oh shit i can't hedge

I had a conversation with a brilliant xVA desk guy this week. Never realised how interesting hedging is from the perspective of a financial institution. A lot of that was common xVA knowledge, but what really got me was the real-world obstacles involved in reducing risky exposures. Namely, liquidity. Not everything has the perfect CDS. Duh!

(Unless you're 2008 Michael Burry and pay 100+M in negative carry haha.)

Of course, that meant I spent all of last night looking at these concepts. Interestingly, there's not a lot of clear literature on non-hedgeable exposures. Fortunately, this one (quite recently published!) paper caught my eye:

When defaults cannot be hedged: an actuarial approach to xVA calculations via local risk-minimization

We consider the pricing and hedging of counterparty credit risk and funding when there is no possibility to hedge the jump to default of either the bank or the counterparty. This represents the situation which is most often encountered in practice, due to the absence of quoted corporate bonds or CDS contracts written on the counterparty and the difficulty for the bank to buy/sell protection on her own default. We apply local risk-minimization to find the optimal strategy and compute it via a BSDE.

arXiv.orgFrancesca Biagini

What do you do when you can’t actually hedge defaults?

In real life, many counterparties don’t have tradable CDSs or bonds, so a bank can’t hedge their jump-to-default risk. Similarly, a bank can’t hedge its own default (you can’t buy CDS on yourself).
So the usual “replication” approach (like in Black-Scholes, where every risk is perfectly hedge) doesn’t work. The market is incomplete.

The maths in the paper is a bit beyond me, but (with a generous amount of help from GPT), the ideas are pretty cool. In particular, local risk minimization (Schweizer, Föllmer & Sondermann, 1980s–1990s):

Choose the portfolio that minimizes the expected squared change in your hedging error at each instant in time, given current information.

The paper models this out as a Backward Stochastic Differential Equation (BSDE). Here's a nice explanation by Gemini (AI caution!):

A Backward Stochastic Differential Equation (BSDE) is a type of equation that evolves randomly, but it's solved backward in time from a known endpoint, rather than forward from a known starting point.

1. The Forward View: Standard SDEs

think about a regular (Forward) Stochastic Differential Equation (SDE). It describes the path of something that moves randomly over time, like a stock price or a particle diffusing in a liquid.

Knowns: You know the starting point (Y_0) at time t=0.

Equation: It looks like dY_t = a(t, Y_t)dt + b(t, Y_t)dW_t.

dY_t is the tiny change in the value Y over a tiny time step dt.

a(...)dt is the drift, or the predictable part of the movement.

b(...)dW_t is the diffusion, or the random shock, driven by a random process W_t (called a Wiener process or Brownian motion).

Goal: To figure out the ending point Y_T at a future time T. Since the path is random, you're really finding the probability distribution of where Y_T could end up.

Think of it like dropping a drop of ink in water. You know where it starts (Y_0), and the SDE describes how it spreads out randomly over time. You want to know where the ink particles might be after 10 seconds (Y_T).

2. The Backward View: BSDEs

Now, let's flip this entire concept on its head. This is the core idea of a Backward Stochastic Differential Equation (BSDE).

Knowns: You don't know the starting point. Instead, you know a specific condition that the ending point Y_T must satisfy at a future time T.

Equation: It's defined by a pair of processes, (Y_t, Z_t), and is written as: dY_t = f(t, Y_t, Z_t)dt + Z_t dW_t with a terminal condition Y_T = ξ.

Goal: To find the starting value Y_0.

What do these terms mean?

Y_T = ξ: This is the crucial part. We know the value at the end. ξ (xi) is a random variable, meaning its exact value depends on the random path taken up to time T.

Y_t: This is the value of our process at any time t. Our main goal is to find Y_0.

Z_t: This is a new, essential component. It represents a control or hedging strategy. It tells you how to manage the randomness dW_t at every step to ensure you meet the final condition Y_T = ξ.

f(...): This is called the "driver" and can represent things like interest rates, costs, or dividends being earned or paid over time.

So, a BSDE solves for a pair of processes (Y_t, Z_t) that work together. Y_t is the value, and Z_t is the strategy to maintain that value, all while working backward from a known future outcome.

A Powerful Analogy: Option Pricing

The most intuitive application of BSDEs is in financial derivatives pricing.

Imagine you want to find the fair price of a European call option today.

Terminal Condition (Y_T = ξ): You know the exact value of the option on its expiry date, T. The value will be max(S_T - K, 0), where S_T is the stock price at expiry (a random variable!) and K is the fixed strike price. This known future payoff is our terminal condition ξ.

The Big Question (Find Y_0): What is the fair price of this option today, at time t=0? This is the unknown Y_0 we need to find.

The BSDE solves this perfectly.

Y_t becomes the fair price of the option at any time t.

Z_t becomes the hedging strategy—specifically, how many shares of the underlying stock you need to buy or sell at time t to perfectly replicate the option's payoff and eliminate risk.

The BSDE works backward from the known payoff at expiry (Y_T) to tell you the price (Y_0) and the required hedge (Z_0) today.

Let's see if I can actually solve the BSDE for a simple setup: one risky asset, one counterparty, deterministic rates. Possibly see what the difference is between a hedgeable and non-hedgeable solutions...