Equity and debt as options on the firm
If a firm's debt has face value D, shareholders receive max(0, V_T − D): a call on the firm's assets with strike D. Lenders receive min(V_T, D): risk-free debt minus a put on the assets. Parity then gives V₀ = c₀ + PV(D) − p₀, where PV is present value.
Before this: Put–call parity: fiduciary call = protective put, Building synthetic positions with put–call parity
On the exam
- Explain put–call parity for European options
- Explain put–call forward parity for European options
Explain put–call parity applied to firm value. Expect 'equity is equivalent to…', 'the credit risk in debt is like…', one calculation of risky debt or equity value, and the effect of asset volatility on each claim.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
How can shareholders’ and lenders’ claims be seen as options?
Treat the firm’s assets as the underlying. Suppose the firm owes lenders a single zero-coupon payment with face value D at time T. Write the firm value (the value of its assets) as V₀ today and V_T at maturity. What each side gets depends on whether the assets cover the debt.
| At debt maturity | Firm solvent (V_T ≥ D) | Firm insolvent (V_T < D) |
|---|---|---|
| Shareholders | V_T − D | 0 |
| Lenders | D | V_T |
When V_T ≥ D the firm is in solvency : lenders are paid in full and shareholders keep the rest. When V_T < D the firm is in insolvency : lenders take all the assets and shareholders get nothing. Shareholders never owe more than they put in, because of limited liability .
The shareholders’ payoff has the shape of a call: nothing below D, then one-for-one gains above D. That makes the equity as a call option view: a call on the firm’s assets with exercise price D. The lenders’ payoff is D, capped, minus a shortfall when V_T falls below D. That is a risk-free bond paying D minus a put on the firm’s assets with strike D. This is risky debt : a risk-free bond with a short put attached.
The same claims can be described from the put side. Shareholders own the assets, owe D and hold a put on the assets with strike D that the lenders have sold them: if the assets fall short, they hand the firm over in full settlement. Assets plus that put, less the D owed, pays max(0, V_T − D), which is exactly the call.
- Shareholders Memorize long call on V, strike D; payoff max(0, V_T − D)
- Solvent (V_T ≥ D) receive V_T − D
- Insolvent (V_T < D) receive 0 (limited liability)
-
- Lenders Memorize long risk-free bond paying D + short put on V; payoff min(V_T, D)
- Solvent (V_T ≥ D) receive D
- Insolvent (V_T < D) receive V_T
-
Try this 0 of 3
- Raise volatility to 40% or more and watch equity and debt
- Drag V_T below D
- Borrow more: raise D to 550 or above
A firm worth V₀ today owes lenders D in two years (zero-coupon, r = 5%). Drag V_T to read who gets what at maturity. Raise volatility to widen the spread of possible outcomes and see who gains.
Shareholders receive max(0, V_T − 400), a call on the firm with strike 400. Lenders receive min(V_T, 400): 400 minus a put at 400. The two add up to V_T. At V_T = 520: equity 120, debt 400. The shaded hump is the spread of possible V_T at 25% volatility.
Today: equity (call) 151.97 + risky debt 348.03 = V₀ 500.00
V_T is at or above D: lenders get 400 in full and shareholders keep the other 120. The put expires worthless.
The option values come from an illustrative model (Black–Scholes–Merton, 5% rate, two years) to show the direction of the effects; the exam gives you the put or call price.
Show working
At maturity: equity = max(0, 520 − 400) = 120; debt = min(520, 400) = 400
PV(D) = 400 ÷ 1.05^2 = 362.81
Risky debt = PV(D) − put = 362.81 − 14.78 = 348.03
Parity: V₀ = c₀ + PV(D) − p₀ → 151.97 + 362.81 − 14.78 = 500.00
Debt yield = (400 ÷ 348.03)^(1/2) − 1 = 7.21%, a spread of 2.21% over the 5% risk-free rate
- 1.05x²1/x×400=362.81, PV(D)
- −14.78=348.03, risky debt
x² square · 1/x reciprocal
- 400ENTER1.05ENTER2y^x÷362.81, PV(D)
- 14.78−348.03, risky debt
ENTER enter (on the HP, puts the number on the stack) · y^x power
Display: Risky debt EUR 348.03m
What does put–call parity say about firm value?
