Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

Filter by
Sorted by
Tagged with
1
vote
0answers
62 views

Likelihood ratio and pathwise sensitivity method for coupled SDEs

I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $W^{(1)}$ and $W^{(2)}$ are independent Brownian motions, initial input are $S_0$ ...
0
votes
0answers
51 views

What is the relationship between Vanna and Gamma?

I'm trying to build a crude model for the effects of delta hedging on major indices like the S&P 500. My background is more in pure mathematics so a lot of this stuff is new to me. That said I ...
2
votes
0answers
109 views
+100

Implementing a Variance Swap Hedging in R

I am trying to compute a hedge for a variance swap, in a simulation. Fo that I am using the following equation:\begin{align*} E^Q\bigg(\sum_{i=1}^n \bigg(\frac{S_{t_{i}}-S_{t_{i-1}}}{S_{t_{i-1}}}\bigg)...
3
votes
1answer
91 views

What is the Radon-Nikodym derivative in the Heston model?

It is clear to me that $$ \frac{dQ}{dP} = e^{-\lambda W_T-\frac{\lambda^2}{2}T}$$ is the Radon-Nikodym derivative that defines the change of measure in the framework described by Black and Sholes. But ...
0
votes
0answers
19 views

Is the market price of risk deterministic or stochastic in the Heston model?

I am recently digging into the Heston model and I have noticed that every author refers to the market price of risk simply as $\lambda$, or sometimes it is more clearly specified to be bi-dimensional ...
0
votes
0answers
28 views

The put-call parity have to be fulfilled by an asian option

Coming from here: https://quant.stackexchange.com/a/7616/43679 we have that for a European option, and due to the put-call parity, due to the non arbitrage rule, the volatility for a put and a call ...
0
votes
0answers
44 views

Can you calibrate the Heston model using stock price trajectories?

I'm interested in calibrating the Heston model so I was reading about it online. All procedures I could find was using market prices for European call options and using the (semi-)closed-form ...
-4
votes
0answers
34 views

Which path should I follow to understand pricing? [closed]

The world of finance is very new to me. That been said, I would like to know what are your opinion on how to study (by myself) finance in general. My interests are cryptocurrencies (mainly because of ...
0
votes
0answers
41 views

Price of european call option for different strike prices

Consider two european put options with strike prices $K, J$ with $K<J$ and maturity $T$. Then the no arbitrage assumption implies $P_{K}(0)<P_J(0)$, where $P_K(0)$ denotes the price of the put ...
1
vote
0answers
44 views

Replicating portfolio in the Heston model

Given the Heston model $$dS_t=\mu S_tdt+\sqrt{\nu_t}S_tdB_{1,t}\\ d\nu_t=k(\theta-\nu_t)dt+\eta\sqrt\nu_tB_{2,t}$$ how should the replicating portfolio $V_t$ for the derivative $F_t$ be composed? I ...
2
votes
0answers
48 views

Breeden and Litzenberger formula for pricing state-contingent claims

I am reading these two papers Prices of State-Contingent Claims Implicit in Option Prices and Implied Risk-Neutral Distribution: A Comparison of Estimation Methods. I understand how we get the formula ...
1
vote
0answers
41 views

Log Moneyness vs Log Strike

In How to calibrate a volatility surface using SVI, is said: "(log-moneyness would be more accurate) ". First, why do we talk about "moneyness", is it a reference of "being in ...
3
votes
2answers
242 views

Heston stochastic volatility, Girsanov theorem

How can we apply Girsanov's theorem to a stochastic volatility model? In Heston's model the dynamics are given by \begin{align*} dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
2
votes
0answers
119 views

Is implied volatility really all that usefull?

I take implied volatility as the positive floating point number which lets the BS formula match an observed option price (assuming we have some useful interest rate, some underlying, etc). How useful ...
1
vote
0answers
52 views

Future price in continous time

I am in the following continuous time market: $S_t^0 = rS_t^0dt$ $S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$ where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$...
1
vote
1answer
145 views

What's the intuition behind there being a perfect linear relationship between option value and expected volatility?

