Black-Scholes Assumptions, Greeks, and Short Straddle P&L

Options Pricing · Medium · Free problem

Three-part options fundamentals question:

  1. State the two core assumptions of the Black-Scholes-Merton (BSM) model.
  1. Define vega and gamma.
  1. You sold a straddle for a total premium of $\
0$. If the underlying stock's realized volatility moves slightly higher than expected, will your position make or lose money at maturity? Explain why.

Hints

  1. For BSM: one assumption is about the price process (what dynamics does $S$ follow?), the other is about the market structure (what trading frictions and arbitrage opportunities exist?).
  2. Vega and gamma are both positive for long options -- think about why intuitively before writing formulas. Higher vol makes an option more valuable (vega). Larger moves in the underlying make a long option more valuable (gamma).
  3. You sold the straddle -- you are short both calls and puts. Since long options have positive vega and gamma, your short position has negative vega and gamma. What happens to the value of your short position when vol increases?

Worked Solution

How to Think About It: These are three distinct conceptual questions that a trader would expect any junior quant to answer cold. The BSM assumptions are the foundation -- everything breaks down if you forget them. The Greeks are the language of options risk. And the straddle question is testing whether you understand the sign of your exposure: you sold options, so you are short all the positive convexity.

Quick Sanity Check on the Straddle: You sold a call and a put at the same strike. Both have positive vega (as a buyer). Since you are short both, you are short vega. Higher vol means both options became more valuable -- you owe more. You lose.

Part 1 -- BSM Assumptions:

The two core assumptions are:

  1. The underlying follows geometric Brownian motion with constant volatility: $dS = \mu S\,dt + \sigma S\,dW_t$, where $\sigma$ is a constant. This means returns are log-normally distributed with no jumps.
  1. Frictionless, arbitrage-free markets: Continuous trading is possible, there are no transaction costs, and there is a constant risk-free rate $r$ at which you can borrow and lend. Together with GBM, this allows dynamic delta-hedging that replicates the option payoff exactly.

(Other standard assumptions sometimes listed: no dividends, European exercise. The two most fundamental are GBM dynamics and no-arbitrage with frictionless markets.)

Part 2 -- Greeks:

Vega ($\mathcal{V}$): The sensitivity of an option's price to a change in implied volatility: $\mathcal{V} = \frac{\partial V}{\partial \sigma}$ Vega is always positive for long options (calls and puts) -- higher volatility increases the value of the right but not the obligation. Vega is largest for at-the-money options and declines for deep in- or out-of-the-money options.

Gamma ($\Gamma$): The rate of change of delta with respect to the underlying price, i.e., the second derivative of option value with respect to $S$: $\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}$ Gamma measures convexity. It is positive for long options, largest for ATM options near expiry, and captures how much delta-hedging you must do as the stock moves.

Part 3 -- Short Straddle P&L with Higher Realized Vol:

A short straddle is short one call and short one put at the same strike and expiry. By selling the straddle: - You collected $\ 0$ in premium upfront. - You are short gamma (you lose money from large moves in either direction). - You are short vega (you lose money if implied volatility rises).

If realized volatility moves slightly higher: - If the question refers to realized vol: larger price moves hurt you because your short gamma position loses money proportional to $\Gamma \cdot (\Delta S)^2$. - If the question refers to implied vol: higher implied vol increases the mark-to-market value of both options you are short, causing an immediate loss of approximately $\mathcal{V} \cdot \Delta\sigma$ per option.

In either interpretation, you lose money. The $\ 0$ premium you collected was compensation for taking on exactly this risk. You profit only if the stock stays near the strike and volatility does not increase.

Answer: 1. BSM assumptions: (i) GBM with constant $\sigma$; (ii) frictionless, arbitrage-free markets with continuous trading. 2. Vega $= \partial V / \partial \sigma$ (sensitivity to vol, always positive for long options); Gamma $= \partial^2 V / \partial S^2$ (sensitivity of delta to price, always positive for long options). 3. Short straddle is short gamma and short vega. Higher volatility causes losses. You lose money.

Intuition

The short straddle is one of the most instructive positions in options trading because it makes the vega/gamma tradeoff concrete. You collect premium (theta decay works in your favor), but you are exposed on both dimensions: large moves (gamma) and volatility spikes (vega). In BSM with constant vol, gamma and vega are perfectly hedgeable -- the premium you collect is exactly fair compensation for the expected delta-hedging cost. In the real world, vol is not constant, so the short straddle is a bet that realized vol will be below implied vol.

The BSM assumptions are worth remembering not just for their content but for what they imply when violated. Constant volatility is the most egregious assumption in practice -- vol smiles, term structure, and mean-reversion all violate it. GBM with no jumps is violated by earnings announcements and macro events. Every extension of BSM (stochastic vol, jump-diffusion, local vol) is essentially a targeted relaxation of one of these assumptions.

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