Bonus Time: The Math of Wall Street Bonuses
Every year around December, Wall Street gets really interesting. Bonus season. Traders who made money want a fat check. Banks want to keep their best people but also not go broke. And nobody really knows if a trader who made a fortune this year was skilled or just lucky.
Chapter 75 of Wilmott’s book takes this very human problem and turns it into math. How should you compensate a trader? What bonus structure maximizes the bank’s profit while keeping the trader happy? And most importantly, can you actually measure whether a trader has skill?
The Basic Problem
Most traders get a base salary plus a bonus. The bonus is typically some percentage (say 5% or 10%) of the profit they generated for the bank that year. Simple, right?
The problem is that this structure encourages bad behavior. If a trader is in the red halfway through the year, what does the optimal strategy look like? Take massive risks. If the gamble pays off, they end up positive and get a bonus. If it does not, the bank loses more money, but the trader was already getting zero bonus anyway. Heads I win, tails you lose.
Banks put limits on this: position limits, capital constraints, restrictions on instrument types. But those are blunt tools. The question Wilmott asks is more fundamental: can we design a bonus structure that naturally discourages excessive risk-taking?
One Bonus Period
Let us start with the simplest case. A trader has one year. They trade a single asset that follows the usual random walk with drift mu and volatility sigma. The trading account tracks their profit and loss. At the end of the year, they get a bonus based on their performance.
The standard bonus is a fraction of positive profit: you get paid if you make money, you get nothing if you lose money. This is mathematically identical to the passport option from earlier chapters. The trader’s bonus is basically a call option on their own P&L.
Things get more interesting when the bonus depends on the Sharpe ratio, not just raw profit. The Sharpe ratio is the profit divided by the volatility of the trading account. Why would a bank want this? Because it controls risk. A trader who makes $10 million with wild swings in their P&L is less valuable than one who makes $10 million with a smooth upward curve.
The Math of Sharpe-Based Bonuses
Wilmott sets up the problem as an optimization. The trader wants to maximize their expected bonus. If V(S, pi, I, t) is the expected bonus (where S is the asset price, pi is the trading account, and I tracks the variance), then the trader picks their position q to maximize this.
The resulting PDE has the trader’s position choice embedded in it:
$$\frac{\partial V}{\partial t} + \max_{|q| \le C} \left[ \text{terms involving } q \right] = rV$$
The position limit C constrains how much the trader can buy or sell. The equation can be simplified by introducing the ratio variables z1 = pi/S and z2 = I/S^2, which reduces the dimensionality.
Numerical results show something really interesting about trader behavior. With Sharpe-ratio-based bonuses, the optimal strategy changes near the end of the year. Early on, the trader just holds a maximum long position (assuming positive drift). But as bonus time approaches, the trader starts reducing their position. Why? They are trying to lock in their Sharpe ratio. Less volatility near the end means a higher Sharpe ratio and a bigger bonus.
This is actually rational and exactly what the bank wants. The bonus structure naturally causes the trader to reduce risk when they are ahead.
Running the Numbers
Wilmott runs 10,000 Monte Carlo simulations with 10% drift, 20% volatility, and zero interest rate. The results:
- Trader’s mean profit: 0.51, standard deviation: 0.79
- Bank’s mean profit: 4.85, standard deviation: 9.90
The bank makes roughly 10 times what the trader makes. The distributions are skewed: the bank has a wide spread of outcomes, but the mean is solidly positive. The trader’s distribution is tighter and also positive.
The Skill Factor: Separating Signal from Noise
Here is where the chapter gets really fun. Wilmott introduces a formal model for trader skill. How do you know if a trader is actually good, or just riding a bull market?
The model works like this. Assume the trader receives correct information about the market’s next move a fraction p of the time. When they get this information, they trade in the correct direction up to their position limit. The remaining (1-p) of the time, they trade randomly.
The probability of being correct on any given trade is p + (1/2)(1-p) = 1/2 + p/2. When p = 0, you are right half the time (pure coin flip). When p = 1, you are right every time (perfect information).
The P&L distribution under this model is a mixture distribution. There is a normal component (from the random trades) and a guaranteed profit component (from the informed trades). The mean profit is proportional to p, and the standard deviation depends on both p and the underlying volatility.
Measuring a Real Trader’s Skill
Wilmott has a trader friend (a woman, he notes, not out of political correctness but because the data is actually from a female trader). He takes her P&L time series from 1998 and 1999 and fits the skill model.
Her Sharpe ratios were 3.7 in 1998 and 4.7 in 1999. These are exceptional numbers. By fitting the model to her P&L distribution, Wilmott backs out a skill factor p of 30% in 1998 and 38% in 1999.
What does p = 38% mean in practice? The probability of being correct on any trade is 1/2 + 38%/2 = 69%. The actual number from counting her trades? 68%. The model fits almost perfectly.
Getting the direction right 69% of the time does not sound like much. But compounded over hundreds of trades per year, it produces spectacular returns. This is the magic of edge: you do not need to be right most of the time, you need to be right slightly more than half the time, consistently.
Putting Skill into the Bonus Model
The final piece combines the skill model with the bonus optimization. Now the trader does not just hold a position and hope. A fraction p of the time, they get correct information and trade on it. The rest of the time, they trade to maximize expected bonus as before.
The PDE changes to incorporate the skill factor. The drift term is now a mix of the informed trading (certain profit) and the random trading (standard drift). The numerical solution shows:
With skill factor p > 0, the trading strategy looks different. The position occasionally jumps to -1 (selling) when the trader receives information that the market will fall, even if the general strategy would be to stay long. The trading account grows faster with skill: without skill the simulated trader made 17, with skill it jumped to 25.
The distributions of profit shift:
- Skilled trader’s mean profit: 1.13 (vs 0.51 without skill), std: 1.12
- Bank’s mean profit: 11.94 (vs 4.85 without skill), std: 10.34
Both the trader and the bank benefit from skill. The bank benefits more in absolute terms. This makes sense: the bonus is a fraction of the profit, so the bank keeps the lion’s share.
What Else Could Be Done
Wilmott ends by noting several extensions worth exploring. What if the trader gets fired when losses exceed some threshold? That would discourage excessive risk-taking even more. What if the trader can walk away after receiving their bonus? Then you need multi-year lock-up structures. What if the bonus from one year only pays out conditional on the next year’s performance? Hedge funds often use this kind of path-dependent compensation.
The framework handles all of these. Each modification just changes the boundary conditions or the optimization constraint in the PDE. The approach is the same.
The Takeaway
The compensation problem in finance is not just about paying people. It is about designing incentive structures that align everyone’s interests. Pay too little and you lose good traders. Pay a flat percentage of profit and traders take insane risks. Pay based on the Sharpe ratio and traders naturally moderate their risk near bonus time.
The skill factor model is a gem. Most people talk about trader skill in vague terms. Wilmott puts a number on it and shows that even a modest edge (knowing the direction 69% of the time vs 50%) produces outsized returns. The hard part is telling the difference between a skilled trader and a lucky one after just one year. You need the math to separate signal from noise.
And if any reader wants their skill factor calculated, Wilmott says to email him a spreadsheet of your P&L. That is the kind of offer that makes quantitative finance fun.
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