I modelled option prices using the BS model at different levels of volatility. Surprisingly, I came out with a perfectly linear relationship. As volatility rises, so does the option value, which is ...
1
vote
0answers
24 views

When is the effect of skew most potent for an early exercise option?

Let us say I have a Bermudan option which I can terminate at 3 possible dates. When can I expect the discrepancy between a local vol and a stochastic vol model to be highest (assuming both are ...
3
votes
1answer
143 views

Conditional probability of Brownian motion (with drift and scaling) hitting barrier

I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to: derive the ...
0
votes
0answers
34 views
2
votes
2answers
92 views

Boundary Conditions for Call Option in Black Scholes Model

Let $C(t,S)$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $0=t_0<t_1<...<t_{N-1}<...
0
votes
1answer
39 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-...
2
votes
1answer
143 views

How do you finance theta decay when replicating an option?

When constructing a replicating portfolio for a short position in a call option under Black Scholes, I am not able to pinpoint the source of gains from theta decay. When theta decay materializes, I ...
2
votes
1answer
55 views

Intrinsic Value of European Options [closed]

I have a question regarding the intrinsic value of an European option. I use the following notations: $S_t$ price of the non dividend paying stock at time $t<T$, $T$ is the maturity, $r$ risk-free ...
2
votes
1answer
236 views

What is the difference between a volatility smile and a correlation smile?

I understand to plot correlation and volatility smiles, we have to plot the implied normal vol vs strike and observe a U-shaped relationship. How are these smiles different? Does a vol smile plotted ...
2
votes
0answers
26 views

Binomial Option Pricing Model gives increasingly higher value for out-of-the-money options

I was developing the binomial option pricing model via Python, according to the explanation given on Wikipedia. After computing the errors against the pricing of real options, I find an interesting ...
3
votes
1answer
103 views

Pricing Call Option on Coupon Bond under Vasicek

Consider a the Vascicek model, and let A and B denote the functions such that $P(t,T)=\exp(A(t,T)-B(t,T)r(t))$. We now look at a coupon bond that makes deterministic payments $\alpha_1,...,\alpha_N$ ...
5
votes
2answers
232 views

Strategy of replicating a portfolio with payoff $\int_0^T \frac{dS_t}{S_t}$

Given the asset price $S_t$ which is defined as follows $$\frac{dS_t}{S_t}= r_tdt+\sigma_tdW_t$$ where $r_t$ is not necessarily deterministic. What is the strategy of replication of the portfolio with ...
1
vote
0answers
22 views

Local vol vs stochastic vol in the context of American digital options

I have two models of some spot. One is under local vol and the other is under stoch vol. Both are calibrated to the prevailing vanilla prices. I then consider the option that pays $1$ if the spot ...
2
votes
1answer
65 views

COS Method and existence of density

Hey in the COS method we use characteristic function of $\ln{S_T}$ to price european options (by recovering density from characteristic function). But how do we know that density exists? For example I ...
1
vote
0answers
97 views

Quanto put hedge\ replication with a brownian motion

Consider $d B_{us}(t)=r_{us} B_{us}(t) dt\\dX(t)=X(t)(r_{us}-r_J)dt+X(t)\sigma^T_J dW(t)\\d B_J(t)=r_{J} B_{J}(t) dt\\dS_J(t)=S_J(t)(r_J-\sigma^T_X\sigma_J)dt+S_J(t)\sigma^T_J dW(t)$ where the $\sigma$...
1
vote
1answer
62 views

Arbitrage-free prices in incomplete markets

Hey where I could find theory of option pricing in incomplete markets? I know that there we have not one price, but interval of arbitrage-free prices and I would like to read more about it and I need ...
1
vote
2answers
92 views

What's the difference between shorting “borrowed” shares and “fake” shares [closed]

Like it or not, millions of people are now looking to r/wallstreetbets for not only memes but to view shared investment research. I've learned a lot about stock options in the past year, but I'd like ...
0
votes
1answer
79 views

Misconception about replicating portfolio [closed]

I am solving a problem in which following payoff is provided: With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$)...
0
votes
0answers
50 views

eurodollar future options basics

I am trying to understand how to calculate the P&L on a eurodollar futures options position. Suppose I am looking at say Dec-2023 99.125 strike put options with a bid-ask of 0.2250 - 0.415. Since ...
0
votes
1answer
114 views

