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Binary Options: Replication ad Skew Sensitivity The final payoff of a binary (or digital) call option is either one if the asset price is above the strike price at the expiry of the option or nothing.  It can be replicated as the limit of a call spread. If we assume that the only changing variable is the strike price, the price of the binary call is equal to the opposite of the derivatives of the call price with respect to the strike price which can be approximated by finite central difference. In the Black-Scholes framework there is a closed-form solution for the price of the binary call option, it is equal to the probability that the option will be exercised times the discount factor. But what if the implied volatility is not constant?  If we assume that the implied volatility is no more constant and is a function of the strike price, we can approximate the price of the binary call option as the sum of two terms.  The first one corresponds to the Black-Scholes price of the binary option, assuming the implied volatility is constant.  The second term is a correction to the Black-Scholes price as the implied volatility is actually not constant. It is the opposite of the product of the vega of a call option and the derivatives of the implied volatility with respect to the strike price, also named volatility skew. In the equity market we observe in general a downward sloping volatility as a function of the strike price, meaning a negative skew.  Lower strikes tend to have a higher volatility than higher strikes making the price of the call spread which is long on the lower strike and short on the higher strike more expensive. The price of the digital option increases when the skew becomes more negative. The impact is far from being negligible. ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Test Your Knowledge We assume that the implied volatility is a downward sloping function of the strike price. What is the impact on the price of a binary call option compared to the Black-Scholes price? More about it with 50 possible Q&A for quant and trading interviews about option Greeks: https://lnkd.in/eVpb2UTM ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Quant and Trading Interview Questions about Option Greeks It is key for quant and trading positions to have a strong understanding of the main option Greeks and how they can be used for risk management purposes. You will find in this article 50 questions you could be asked in an interview with possible answers. https://lnkd.in/eVpb2UTM ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Credit Risk Modelling: Pricing of a Defaultable Bond with Reduced-Form Models Part I We just released our latest video on credit risk modelling. In this video we will see how to price a risky zero coupon bond assuming zero recovery rate with a reduced-form model for credit risk. https://lnkd.in/eUV6MMEW More videos about credit risk modelling: - Credit Risk: an Introduction https://lnkd.in/e3gAqhxy - Credit Risk Modelling: The Probability of Default https://lnkd.in/ewSFwFUm - Credit Risk Modelling: Default Time Distribution https://lnkd.in/gfUDNH5q - Credit Risk Modelling: an Introduction to Reduced-Form Models https://lnkd.in/ejaDvgsM Please do not hesitate to like it, share it, comment it. Quant Next ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Pricing of European options using the characteristic function The characteristic function of a stochastic variable is the conjugate of the Fourier transform of its density function. If we know the characteristic function we can recover the density function by applying the Fourier inversion theorem. While it is often difficult to obtain the density function for many processes, it is more simple to get their characteristic function, as it can be derived as the solution of a partial differential equation using the Feynman Kac theorem. If we know the characteristic function, we can recover the density function by Fourier inversion and we can derive the price of european call options with a semi analytic solution. The price of the call option has a similar expression as in the Black-Scholes formula. We use the characteristic function in the integral instead of the density function. This is typically the case for the Heston model, which is one of the most popular stochastic volatility model. More about it in our training course Options, Pricing and Risk Management Part III: https://lnkd.in/gmW3TvEs Please do not hesitate to like it, share it, comment it. Quant Next ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Summer comes to an end, you still have a few days to benefit from our additional 25% summer discount on our training program Options Pricing and Risk Management Part I, II and III. You will be able to enhance your skills on these key topics of quantitative finance at your own pace. You will get a 1-year access to training courses which are 100% digital, with multiple videos, tutorials and exercises in Python, interactive quizzes and evaluations. 25% discount coupon: Summer25 Link to the courses: https://lnkd.in/em-RYMQR Quant Next

