Volatility Smiles Chapter 18 Put-Call Parity Arguments

Volatility Smiles Chapter 18

Put-Call Parity Arguments

• Put-call parity p +S0e-qT = c +X e–r T holds regardless of the assumptions made about the stock price distribution

• It follows that

• pmkt-pbs=cmkt-cbs

Implied Volatilities

• The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity

• The same is approximately true of American options

Volatility Smile

• A volatility smile shows the variation of the implied volatility with the strike price

• The volatility smile should be the same whether calculated from call options or put options

The Volatility Smile for Foreign Currency Options

Implied Distribution for Foreign Currency Options

• The implied distribution is heavier in both tails than for the lognormal distribution.

• It is also “more peaked” than the lognormal distribution

The Volatility Smile for Equity Options

Implied Distribution for Equity Options

Other Volatility Smiles?

• What is the volatility smile if

• True distribution has a less heavy left tail and heavier right tail

• True distribution has both a less heavy left tail and a less heavy right tail

Possible Causes of Volatility Smile

• Asset price exhibiting jumps rather than continuous change

• Volatility for asset price being stochastic

• (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)

Volatility Term Structure

• In addition to calculating a volatility smile, traders also calculate a volatility term structure

• This shows the variation of implied volatility with the time to maturity of the option

Volatility Term Structure

• The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low (mean reversion)

Example of a Volatility Surface

Brief Review

Consider a 3-month put (European) on a stock with K = $20. Over the next 3 months the stock is expected to either rise by 10% or drop by 10%. The risk-free rate is 5%.

• Consider a 3-month put (European) on a stock with K = $20. Over the next 3 months the stock is expected to either rise by 10% or drop by 10%. The risk-free rate is 5%.

• Q1: What is the risk-neutral probability of the up-move?

• Q2: what if the stock pays a dividend yield of 3% per year?

Q3: What position in the stock is necessary to hedge a long position in 1 put option (assume no dividends)?

• Q3: What position in the stock is necessary to hedge a long position in 1 put option (assume no dividends)?

• Buy 0.5 shares

• Q4: How would the answer to Q3 change if the stock paid a dividend yield of 3%?

• Q5: What is the value of the put?

Assume now that the put has 6 months to maturity and there are 2 periods on the tree (each period is 3-moths long). Everything else is the same.

• Assume now that the put has 6 months to maturity and there are 2 periods on the tree (each period is 3-moths long). Everything else is the same.

• Q6: Compute the value of the put option.

• Q7: Answer question 6 for an American put.

• Q8: The cost of hedging which option is higher – American or European put? Compute it for both. Compute Gamma of the European put option. Can you hedge your Gamma-exposure by trading futures?

Consider a European call option on FX. The exchange rate is 1.0000 $/FX, the strike price is 0.9100, the T is 1 year, the domestic risk-free rate is 5% the foreign risk-free rate is 3%. If the call is selling for 0.05 $/FX, is there arbitrage and, if so, how would you exploit it?

• Consider a European call option on FX. The exchange rate is 1.0000 $/FX, the strike price is 0.9100, the T is 1 year, the domestic risk-free rate is 5% the foreign risk-free rate is 3%. If the call is selling for 0.05 $/FX, is there arbitrage and, if so, how would you exploit it?

• Hence, borrow and sell units of FX, invest the PV of K and buy the call today.

• At maturity, you have $K which is more than enough to buy the currency.

A trader in the US has a portfolio of derivatives on the AUD with the delta of 450. The USD and AUD risk-free rates are 5% and 7%.

• A trader in the US has a portfolio of derivatives on the AUD with the delta of 450. The USD and AUD risk-free rates are 5% and 7%.

• Q1: What position in the AUD creates a delta-neutral position?

• Short 450 AUD

• Q2: What position in 1-year futures contract on the AUD creates a delta-neutral position?

• Redo Q2 for the case of forwards.

A portfilio of derivatives on a stock has a delta of 2000 and a gamma of -100.

• A portfilio of derivatives on a stock has a delta of 2000 and a gamma of -100.

• Q1: What position in the stock would create a delta-neutral position?

• Short 2,000 shares

• Q2: If an option on the stock with a delta of 0.6 and a gamma of 0.04 can be traded, what position in the option and the stock creates a portfolio that is both gamma and delta neutral?

• Short 3,500 shares and buy 2,500 options