Pricing of $(S(T_0)-S(T))^+$

Problem: Consider a new derivative that at time $T$ pays $Y =(S(T_0) − S(T))^+$ where $0 < T_0 < T$ is a fixed date. (i) Show that the arbitrage-free of Y at time $t = T_0$ is given by $\pi_{...
0
votes
2answers
81 views

Prove the Euro call option value has positive relationship with the risk-free rate under discrete time model (Binomial tree model)

Could anyone show me how to prove that the European call option value has a positive relationship with the risk-free rate in a two-step binomial model with strike price K and different risk neutral ...
2
votes
2answers
104 views

Risk-free interest rate for option pricing from treasury yield curve rates

I am experimenting with an implementation of the Black-Scholes valuation for call options, and ran into the following questions: Black-Scholes pricing requires a risk-free interest rate. What is '...
0
votes
0answers
47 views

Greeks for Asian options on futures

I'm trying to get the Greeks for the PDB Option Contract (Crude Outright - Dated Brent (Platts) Average Price Option): https://www.theice.com/products/26535747/Crude-Outright-Dated-Brent-Platts-...
2
votes
1answer
74 views

Martingale pricing with time-dependent risk-free rate

I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e. $$ d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du} $$ I ...
0
votes
1answer
153 views

Calculation of market price for option at underlying strike price at some point in future

Would appreciate clarification on the below scenario. If a put option was sold at the start of the week, when the broker (Interactive Brokers) calculates the cost basis (the premium collected) are the ...
3
votes
1answer
128 views

Does CRR Model lose completeness if we add another instrument?

Consider the multiperiod binomial/CRR model with one risky asset $S^{1}$ and a numeraire $S^{0}$. By seeing that the equivalent martingale measure is uniquely determined, we obtain that the market is ...
2
votes
1answer
118 views

What's the point of complex option formula adjustments?

I had a discussion with a colleague of mine about the implications of risk modelling for the Gamestock/Wirecard cases in the recent weeks and that stock borrow rates could be an important risk factor ...
0
votes
0answers
41 views

How to calculate approximate historical price of options?

Assuming that I have access to the historical daily closing price, it is possible to approximately calculate the daily price of an option? I understand that one pricing model is the Black-Scholes ...
1
vote
2answers
518 views

How to prove no-arbitrage when a long butterfly is strictly positive?

I want to prove why there are no arbitrage opportunities when a long butterfly is strictly positive. I know there is a similar topic out there, but it seems it doesn't solve my question: Prove that ...
1
vote
1answer
99 views

Risk free rate in black-scholes model

Currently reading A. Damodaran‘s book Investment Valuation. In chapter 5 in order to value an option using black-scholes model he adjusts risk free rate using the following formula: $1-e^{-r}$ I. E. ...
1
vote
1answer
62 views

Black (1976) model growth rate input for futures price

When using the Black 76 model for pricing European index options I've often seen people use 2 different rates: the typical risk free rate used to get the discount factor, and a growth rate used to get ...
1
vote
1answer
127 views

Option pricing using discrete fourier transform (python)

I am trying to implement the pricing formula for a European (call) option given in Ales Cerny's paper "Introduction to Fast Fourier Transform in Finance" (paper can be found here), as ...
0
votes
0answers
24 views

Swaption with KO in FX

Curious to know if anyone has much experience modelling such products. I assume a three factor model (domestic/foreign interest rates/FX + correlation) would be necessary for these - but if the ...
0
votes
0answers
29 views

What is the expected return of a “correctly” priced option if volatility stays constant?

For example, if we take a call option on some stock priced using an option pricing model such as Black-Scholes and assume that volatility stays constant and the underlying stock moves according to the ...
1
vote
1answer
32 views

Valuation of the minimum guaranteed return that (some) pension funds provide - how would you do it?

Let's say a pension fund guarantees an annual return of at least 5% to their customers/investors, such that the investors face a payoff like the one of a call option (no downside). For this guarantee ...

1
2 3 4 5
29