Substitute the firm’s value V₀ for the share price and the debt D for the exercise price in S₀ + p₀ = c₀ + PV(X), where PV(X) is the present value of the exercise price. Shareholders hold the call. Lenders hold the bond and are short the put. Add the two claims and you get the firm.
The put is the price of the credit risk. The more it is worth, the more value lenders give up compared with risk-free debt, so the bigger the credit spread.
A firm's assets are worth 500.00. Its only debt is a zero-coupon bond with a face value of 400.00 due in 2 years, and the risk-free rate is 5.00%. A European put on the firm's assets with an exercise price equal to the face value is worth 15.00. The value of the firm's equity is closest to:
500.00 − 400.00 × (1 + 0.05)^−2 + 15.00 = 500.00 − 362.81 + 15.00 = 152.19
Call on the assets (equity), c₀152.19
- 1.05y^x2+|-=×400=362.81
- +|-+500+15=152.19
y^x power · +|- change sign
- 1.05ENTER2CHSy^x400×362.81
- CHS500+15+152.19
ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power
How does asset volatility move value between lenders and shareholders?
Higher asset volatility raises the value of both the call and the put. Shareholders hold the call, so they gain. Lenders are short the put, so risky debt loses value. The total firm value stays the same: value moves from lenders to shareholders.
Exam tip Equity is a call, lenders are short a put. A rise in volatility is good for the call and bad for the short put.
Go deeper Why shareholders may take on riskier projects
If a firm is close to insolvency, shareholders lose little from a bad outcome (equity cannot go below zero) but gain a lot from a good one. That asymmetry can push shareholders toward riskier projects, at the expense of lenders. Lenders respond with covenants and higher yields.
Go deeper The put as a measure of credit risk
The put’s value measures how much credit risk the debt contains. It rises with asset volatility, with the face value D and as V falls toward D, and it usually rises with the time to maturity. The same idea sits behind structural models of credit risk in the fixed income topic.
Memorize this
MemorizeOption view of the firm
| Claim | Option position on firm value | Payoff at maturity | Higher asset volatility |
|---|---|---|---|
| Equity | long call, strike | raises value | |
| Risky debt | risk-free debt put, strike | lowers value | |
| Firm | equity + risky debt | unchanged |
Memory hook Shareholders own the upside (call). Lenders own the bond and have sold a put: capped at D, exposed below D.
Know it cold . The put is the credit-risk discount (the credit spread). Equity is a call, never a put; lenders are short the put.
- Shareholders Memorize long call on V, strike D; payoff max(0, V_T − D)
- Solvent (V_T ≥ D) receive V_T − D
- Insolvent (V_T < D) receive 0 (limited liability)
-
- Lenders Memorize long risk-free bond paying D + short put on V; payoff min(V_T, D)
- Solvent (V_T ≥ D) receive D
- Insolvent (V_T < D) receive V_T
-
Quick example
Debt face value D = 400. If V_T = 520, equity gets 120 and lenders 400. If V_T = 350, equity gets 0 and lenders 350.
Quick example
Equity (call) 38.37, debt D = 150 due in one year, r = 3%, put on the assets 4.00. PV(D) = 145.63. V = 38.37 + 145.63 − 4.00 = 180.00.
- 150÷1.03= 145.63
- +38.37−4= 180
- 150ENTER1.03÷ 145.63
- 38.37+4− 180
ENTER enter (on the HP, puts the number on the stack)
Quick example
D = 150, r = 3%, one year, put 4.00. Risky debt = 145.63 − 4.00 = 141.63; the put is the discount for credit risk.
