Options Basics

Table of Contents

Basics

Equal price of Call and Put

Using the put call parity, not difficult to find that when $Se^{rT}=K$. That is, when the strike price equal to the forward price of the asset. Then Call and put have the same price.

Delta

Approximation of Delta for ATM Call

Delta for Non-dividend call option: $\Delta(C)=N(d_1)$ where, $$ d_{1}=\frac{\ln \left(\frac{S}{K}\right)+\left(r+\frac{\sigma^{2}}{2}\right)(T-t)}{\sigma \sqrt{T-t}} $$

1. When ATM, $d_1=\left(\frac{r}{\sigma}+\frac{\sigma}{2}\right) \sqrt{\tau}>0$. then $\Delta (C)=N(d_1)>0.5$ all the time.

2. But as time to maturity $\tau$ decreasing, $d_1$ is decreasing to 0, which means an delta close to 0.5.

3. Further assume that $r=0$, then $d_1=\frac{\sigma}{2}\sqrt{\tau}$, approximate $N(x)$ at $x=0$, then we have $N(x)\approx0.5+x/\sqrt{2\pi}\approx 0.5+0.4x$.

$$ \Delta=N(d_1)=N(\frac{\sigma\sqrt{T}}{2})\approx0.5+0.2\sigma\sqrt{T} $$

Delta Hedging

Suppose you have a long position in a call option. Then to protect this delta, need to short stocks, and invest the cash. If the option price follows the BS formula exactly, then cash is $Ke^{-r\tau }N(d_2)$ for each share of the option.

Estimate value of ATM Call

The solution is,

$$ C\approx 0.4\sigma S\sqrt{T} $$

Gamma

Small gamma portfolio would need to be rebalanced less frequent. Being delta and gamma neutral can make the portfolio well hedged to small changes of stock price movements.

Theta

On different books, people may refer theta to different derivatives, maybe $t$ or $\tau$, which shows different directions. But here, we are using the derivatives w.r.t. $t$. The option time premium decays as time elapses.

Theta of non-dividend European call is always negative.

1. The closer to maturity, the higher the kurtosis.
2. The closer to ATM, the higher the absolute value.

A list of possible theta values.

	Dividend	No Dividend
European Call	Positive when Deep ITM	Negative
European Put	?	Positive when Deep ITM
American Call	?	?
American Put	?	?

• The above positive theta cases also provide reasons why American options might be able to early exercised
  • American call with dividends
  • American put regardless of dividend amounts

Recall the Black-Scholes PDE, and understand it as the greeks representation.

$$ \frac{\partial V}{\partial t}+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}}+r S \frac{\partial V}{\partial S}-r V=0 $$

Which indicates $\Theta+\frac{1}{2}\sigma^2S^2\Gamma=rV-rS\Delta$. Delta here is 0, theta and gamma have opposite signs. The positive gain from long gamma can be offset by the negative theta. So there is no arbitrage opportunity.

Vega

1. $S_t$. ATM options most sensitive.
2. $\tau$. Longer time to maturity, higher the vega.

Implied volatility and volatility smile

1. Currency options: volatility smile
2. Equity Options: volatility skew, volatility decrease as strike price increases

Volga

This question depends on whether the price function is convex respect to $\sigma$ or not. If it is convex, then $c(E[\sigma]) \leq E[c(\sigma)]$, the constant volatility option should offer lower price.

To see if the function is convex, we need to use second order derivatives to volatility, which is Volga.

$$ \begin{aligned} v=\frac{\partial c}{\partial \sigma}=S \sqrt{\tau} N^{\prime}\left(d_{1}\right)=\frac{S \sqrt{\tau}}{\sqrt{2 \pi}} \exp \left(-d_{1}^{2} / 2\right)\\ \frac{\partial^{2} c}{\partial \sigma^{2}}=\frac{S \sqrt{\tau}}{\sqrt{2 \pi}} \exp \left(-d_{1}^{2} / 2\right) \frac{d_{1} d_{2}}{\sigma}=v \frac{d_{1} d_{2}}{\sigma} \end{aligned} $$

The sign of volga is determined by $d_1d_2$. Recall the numerators of them has $\ln(\frac{S}{K})$, thus for deep ITM and deep OTM options, the sign of $d_1$ and $d_2$ should be the same. In these cases, the function is convex, and stochastic volatility will need higher price compensation.

While, things might change with the option is around the strike price, and $d_1d_2<0$ is possible. Then constant volatility can also leads to higher price.

Early Exercise

	Dividend	No Dividend
American Call	Not optimal except for time before ex-dividend date	NEVER OPTIMAL
American Put	Can be early exercised	Can be early exercised

The reasons for the early exercise can be found in the “Theta” discussion part.

An argument of intrinsic value

For a European option, consider the put call parity, $$ C=S-Ke^{-r\tau}+p\ C=(S-K) + (K-Ke^{-r\tau}) + p $$ The second and third parts of the RHS are always positive. So the option value should be greater than $S-K$ at least (the value of immediate exercise). The American call is worth greater than the European call, so this relationship still holds. There is no value to early exercise.

Questions

This really depends on the current price of the underlying.

1. If the call is deep in the money, then the put is deep out of the money and the value if trivial. Then from the put call parity, we know $C\approx S-Ke^{-rT}$, so the call price is mainly driven by the change in $S$.
2. If the call is deep out the money, the original value of the call is tiny, and dramatic change in the $S$ can lead to call become moderate valuable, this will usually lead to several order of magnitude increase in value.
3. If the call is ATM, the percentage change generated by doubling the stock price is about one order of magnitude larger since the option will become deep in the money.