2024-02-15

Option Pricing

I'm trying to get into volatility trading and the first thing you need to understand is how to value options with a model. It's impossible to find a model that 100% accurately reflects the price of an option; the variables are just too complex. The map is not the territory.

To get mispriced options, we need to approach the model backwards. First, we assume that the trader holds a delta-hedged portfolio, consisting of one call option and a delta amount of stock sold short.

Some known facts:

  • A call (put) becomes more valuable as the underlying rises (falls), as it has a greater chance of becoming intrinsically valuable.
  • An option loses value as time passes, as it has less time to become intrinsically valuable.
  • An option loses value as rates increase. Since we have to borrow money to pay for options, as rates increase, our financing costs increase, ignoring for now any rate effects on the underlying.
  • The value of a call (put) can never be more than the value of the underlying (strike).
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    With this formula, we are delta neutral. Now, over time, the underlying changes, which we define as St+1​. To define the change for C, it is now...

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    and for delta St it is now

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    Now, we also need to add any financing charges we incur for borrowing the money we need for our position. This is defined as...

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    In total:

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    Now we try to estimate the price of the option by using the change of the underlying through second-order Taylor expansion

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    The first term is the gamma and is proportional to half the square of the underlying price. The second is the theta; the holder loses money with the passing of time. The third term gives the effect of financing. Holding a hedged long option portfolio is equivalent to lending money.

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    Sigma is the standard deviation of the underlying return, known as volatility, so we can rewrite it as...

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    We can set the equation to 0 because it is riskless and we finance it with borrowed money.

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    Now, the price change is not in the equation, but is being expressed through sigma or volatility. You might say that we need to include the drift, but it does not matter due to it being hedged away by the right amount of shares. You can use the equation for European, American, or even exotic options. We have two options when going on:

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    If we estimate the volatility and it differs strongly from the volatility implied by the option market, we have a trade. We buy the option and hedge by selling the stock. Our profit will be the difference between the implied volatility and the forecasted volatility if we are right.

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    You can think of this as Vega, which is how much an option price changes when the implied volatility changes from, let's say, 18% to 19%. In other words, the profit of a hedged option.

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    Now, if we hold the option and the realized volatility is sigma, we will make this amount of Vega profit. Now note that we need to rebalance our hedge from time to time. This can be accounted for by noting the relationship between Vega and gamma.

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    or

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    now being

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    Here are some statements about the model:

  • We assume that the underlying is a tradable asset, such as equities or futures. Tradable assets can also include underlyings that are really illiquid.
  • The underlying pays no dividend.
  • We assume that the underlying is shortable. This is no problem with futures but can be an issue with some stocks. We account for the borrowing cost by stating that it cancels out with the dividend yield.
  • Interest rates have a bid-ask spread; we negate this by stating that the interest rate risk charge (rho) is insubstantial compared to other risks associated with the option.
  • We assume that volatility is constant, so it is neither a function of time nor the change of the underlying. This is not true, but since we are only trying to trade the volatility changes, it is acceptable. While there are models that take this into account, we try to use a framework that organizes our thoughts and what we are trying to achieve rather than accurately depicting reality.
  • We assume that volatility is the only parameter needed to specify the return of the underlying. This would require assuming that returns and prices are normally or log-normally distributed. Essentially, we are using the wrong number in the wrong formula to get the right price.
  • We assume that the price of the underlying changes continuously. This is not true; sometimes, on some names, the price can jump 90% or more in a day. To account for these jumps and the inefficiency in hedging, we need to use semi-static hedging where we hedge the risk with other options.

    • The Black-Scholes-Merton model is only useful for controlling the fast-moving option price as a slow-moving parameter, which is the implied volatility; it is not for risk control. We also need to be aware of black swan events that supposedly never happen. This tail risk can be taken care of by either buying out-of-the-money options or keeping individual position sizes small. But generally, we get paid for taking risk. So never use the model for which you assumed the price for assuming risk.

    Always remember that our model is not an accurate reflection of reality:

  • The drift of the underlying can be hedged away.
  • The magnitude of the underlying price moves cannot.
  • The BSM model is a model for finding trades, not a model for controlling risk.
  • The assumptions used in the derivation of the model need to be remembered at all times.

    • Most of this is from the book Volatility Trading

    Thanks,

    Finn