Downward Jumps, Put-Call Parity, and Call Option Value

# Price of Call Option with or without jumps


# Price of Call Option with or without jumps












> Suppose two assets in the Black Scholes world have the same volatility, but different drifts and that one has downward jumps at random times. How does this affect the option prices?

I would have thought that downward jumps would decrease the value of the call option because you have more chance of being out of the money (ie below the strike). Apparently, the answer is the reverse.

Does anyone have an explanation for this?

## Answer by Robert McDonald (score 6, accepted)

https://quant.stackexchange.com/a/55448

The call for the stock that can jump downward will be more valuable due to put-call parity. Suppose you have two stocks, both with a price of $100 and the same diffusive volatility. Stock A does not jump, whereas stock B can at some random time jump (for example) to zero. Clearly a put on stock B will be worth more, but the call must therefore also be worth more due to parity:

$$\text{Call}(S_{0}, K, T) = \text{Put}(S_{0}, K, T) + S_0 - K e^{-rT}$$

The economic explanation for this is that both stocks have the same price. If the stock that can jump downward is worth the same as the stock that cannot jump, it must have more probability mass on the upside. In the Merton jump model, the stock that can jump to zero has a risk-neutral drift, conditional on no jump, of $r + \lambda$, where $\lambda dt$ is instantaneous probability of the jump to zero. With this drift, the stock's unconditional drift is $r$. The call price in this case is obtained by replacing $r$ with $r+\lambda$, which results in a higher call price. (Merton discusses this case specifically in his 1976 JFE paper.)

## Answer by Brian B (score 4)

https://quant.stackexchange.com/a/55394

There's a lot left unspecified in this question, since it is stated without precision, but the effective idea of the answer given here is that those jumps introduce extra variation into the forward distribution of the underlying. And such variation is the bread-and-butter of option value.

That said, the ambiguity in the question leaves room for other interpretations. In particular if you as a market-maker sold an at-the-money call for $100, and them immediately after your sale everyone found out that the underlying had a 50-50 probability of jumping down by half tomorrow, you would be very happy, because the underlying would drop in value by 25% or so and the option would go far out of the money.

So, what the person who said the value increases meant was, given two ATM options on separate underlyings with the same continuous volatility, and where the second underlying also had some downward jumps, the latter option will have higher fair value.

Mathematically, this ends up being associated with the second underlying having higher risky drift.

## Answer by Kupoc (score 2)

https://quant.stackexchange.com/a/55447

You can check out those discussions in Merton paper when introducing jumps "Option pricing when underlying stock returns are discontinuous". In the very last part he discusses the influence of considering jumps compared to the usual Black Sholes model. From what i remember it'sall about considering your option is ATM or not , that will usually make the BS model call prices higher.

## Answer by Arshdeep (score 0)

https://quant.stackexchange.com/a/55607

I'm not too sure if I interpret the question correctly, but I am inclined to say that as the call is long gamma, 'jumps' (second order moves) would always result in higher value in the delta hedged portfolio, and therefore should be built into the price of the option. So the call should be more expensive if it has jumps.