# Delta Hedge - Risk Free

Can someone help me understand a couple of things: •Is the Ct in schweser the call premium or the call payoff •How by creating a delta hedge on a written call the portfolio earns the risk free rate? In schweser they have an example: Initial call value \$1.40 30 days left to maturity Delta 0.5739 20,000Calls written \$100 share of stock Thus, the dealer will purchase: 20,000*0.5739 = 11,478 shares so that if the stock rises by \$1 then the liability on the short call is \$11,478 and this will be offset by the \$11,478 increase in value on the long stock. The portfolio init val is : 11,478*100 - 1.4*20,000 = \$1,119,800 Now a day later: Call value is \$1.39 Stock has not changed: The portfolio final val is: 11,478*100 - 1.39*20,000 = \$1,120,000 Thus the increase in port. value is \$200 and this is equivalent to the growth at the risk-free rate. I’m sort of okay until here: Then consider another situation: 1 day has passed S Stock price at \$101 Value of call is \$1.97 Delta is 0.7040 So the portfolio value is now: 11,478*101 - 1.97*20,000 = \$1,119,878 This increase is only \$78 (rel. to init value) which is less than the \$200 earlier and the \$184 if portfolio grows at the risk-free rate of 6%. So the reason apparently why the portfolios did not grow exactly at the risk-free rate is due to rounding. Okay in the case where, the difference is \$16 i.e.\$200 -\$184, I can agree. But, when the diff is \$122 from \$200 -\$78 - is this 'cos of rounding? And how and why should the portfolio value grow at the risk free rate? In this case the portfolio did not grow at the risk-free rate 'cos the port. did not have enough shares to offset the increase in the call liability and that the delta of the port. was not a good predictor.

It’s not just rounding; it’s also gamma. The rounding difference appears to be \$78 (=0.0039*20000) which leaves the rest to gamma. Delta-hedging doesn’t give you a risk free return unless you do it continuously and the stock price has no jumps.

JoeyDVivre Wrote: ------------------------------------------------------- > It’s not just rounding; it’s also gamma. The > rounding difference appears to be \$78 > (=0.0039*20000) which leaves the rest to gamma. > > Delta-hedging doesn’t give you a risk free return > unless you do it continuously and the stock price > has no jumps. or in other words (not trying to trump you Joey) Delta hedging gives you the risk free rate if the volatility you used to price the option is the actual volatility realized over the life of the option (assuming the hedge ratio is adjusted constantly…). If realized volatility exceeds the volatility estimate than the return is greater than the risk free rate, but the return is less than the risk free rate if realized volatility is less than the estimate.

trymybest Wrote: ------------------------------------------------------- > Can someone help me understand a couple of > things: > •Is the Ct in schweser the call premium or the > call payoff > •How by creating a delta hedge on a written call > the portfolio earns the risk free rate? > > In schweser they have an example: > > Initial call value \$1.40 > 30 days left to maturity > Delta 0.5739 > 20,000Calls written > \$100 share of stock > > Thus, the dealer will purchase: > 20,000*0.5739 = 11,478 shares so that if the stock > rises by \$1 then the liability on the short call > is \$11,478 and this will be offset by the \$11,478 > increase in value on the long stock. > > The portfolio init val is : 11,478*100 - > 1.4*20,000 = \$1,119,800 > > Now a day later: > Call value is \$1.39 > Stock has not changed: > > The portfolio final val is: 11,478*100 - > 1.39*20,000 = \$1,120,000 > > Thus the increase in port. value is \$200 and this > is equivalent to the growth at the risk-free rate. > I’m sort of okay until here: > > Then consider another situation: > 1 day has passed S > Stock price at \$101 > Value of call is \$1.97 > Delta is 0.7040 > > So the portfolio value is now: 11,478*101 - > 1.97*20,000 = \$1,119,878 > > This increase is only \$78 (rel. to init value) > which is less than the \$200 earlier and the \$184 > if portfolio grows at the risk-free rate of 6%. > > So the reason apparently why the portfolios did > not grow exactly at the risk-free rate is due to > rounding. Okay in the case where, the difference > is \$16 i.e.\$200 -\$184, I can agree. > > But, when the diff is \$122 from \$200 -\$78 - is > this 'cos of rounding? And how and why should the > portfolio value grow at the risk free rate? In > this case the portfolio did not grow at the > risk-free rate 'cos the port. did not have enough > shares to offset the increase in the call > liability and that the delta of the port. was not > a good predictor. Sorry, I may not have been clear in this answer. The theory assumes that you were buying stock constantly between 100 and 101, so by the end of the day, you own 14,080 shares instead of 11,478… Does that help?

ahahah Wrote: ------------------------------------------------------- > JoeyDVivre Wrote: > -------------------------------------------------- > ----- > > It’s not just rounding; it’s also gamma. The > > rounding difference appears to be \$78 > > (=0.0039*20000) which leaves the rest to gamma. > > > > > Delta-hedging doesn’t give you a risk free > return > > unless you do it continuously and the stock > price > > has no jumps. > > > or in other words (not trying to trump you Joey) > Delta hedging gives you the risk free rate if the > volatility you used to price the option is the > actual volatility realized over the life of the > option (assuming the hedge ratio is adjusted > constantly…). If realized volatility exceeds > the volatility estimate than the return is greater > than the risk free rate, but the return is less > than the risk free rate if realized volatility is > less than the estimate. Actually that’s vega not gamma which is also not a problem if you continually hedge (although harder to see).

come on JDV, I said realized volatility not implied…

Hmm. You did, so the idea is that realized vol > implied vol so this represents gamma? I’ll have to think about that.