# Delta Hedged PnL on Call Spread

Suppose I buy a call and then sell a call one dollar in strike higher. Suppose I get into this position for 10 cents lower than it is theoretically worth. (I.e if this spread is worth 0.50 I just bought it for 0.40). Then I delta hedge the spread to expiry. What will be my PnL? What will be my average PnL?

## 2 Answers

If the actual dynamics are those of Black Scholes and if the vol used in the delta hedge is the actual vol, then the P&L will be 10 cents i.e. not random and not dependent on the path.

• So basically, since you can just add greeks across the calls, we just treat it as one position with implied vol X and the hedged pnl is the same as for one mispriced call since the pnl for the spread is just the sum of the pnls on the 2 calls? – roz Feb 19 '20 at 15:33
• No you can't define the notion of implied vol for a spread of two calls. Notice that neither the question nor my answer made any reference to implied vol. The price of the spread is actually the difference of the two nonnegative call prices, not the sum. – P. Carr Feb 20 '20 at 19:26
• Why can't you define implied vol for a spread? Wouldn't it just be the volatility at which the theoretical value of the spread matches the price quoted in the market? Just solve for sigma in: Spread price = C(sigma, K1) - C(sigma, K2)? – roz Feb 20 '20 at 19:34
• Is it because of the smile? – roz Feb 20 '20 at 19:37
• It's not becauae of the smile. It's because vega of a call spread is not sign definite. You wrote "Wouldn't it just be the volatility" suggesting this vol is unique. There can be two. – P. Carr Feb 21 '20 at 23:17

When you compare the market price to the theoretical price, your difference is model dependent.

Let's say you are using Black Scholes and you're using some other vol that you think is more appropriate (you're not using the IV because that would give you the market price)

Under the Black Scholes assumptions of continuous trading and no execution costs your P/L would be the intrinsic value of the position when you delta hedged. Since you are long, if you delta hedge when the options are OTM, your P/L will probably be close to zero because the optionaliy will eventually expire and its as if the settlement price is the initial hedge price. If the options are ITM you would guarantee the intrinsic value.

This is obviously very theoretical, since these BS assumption are not observed in the real world. Also, don't forget from the start that your delta depends on the vol you considered.

• Wouldn't the pnl depend on the gamma and the price path? I am assuming dynamic delta hedging, not just a single delta hedge after I initially buy the call spread. Seems like the pnl should be a function of the difference in implied volatility and realized volatility of the spread. – roz Feb 18 '20 at 18:19