Why Delta Hedging Leaves Jump Risk in Options

# delta-hedging is failing


# delta-hedging is failing












While I was recently testing a delta-hedging on a few products, I got a P&L result of 20% for some of them.

First, I thought that the implementation was incorrect. But I couldn't find an outright error.

After a deeper look in my (back)test-data, I found that delta-hedging might not be enough even for simple products like 'vanilla calls'.

Here's an example (with payoff = (S(T) - K )+): (T = 20 days, K =50.5, (hypothesis : Black-Scholes model))

S:

day 0 = 50 ;

day 1 = 50.5 ;

day 2 = 50.3 ;

... ;

day 15 = 52.7 ;

day 16 = 49.3 ;

day 17 = 37.5 ;

day 18 = 36.4 ;

day 19 = 36.8;

day 20 = 37.7.

I rebalance every day and because of the huge price fall on day 17, and because my delta was close to 1, I lost a lot of money (call price is almost 0, while my porfolio is about -15), causing a big P&L.

So, my questions are :

1/ Is it correct that : "delta-hedging is ok for continuous data (and continuous heding). Price-jumps are not hedged with delta-hedging"..?

2/ Assuming that we can't hedge more than once a day, and having recurrent (there are a few) prices falls in my data, what do you suggest for a hedging ?

Thanks a lot,

## Answer by Daneel Olivaw (score 17, accepted)

https://quant.stackexchange.com/a/33290

Regarding your 1st question, jumps are indeed unhedgeable. From a theoretical point of view, you might want to look at Merton's "Option pricing when underlying stock returns are discontinuous", the original paper that adapted Black-Scholes framework to include jumps. If you look at page 7, just after equation $(9)$:

> Unfortunately, in the presence of the jump process, $dq$, the return on the [hedging] portfolio [...] will not be riskless. Moreover, inspection of [equation] $(7c)$ shows that there does not exist a set of portfolio weights [...] that will eliminate the 'jump' risk [...]. [...] Note: the return on the portfolio is a 'pure' jump process because the continuous parts of the stock and the option price movements have been 'hedged' out.

So, as regards to question 2, you can for example increase the frequency of your hedging portfolio reallocation to get closer to purely continuous trading; but even there, if you assume a Merton framework, you will still be exposed to the jump risk.

Edit: jump risk is unhedgeable because there isn't a tradeable asset allowing to hedge against it. For example, Heston's original stochastic volatility model was incomplete (as is Merton's) because the volatility risk couldn't be hedged out. However, if you include a volatility derivative in your model, then volatility risk becomes hedgeable. If you had at your disposal a tradeable asset allowing you to hedge against jump risk, then the 'model' would be complete and you could hedge out the risk.

Edit 2: I am writing down some additional thoughts related to your comment "[...] how banks hedge their (for instance) long term structured products [...]?" @StudentInFinance.

Going back to Merton's article, consider equation $(10)$:

$$ \frac{dP}{P} = \left(\alpha_P - \lambda k_P \right)dt + dq $$

$\frac{dP}{P}$ is the return of the hedging portfolio (stock, option and zero-coupon bond), $\alpha_P$ its instantaneous return, $\lambda$ the mean number of jumps per unit of times, $k_P$ the expected percentage change in the portfolio value if a jump occurs and $q$ the Poisson process modelling jumps.

The key feature is that Merton argues that the random jump component of the portfolio is uncorrelated to the market. Emphasis mine:

> The total change in the stock price is posited to be the composition of two types of changes: (1) the 'normal' vibrations in price, [...]. (2) The 'abnormal' vibrations in price are due to the arrival of important new information about the stock that has more than a marginal effect on price. Usually, such information will be specific to the firm or possibly its industry. [...] [...] the stock price dynamics were described as the result of two components: the continuous part [...] and the jump part which is a reflection of important new information that has an instantaneous, non-marginal impact on the stock. If the latter information is usually firm [...] specific, then it may have little impact on stocks in general (i.e. the 'market'). [...] If the source of the jumps is such information, then the jump component of the stock's return will represent 'non-systematic' risk, i.e. the jump component will be uncorrelated with the market.

Now, consider that some investment bank has a position in 2 different options written on 2 different stock prices, $S_t^{(1)}$ and $S_t^{(2)}$, with different jump risk sources. The return of each hedging portfolio will be:

$$ \begin{align} & R_1 \equiv \frac{dP_1}{P_1} = \left(\alpha_{P,1} - \lambda_1 k_{P,1} \right)dt + dq_1 \\[12pt] & R_2 \equiv \frac{dP_2}{P_2} = \left(\alpha_{P,2} - \lambda_2 k_{P,2} \right)dt + dq_2 \end{align} $$

Hence:

$$ \mathbb{C}\text{ov}\left[R_1,R_2\right] = \mathbb{C}\text{ov}[dq_1,dq_2] $$

From Merton's comments we can posit that:

$$ \mathbb{C}\text{ov}[dq_1,dq_2]=0 $$

Hence, letting $\sigma_i^2 \equiv \mathbb{V}\text{ar}[R_i]$, $i \in \{1,2\}$, the bank's return variance of these 2 portfolios is $-$ if held in proportions $w_1, w_2$ of its total book, such that $w_1, w_2>0$ and $w_1+w_2=1$:

$$ \mathbb{V}\text{ar}\left[w_1R_1+w_2R_2 \right] = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 < w_1\sigma_1^2 + w_2\sigma_2^2$$

You see there is some "diversification effect" kicking in: the book's risk is lower than the weighted risk of each individual portfolio. Non-systematic risk is lowered by including additional (uncorrelated) jump risks in your portfolio, hence based on this interpretation banks, by writing options on multiple stock underlying, are "naturally" hedging out part of their jump risk.