- 150÷1.03=−4= 141.63
- 150ENTER1.03÷4− 141.63
ENTER enter (on the HP, puts the number on the stack)
How it is asked
-
Equity is equivalent to…
Shareholders of a firm with zero-coupon debt hold…?
A call option on the firm assets, strike = face value of debt. Trap: a put.
-
Value of risky debt
V = 420, D = 350 in one year, r = 6%, put 11. Debt?
PV(D) − p = 330.19 − 11 = 319.19. Trap: 330.19 (put ignored).
-
Lenders hold…
Which option position do lenders hold?
Short put (plus a risk-free bond). Trap: long put.
-
Volatility effect
Asset volatility rises. Equity and debt?
Equity up (call gains), risky debt down (put gains, credit risk rises).
Exam traps
- Calling equity a put on the firm: shareholders hold a call.
- Saying lenders hold a long put: they are short the put.
- Thinking higher volatility helps lenders: it raises the put value, so risky debt falls.
- Subtracting the put from the face value D instead of from PV(D).
Checkpoint
5 exam-style questions. About 90 seconds each.
1In the option view of a firm with zero-coupon debt of face value D, the shareholders' claim is best described as:
2A firm's assets are worth GBP 300 million. Its only debt is zero-coupon debt with a face value of GBP 250 million due in one year. The risk-free rate is 4% and the put on the firm's assets with an exercise price of GBP 250 million is worth GBP 8 million. The value of the risky debt is closest to:
3An increase in the volatility of a firm's assets, all else equal, most likely:
4A firm's assets are worth GBP 300 million. Its only debt is zero-coupon debt with a face value of GBP 250 million due in one year. The risk-free rate is 4% and the put on the assets with an exercise price of GBP 250 million is worth GBP 8 million. The value of the firm's equity (the call) is closest to:
5In the option view of a firm, the lenders' position is best described as:
Key takeaways
- Equity = call on firm assets with strike D (limited liability caps the downside at zero).
- Risky debt = risk-free debt − put on firm assets; the put is the credit spread.
- V₀ = c₀ + PV(D) − p₀: equity plus risky debt.
- More asset volatility raises equity and lowers risky debt: value shifts from lenders to shareholders.
Quick example
Debt face value D = 400. If V_T = 520, equity gets 120 and lenders 400. If V_T = 350, equity gets 0 and lenders 350.
Quick example
Equity (call) 38.37, debt D = 150 due in one year, r = 3%, put on the assets 4.00. PV(D) = 145.63. V = 38.37 + 145.63 − 4.00 = 180.00.
- 150÷1.03= 145.63
- +38.37−4= 180
- 150ENTER1.03÷ 145.63
- 38.37+4− 180
ENTER enter (on the HP, puts the number on the stack)
Quick example
D = 150, r = 3%, one year, put 4.00. Risky debt = 145.63 − 4.00 = 141.63; the put is the discount for credit risk.
- 150÷1.03=−4= 141.63
- 150ENTER1.03÷4− 141.63
ENTER enter (on the HP, puts the number on the stack)
All Derivatives formulas, every lesson
Practice exam-style questions on Put–Call Parity and Option Replication
Questions about this lesson?
Where this comes back
- Credit Risk Fixed Income
- Capital Structure Corporate Finance
Questions candidates ask
Why is equity a call option on the firm?
Shareholders receive what is left after debt is repaid: max(0, V_T − D). They gain without limit if assets exceed D and lose nothing more than their stake if assets fall short. That payoff is a call on the firm's assets with strike equal to the debt's face value.
How is risky debt like a short put?
Lenders are paid D if the firm is solvent and only V_T if it is not. That is a risk-free bond paying D minus a put on the firm's assets with strike D. The put's value is the discount for credit risk, so it reflects the credit spread.
How does volatility affect shareholders and bondholders?
Higher asset volatility raises both the call and the put. Equity holds the call, so it gains. Lenders are short the put, so risky debt falls. Total firm value is unchanged: value moves from lenders to shareholders.
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