with stochastic volatility for pricing and hedging exotic (e.g. path-dependent) interest rate options with smile. However let us consider the following approach: ...
0 votes
1 answer
91 views

Why do we worry about the bid/ask spread when pricing option in incomplete market?

Several resources I saw introduce the notion of bid/ask spread when trying to price options in incomplete market, I don't understand why the notion is introduced since we are interested on the price ...

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