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 […]
We will introduce in this post stochastic volatility models. They assume that the asset price but also 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 […]
The buyer of an American Option has the right to exercise the option at any time before and including the maturity date of the option while the buyer of a European Option has the right to exercise the option at the maturity date only. We assume that there is no arbitrage and we place ourself […]
Finite-difference methods is the generic term for a large number of techniques that can be used for solving differential equations, approximating derivatives with finite differences. It consists of a discretization of both spatial (underlying price in our case) and time intervals to obtain a finite grid. We find solutions for a differential equation by approximating […]
The Fourier Transform The spectral decomposition of a period function via Fourier series can be generalised to any integrale function via the Fourier transform. And we can recover the initial function from its frequency domain representation by applying the Fourier inversion theorem. Why is it Useful for the Pricing of Options? The characteristic function of […]
P&L Attribution & Greeks: Vanna and Volga Risks Greeks are used to understand and manage the different dimensions of risk involved when trading options. With a second order Taylor expansion of the option value over a short time interval, we can decompose the change of the price of the option with first order greeks on […]
In the Heston model, both the dynamic of the asset price and its instantaneous variance nu are stochastic. The model assumes that the variance follows a mean-reverting Cox-Ingersoll Ross process, and it is correlated with the asset price. The Heston model has five unknown parameters:– ν0: the initial variance– κ: the speed of reversion– θ: […]
The Black-Scholes Model The Black-Scholes model is a pricing model used to determine the theoretical price of options contracts. It was developed by Fischer Black and Myron Scholes in 1973 and later refined by Robert Merton. The model assumes that the financial market has two assets, one risky asset such as a stock price and […]
The Black-Scholes model was developed by Fischer Black and Myron Scholes in 1973 and later refined by Robert Merton. It was the first arbitrage free model relating the option price to the hedging strategy consisting of buying or selling the underlying asset to eliminate the risk. It is well known that some of the assumptions […]
It is crucial for quant or 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 below several questions you could be asked in an interview for a quant or a trading role, with possible answers. Save 10% on All […]
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