Deep Hedging

The The Black-Scholes Model hedge ratio is exactly right in a world that does not exist: one where you can trade continuously, in any size, at zero cost, in a stock whose volatility is a known constant. Real desks rebalance a few times a day, pay a spread on every trade, and face volatility that moves. There is no clean formula for the best hedge under those frictions — the moment you add proportional costs, the problem stops having a closed-form solution. Deep hedging sidesteps the derivation entirely: simulate millions of possible market paths, let a network choose a position at every step, and score it on the outcome the desk actually cares about.

The analogy: train the trainee, don't hand them the recipe

Two ways to teach someone to hedge. The first is to hand them a formula derived under assumptions you both know are false, then add a folder of manual overrides — do not rebalance for a move under 10 cents, widen the band before the close, cut gamma into earnings. Each override is a patch on the same idealised recipe.

The second is to sit the trainee in front of a simulator, let them hedge a million fake weeks, and grade them at the end of each one on a single number that already includes the commissions they paid. They never learn a hedge ratio. They learn a policy: given what I see right now, how much stock should I hold? Deep hedging is the second approach, with a neural network as the trainee.

Setting up the objective

Say you have sold a derivative that pays $Z$ at maturity. You hold $\delta_t$ shares between step $t$ and $t+1$, and each trade costs a fraction $c$ of the notional traded. Terminal profit and loss is

$$\text{PL} = -Z + \sum_{t=0}^{T-1} \delta_t \, (S_{t+1} - S_t) \; - \; c \sum_{t=0}^{T-1} \left| \delta_t - \delta_{t-1} \right| S_t .$$

In words: you owe the payoff, you collect whatever your stock position earned along the way, and you pay for every share you traded. Three terms, nothing hidden.

The network with parameters $\theta$ produces the positions, $\delta_t = f_\theta(\text{state}_t)$, and training minimises

$$\min_{\theta} \; \rho\big(\text{PL}(\theta)\big),$$

where $\rho$ is a risk measure — a single number summarising how bad a distribution of outcomes is. Plain variance is one choice; conditional value at risk, the average of the worst outcomes, is the common one because a desk cares about the bad tail rather than symmetric wiggles. Everything is differentiable, so ordinary gradient descent works.

Below is the loss surface in miniature. Push the learning rate up and watch the steps overshoot, pull it down and watch them crawl — training a hedging policy is exactly this, with millions of parameters instead of one.

Worked example: the frequency you rebalance is a real choice

Sell a one-month at-the-money call on a $100 stock at 20 percent volatility. Its price is about $2.31. You hedge once a day for 21 trading days, and the round-trip cost is 20 basis points of whatever notional you trade.

An at-the-money option has gamma near $0.069$ per dollar of spot move, and a daily move of 20 percent annual volatility is about $1.26. So the delta shifts by roughly $0.069 \times 1.26 = 0.087$ each day, and the average absolute shift is around $0.07$.

• Notional traded per rebalance: $0.07 \times 100 = 7$ dollars.
• Cost per rebalance: $7 \times 0.0020 = 0.014$ dollars.
• Over 21 days: $21 \times 0.014 = 0.294$ dollars.

That is 29 cents against a $2.31 premium — 13 percent of the option's value, gone to commissions.

Now rebalance every second day instead. Delta drifts further between trades, by $\sqrt{2}$ as much on a random walk, so each trade moves $0.099$ shares and costs $0.0198$ — but there are only 10.5 trades:

$$10.5 \times 0.0198 = 0.208 .$$

Costs fall to 21 cents, a saving of 29 percent, because total cost scales like $\sqrt{m}$ in the number of rebalances $m$ while hedging error scales like $1/\sqrt{m}$. Trading half as often saves 29 percent of the costs and increases residual risk by 41 percent.

That is the entire tension, and there is no formula telling you where to sit on it. Worse, the right answer is state-dependent: trade eagerly when gamma is large, sit still when it is small. A deep hedger discovers that band by itself, which is why its position path looks like a staircase rather than a wiggle.

Worked example: scoring two policies, and what to charge

Suppose the simulator produces five equally likely terminal outcomes per policy, in dollars per option. Policy A gives $+0.40,\ +0.15,\ -0.05,\ -0.30,\ -1.10$; policy B gives $+0.10,\ +0.05,\ -0.02,\ -0.15,\ -0.48$.

Mean profit and loss. A: $(0.40 + 0.15 - 0.05 - 0.30 - 1.10)/5 = -0.18$. B: $(0.10 + 0.05 - 0.02 - 0.15 - 0.48)/5 = -0.10$.

Conditional value at risk at the 80 percent level — the average of the worst 20 percent, which with five scenarios is simply the worst one. A: $-1.10$. B: $-0.48$.

B wins on both counts. Now the useful part. The indifference price is the premium you would have to charge for this risk measure to come out at zero:

$$\text{price}_A = 1.10, \qquad \text{price}_B = 0.48 .$$

The better hedging policy lets the desk quote 62 cents tighter on the same option and still sleep at night. That is the commercial argument for deep hedging in one line: a hedge is not just a risk control, it is an input to the price you can show a client.

What this means in practice

Desks use this for exotic books with no clean Greeks, for illiquid underlyings where costs dominate, and for portfolios where hedging every instrument separately is wasteful. Three things decide whether it works:

• The simulator is the model. The network learns the market you sampled from. Train on geometric Brownian motion and you get a policy that has never seen a gap, a vol spike or a liquidity hole.
• Include costs from step one. A policy trained frictionless and then run with costs is worse than the textbook delta, because it learned to trade constantly.
• State design matters as much as architecture. Feed it spot, time remaining, current inventory and a volatility estimate. Omit inventory and the network cannot compute its own trading cost, so it cannot learn a no-trade band.