The intuition behind option valuation in high Credit Risk scenarios

Remember the Jump-to-Default Approach with Hazard Rate modeled with an SDE providing the Stock Price Dynamics:

dS_t = μS_tdt + σS_tdW_t + J_tdN_t

In option trading, volatility plays a pivotal role in pricing. For call options with low strike prices, this volatility is heavily influenced by the jump component (J). Here's why:

The "Brownian" part of the stock's movement (represented by σ) involves small, incremental changes, which are unlikely to bring the stock price down to very low levels.

However, the "jump" part (represented by J) can cause sudden and significant drops in price, particularly in high credit risk situations where the possibility of a company defaulting is more substantial.

The Black-Scholes model assumes a log-normal distribution of stock prices and smooth, continuous movements. However, it doesn't directly incorporate sudden jumps or credit risk factors. This leads to discrepancies:

For low strike call options (aka deep in the money calls), especially in high credit risk contexts, the market often quotes a volatility lower than what Black-Scholes predicts.

The reason is the market's anticipation of the stock's significant price drop in case of default, which is not fully captured by the Black-Scholes model.

In high credit risk situations, there is an increased likelihood that a company might default. A default event can lead to a substantial decrease in the company's stock price, potentially even to zero.

A call option is deep in-the-money when its strike price is significantly below the current stock price. These options have a high intrinsic value under normal conditions because they can be exercised profitably.

However, in a high credit risk environment, the value of these options becomes more uncertain. Despite being deep in-the-money, the risk of the underlying stock plummeting due to potential default makes their future less secure.

The market, anticipating the potential for a sharp decline in the stock's value due to default risk, adjusts its valuation of these call options. The expectation is that in the event of default, these options will lose much of their value.

In this scenario, even though high credit risk suggests higher uncertainty, the market factors in the specific nature of the risk the likelihood of a sharp decline in stock price.

Given the market's expectation of lower prices for these call options, the implied volatility quoted in the market is adjusted downward. This is because the primary risk for these options is not the usual market volatility but the significant chance of a steep drop in stock price due to default.

Lower quoted volatility reflects this market sentiment, differing from the Black-Scholes model, which might suggest higher volatility not specifically accounting for the acute default risk.

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