Introduction to Stochastic Volatility Models Stochastic volatility models assume that both the asset price and its variance follow stochastic processes. Such models are used in finance to simulate the price of the underlying asset and then to value options. They are able to explain with a few additional parameters why the Black-Scholes implied volatilities with different strike prices or different time to maturities are different. They can be very useful to fit the volatility curve from vanilla options and to price more complex exotic options. The choice of the two functions α and β will determine how the volatility behave. If we assume that both α and β are equal to zero, then the variance is no more stochastic, it is constant equal to the initial variance ν0. We are back in the Black-Scholes framework. The initial variance ν0 controls the volatility level. ξ is the volatility of volatility.  A higher volatility of volatility increases the probability to have extreme movements on both sides. It increases the tail risk of the return distribution. This translates into higher risk and higher prices for out-of-the-money options and an increase of the implied volatility on the wings creating a volatility smile. ρ is the correlation between the two brownian motion. It measures the correlation between the asset return and the change of the variance, it controls the spot / vol correlation. A negative correlation would mean that negative returns tend to come with higher volatility. This is typically what we observe on the stock market, with higher volatility in crisis periods. This parameter controls the asymmetry of the return distribution, its skewness. A negative spot / vol correlation will typically increase the probability to have very negative returns and reduce the probability to have very positive returns. This translates into higher risk / higher price and so higher implied volatility for out-of-the-money put options on the left side, and lower risk / lower price and so lower implied volatility for out-of-the-money call options on the right side. The implied volatility is said to have a negative skew in this case, with an asymmetric curve and higher implied volatilities for low strike prices on the left compared to high strike prices on the right. If you wish to deepen your knowledge on stochastic volatility models, have a look at our training course: https://lnkd.in/gmW3TvEs You will be able to learn it at your own path with multiple videos, quizzes and applications in Python. We offer a 25% discount coupon on the three courses "Options, Pricing and Risk Management" Part I, II and III for the summer with the coupon code: Summer25 For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested. contact@quant-next.com Quant Next https://quant-next.com/

1h+ Free Videos on Options, Pricing and Risk Management Exotic Options - Part I https://lnkd.in/e5rK72fz Exotic Options - Part II https://lnkd.in/eCSKVBNt The Cox-Ross-Rubinstein Binomial Option Pricing Model https://lnkd.in/eJZavkFs American Option Pricing with Binomial Tree https://lnkd.in/eb6Gt9K6 Introduction to Stochastic Calculus https://lnkd.in/ehZswrpZ The Black-Scholes Model https://lnkd.in/e3ekaUPi The Black-Scholes Formula Explained https://lnkd.in/eFHnVv26 The Volatility Smile and Skew https://lnkd.in/ePxjnTdh Introduction to Option Greeks and Risk Management https://lnkd.in/e8JfZPgQ The Option Greek Delta Explained https://lnkd.in/eBP7Sbx5 All these videos are part of our training course "Options, Pricing and Risk Management Part I". Here is an overview of the course: https://lnkd.in/e5u689gU ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ We offer a 25% discount coupon during the summer on the three courses "Options, Pricing and Risk Management" Part I, II and III with the coupon code: Summer25 ★★ ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Monte Carlo integration Monte Carlo simulations use random quantities to provide an estimate of a deterministic quantity. It can be for example the probability a specific outcome. We simulate multiple values of a specific random variable and then we average it to obtain an estimate. A useful application of Monte Carlo simulations is to estimate integrals. If you consider the integral I of a function g between a and b, we could rewrite I as the product of b - a and the integral on R of g times a function f which is the density function of the uniform distribution between a and b. So we can rewrite the integral I as b - a times the expectation of g of u with u following a uniform distribution between a and b. We simulate u a large number of time and we obtain an estimation of I by Monte Carlo method. It is a non bias estimator converging to the true value when the number of simulations become large enough. Such method can be particularly useful to estimate multidimensional integrals. More about Monte Carlo simulations and other numerical methods for the pricing of options in our training course Options, Pricing and Risk Management Part II: https://lnkd.in/epFJf6VS Please do not hesitate to like it, share it, comment it. ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ We offer a 25% discount coupon during the summer on the three courses "Options, Pricing and Risk Management" Part I, II and III with the coupon code: Summer25 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★

Credit Risk Modelling: an Introduction to Reduced-Form Models We just released our latest video on credit risk modelling introducing reduced-form credit risk models with constant, non-constant and stochastic default intensity models. https://lnkd.in/ejaDvgsM Please do not hesitate to like it, share it, comment it. ★★ Save 10% on all Quant Next courses with the coupon code: QuantNextLinkedIn10 ★★ ★★ We offer a 25% discount coupon during the summer on the three courses "Options, Pricing and Risk Management" Part I, II and III with the coupon code: Summer25 ★★ ★★ For students and graduates, we offer a 50% discount on all courses, please contact us if you are interested ★★ ★★ Visit us: https://quant-next.com ★★ ★★ Contact us: contact@quant-next.com ★★ ★★ Follow us: https://lnkd.in/eQEp_YM5 ★★