## Volatility Smiles Chapter 18
## Put-call parity *p +S*0*e-qT* = *c +X e–r T* holds regardless of the assumptions made about the stock price distribution ## It follows that * p*mkt-*p*bs=*c*mkt-*c*bs
## 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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