Volatility Smiles Chapter 18
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
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
The implied distribution is heavier in both tails than for the lognormal distribution. It is also “more peaked” than the lognormal distribution
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
Possible Causes of Volatility Smile Asset price exhibiting jumps rather than continuous change (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)
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? 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
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